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"value": "## Task B-1.3.\n\nA ship traveling along a river has covered $24 \\mathrm{~km}$ upstream and $28 \\mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \\mathrm{~km}$ upstream and $21 \\mathrm{~km}$ downstream, or half an hour more than for ... | [
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"value": "## Task B-1.3.\n\nA ship traveling along a river has covered $24 \\mathrm{~km}$ upstream and $28 \\mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \\mathrm{~km}$ upstream and $21 \\mathrm{~km}$ downstream, or half an hour more than for ... | [
"We need to solve a problem about a ship traveling upstream and downstream. The problem states:",
"A ship traveling along a river has covered 24 km upstream and 28 km downstream. For this journey, it took half an hour less than for traveling 30 km upstream and 21 km downstream, or half an hour more than for trave... | [
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"value": "3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a ... | [
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"value": "3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a ... | [
"We are given a problem: A construction company was building a tunnel. When 1/3 of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by 20% and reduced the working hours to 80% of the original. As a result, it took a total of 185 days to complet... | [
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"value": "4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).\n(A) $1<x<7$.\n(B) $5 \\ll x<7$.\n(C) $1<x<\\sqrt{7}$.\n(D) $5<x<7$ or $1<x<\\sqrt{7}$."
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"value": "<think>\n\nWe are given: three sides of ... | [
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"value": "4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).\n(A) $1<x<7$.\n(B) $5 \\ll x<7$.\n(C) $1<x<\\sqrt{7}$.\n(D) $5<x<7$ or $1<x<\\sqrt{7}$."
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"value": "<think>We are given: three sides of an o... | [
"We are given: three sides of an obtuse triangle are 3, 4, x. We need the range of values for x such that a triangle with sides 3, 4, x is obtuse (i.e., has an angle greater than 90 degrees). Also the triangle must be valid (triangle inequality). So we need to find x satisfying triangle inequality and the condition... | [
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"value": "1. Solve the equation: $\\frac{8 x+13}{3}=\\frac{19-12 x}{2}-\\left(16 x-\\frac{7-4 x}{6}\\right)$."
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"value": "<think>\n\nWe are asked to solve the equation:\n\n(8x + 13)/3 = (19 - 12x)/2 - (16x - (7 - 4x)/6)\n\n<latent><latent_1></latent>\n\... | [
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"value": "1. Solve the equation: $\\frac{8 x+13}{3}=\\frac{19-12 x}{2}-\\left(16 x-\\frac{7-4 x}{6}\\right)$."
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"value": "<think>We are asked to solve the equation:\n\n(8x + 13)/3 = (19 - 12x)/2 - (16x - (7 - 4x)/6)\n\nWe need to solve for x. Let's rewr... | [
"We are asked to solve the equation:",
"(8x + 13)/3 = (19 - 12x)/2 - (16x - (7 - 4x)/6)",
"We need to solve for x. Let's rewrite clearly.",
"Equation:",
"(8x + 13)/3 = (19 - 12x)/2 - [16x - (7 - 4x)/6]",
"We need to simplify both sides. Let's start by simplifying the right-hand side (RHS). There is a term... | [
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"value": "Let $a_n\\ (n\\geq 1)$ be the value for which $\\int_x^{2x} e^{-t^n}dt\\ (x\\geq 0)$ is maximal. Find $\\lim_{n\\to\\infty} \\ln a_n.$"
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"value": "<think>\n\nWe are given: Let a_n (n ≥ 1) be the value for which ∫_x^{2x} e^{-t^n} dt (x ≥ 0) is ... | [
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"value": "Let $a_n\\ (n\\geq 1)$ be the value for which $\\int_x^{2x} e^{-t^n}dt\\ (x\\geq 0)$ is maximal. Find $\\lim_{n\\to\\infty} \\ln a_n.$"
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"value": "<think>We are given: Let a_n (n ≥ 1) be the value for which ∫_x^{2x} e^{-t^n} dt (x ≥ 0) is maxi... | [
"We are given: Let a_n (n ≥ 1) be the value for which ∫_x^{2x} e^{-t^n} dt (x ≥ 0) is maximal. Find lim_{n→∞} ln a_n.",
"Interpretation: For each n, define f_n(x) = ∫_x^{2x} e^{-t^n} dt, for x ≥ 0. We need to find the x that maximizes f_n(x) (presumably unique, but we are told \"a_n be the value for which ... is ... | [
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"value": "11. Given that the internal angles $A, B, C$ of $\\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\\sqrt{3} b \\cos \\frac{A+B}{2}=c \\sin B$.\n(1) Find the size of $\\angle C$;\n(2) If $a+b=\\sqrt{3} c$, find $\\sin A$."
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"value": "11. Given that the internal angles $A, B, C$ of $\\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\\sqrt{3} b \\cos \\frac{A+B}{2}=c \\sin B$.\n(1) Find the size of $\\angle C$;\n(2) If $a+b=\\sqrt{3} c$, find $\\sin A$."
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"We are given: √3 b cos((A+B)/2) = c sin B.",
"We need to find angle C first.",
"Recall that in triangle ABC, A+B+C = π, so (A+B)/2 = (π - C)/2 = π/2 - C/2.",
"Thus cos((A+B)/2) = cos(π/2 - C/2) = sin(C/2).",
"So left side: √3 b * sin(C/2).",
"Right side: c sin B.",
"Thus equation: √3 b sin(C/2) = c sin... | [
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"value": "Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\\mathbb{C}$."
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"value": "<think>\n\nWe have to parse the problem: \"L... | [
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"value": "Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\\mathbb{C}$."
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"value": "<think>We have to parse the problem: \"Let n... | [
"We have to parse the problem: \"Let n be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation x^4 = n in the set C.\" So we need to find all complex fourth roots of n.",
"First, what is n? The number 14641 has digits: 1,4,6,4,1. \"Writing... | [
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"value": "6. As shown in Figure 2, let $P$ be a point inside the equilateral $\\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Th... | [
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"value": "6. As shown in Figure 2, let $P$ be a point inside the equilateral $\\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Th... | [
"We are given an equilateral triangle ABC with side length 12. Point P is inside. Perpendiculars PD, PE, PF to sides BC, CA, AB respectively. Distances are in ratio PD:PE:PF = 1:2:3. Need area of quadrilateral BDPF. Quadrilateral BDPF: vertices B, D, P, F? Order: B, D, P, F? Typically quadrilateral BDPF would have ... | [
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"value": "8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation\n$$\nf(x)+f(y)=f(x+y)-x y-1 .\n$$\n\nIf $f(1)=1$, then the number of integers $n$ (where $n \\neq 1$) that satisfy $f(n)=n$ is\n(A) 0.\n(B) 1.\n(C) 2.\n(D) 3.\n(E) infinitely many.\n(30t... | [
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"value": "8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation\n$$\nf(x)+f(y)=f(x+y)-x y-1 .\n$$\n\nIf $f(1)=1$, then the number of integers $n$ (where $n \\neq 1$) that satisfy $f(n)=n$ is\n(A) 0.\n(B) 1.\n(C) 2.\n(D) 3.\n(E) infinitely many.\n(30t... | [
"We are given a functional equation: for all real numbers x, y, f(x) + f(y) = f(x+y) - x y - 1. So f(x+y) = f(x) + f(y) + x y + 1.",
"Given that f(1)=1. We need to find number of integers n (n ≠ 1) such that f(n)=n.",
"We need to determine f for integer arguments possibly from functional equation.",
"This is ... | [
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"value": "Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+x y}+\\frac{z}{1+x+y z}=\\frac{3}{x+y+z} .$$"
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"value": "<think>\n\nWe need to solve the equation:\n\n<latent><latent_1><latent_2></latent>\n\nW... | [
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"value": "Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+x y}+\\frac{z}{1+x+y z}=\\frac{3}{x+y+z} .$$"
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"value": "<think>We need to solve the equation:\n\nFor $0 \\le x,y,z \\le 1$, solve:\n\n$$\\frac{... | [
"We need to solve the equation:",
"For $0 \\le x,y,z \\le 1$, solve:",
"$$\\frac{x}{1+y+zx} + \\frac{y}{1+z+xy} + \\frac{z}{1+x+yz} = \\frac{3}{x+y+z}.$$",
"We need to find all triples (x,y,z) in [0,1]^3 satisfying this.",
"We suspect that maybe the equality holds when x=y=z (some value) or perhaps some of ... | [
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"value": "## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{3 / 2} \\frac{x^{2} \\cdot d x}{\\sqrt{9-x^{2}}}\n$$"
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"value": "<think>\n\nWe need to evaluate the definite integral ∫_{0}^{3/2} x^2 / sqrt(9 - x^2) dx.\n\nFirst, note... | [
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"value": "## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{3 / 2} \\frac{x^{2} \\cdot d x}{\\sqrt{9-x^{2}}}\n$$"
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"value": "<think>We need to evaluate the definite integral ∫_{0}^{3/2} x^2 / sqrt(9 - x^2) dx.\n\nFirst, note tha... | [
"We need to evaluate the definite integral ∫_{0}^{3/2} x^2 / sqrt(9 - x^2) dx.",
"First, note that the integrand involves sqrt(9 - x^2), suggesting a trigonometric substitution: x = 3 sin θ, then dx = 3 cos θ dθ. Also sqrt(9 - x^2) = sqrt(9 - 9 sin^2 θ) = sqrt(9 cos^2 θ) = 3|cos θ|. Since we integrate from x=0 to... | [
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"value": "Example 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person... | [
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"value": "Example 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person... | [
"We need to answer two questions.",
"Game: For each level n (starting presumably from n=1,2,...?), we roll a fair die n times. Let S_n be sum of n rolls (each uniform {1,...,6}). Condition to pass level n: S_n > 2^n.",
"(1) \"What is the maximum number of levels a person can pass in this game?\" That likely mea... | [
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"value": "2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation."
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"value": "<think>\n\nWe are asked:... | [
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"value": "2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation."
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"value": "<think>We are asked: \"2... | [
"We are asked: \"2. (9th Canadian Mathematical Competition) N is an integer, its representation in base b is 777. Find the smallest positive integer b such that N is a fourth power of an integer in decimal notation.\"",
"Interpretation: There exists an integer N. In base b (with b > 7 presumably, because digit 7 ... | [
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"value": "7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a ... | [
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"value": "7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a ... | [
"We need to solve the Diophantine equation:",
"n! / 2 = k! + l! for natural numbers (presumably positive integers? Usually natural numbers include 1,2,3,...). The equation is (n!)/2 = k! + l!. We need to find all triples (n,k,l) satisfying this.",
"We must consider that n! is even for n >= 2, so n! /2 is intege... | [
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"value": "Example 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 1"
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"value": "Example 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 1"
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... | [
"We are asked: \"The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\" Options: A. -1, B. -1/2, C. 1/2, D. 1.",
"We need to find the real root of each equation and sum them.",
"First equation: x^5 + x + 1 = 0. Second: x + \... | [
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"value": "Example 5 Given that $x_{1}, x_{2}, \\cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\\cdots+x_{10}^{2}$."
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"value": "<think>\n\nWe need to find max ... | [
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"value": "Example 5 Given that $x_{1}, x_{2}, \\cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\\cdots+x_{10}^{2}$."
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"value": "<think>We need to find max and ... | [
"We need to find max and min of sum of squares given that x_i are positive integers (positive means >=1) and sum is 2005.",
"We have 10 variables, each positive integer. Sum fixed. Want to minimize and maximize sum of squares.",
"This is a classic integer optimization: For fixed sum, sum of squares is minimized... | [
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"value": "1. If the set\n$$\nA=\\{1,2, \\cdots, 10\\}, B=\\{1,2,3,4\\},\n$$\n$C$ is a subset of $A$, and $C \\cap B \\neq \\varnothing$, then the number of such subsets $C$ is $(\\quad)$.\n(A) 256\n(B) 959\n(C) 960\n(D) 961"
},
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"value": "<think>\n\nWe nee... | [
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"value": "1. If the set\n$$\nA=\\{1,2, \\cdots, 10\\}, B=\\{1,2,3,4\\},\n$$\n$C$ is a subset of $A$, and $C \\cap B \\neq \\varnothing$, then the number of such subsets $C$ is $(\\quad)$.\n(A) 256\n(B) 959\n(C) 960\n(D) 961"
},
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"from": "assistant",
"value": "<think>We need to... | [
"We need to find number of subsets C of A = {1,2,...,10} such that C ∩ B ≠ ∅, where B = {1,2,3,4}. So C is any subset of A that contains at least one element from {1,2,3,4}.",
"Total subsets of A: 2^10 = 1024.",
"Number of subsets with no element from B (i.e., C ⊆ A \\ B = {5,6,7,8,9,10}) is 2^6 = 64.",
"Thus... | [
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"value": "Augusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag... | [
{
"from": "user",
"value": "Augusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag... | [
"We need to solve a problem about cutting a wire and forming squares to minimize sum of areas.",
"First part: We have wire length 10 m. Cut at point x (presumably x is length of first piece, the other is 10-x). From each piece we form a square. So side of first square = x/4 (since perimeter = length = x). Area = ... | [
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"value": "29. Choose any three numbers from $a, b, c, d, e$ and find their sum, exactly obtaining the ten different numbers $7,11,13,14,19,21,22,25,26,28$. Then $a+b+c+d+e=$ ( i). MATHEMATICE YoUTH CLUE A. 25\nB. 31\nC. 37\nD. 43"
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"value": "<think>\n\n... | [
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"from": "user",
"value": "29. Choose any three numbers from $a, b, c, d, e$ and find their sum, exactly obtaining the ten different numbers $7,11,13,14,19,21,22,25,26,28$. Then $a+b+c+d+e=$ ( i). MATHEMATICE YoUTH CLUE A. 25\nB. 31\nC. 37\nD. 43"
},
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"value": "<think>We a... | [
"We are given: \"Choose any three numbers from a, b, c, d, e and find their sum, exactly obtaining the ten different numbers 7,11,13,14,19,21,22,25,26,28. Then a+b+c+d+e= ?\" Options: A.25, B.31, C.37, D.43.",
"Interpretation: We have five numbers a,b,c,d,e. All sums of three distinct numbers (there are 5 choose ... | [
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"value": "Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea?"
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"value": "<think>\n\nWe are asked: \"In the s... | [
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"from": "user",
"value": "Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea?"
},
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"value": "<think>We are asked: \"In the schoo... | [
"We are asked: \"In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea?\"",
"We need to find number of teachers who drink only coffee, and only tea.",
"Given total = 50. Those who dr... | [
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"value": "12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4."
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"value": "<think>\n\nWe need to solve: In ... | [
{
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"value": "12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4."
},
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"value": "<think>We need to solve: In a tr... | [
"We need to solve: In a triangle, side a = 15, sum of other two sides b + c = 27. Inradius r = 4. Find cos A? Actually \"cosine of the angle opposite the given side\". The given side is 15, so that's side a opposite angle A. So find cos A.",
"Given: a = 15, b + c = 27, r = 4.",
"We need cos A. Possibly use form... | [
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"value": "83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\\qquad$. $(a+b$ and $b+a$ are considered the same equation)"
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"value": "83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\\qquad$. $(a+b$ and $b+a$ are considered the same equation)"
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"from": "assi... | [
"We are given: \"Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is ______. (a+b and b+a are considered the same equation)\". We need to interpret.",
"Likely there is an addition... | [
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"value": "4. The number of real solutions to the equation $\\left|x^{2}-3 x+2\\right|+\\left|x^{2}+2 x-3\\right|=11$ is ( ).\n(A) 0\n(B) 1\n(C) 2\n(D) 4"
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"value": "<think>\n\nWe need to find the number of real solutions to |x^2 - 3x + 2| + |x^2 + 2x - ... | [
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"value": "4. The number of real solutions to the equation $\\left|x^{2}-3 x+2\\right|+\\left|x^{2}+2 x-3\\right|=11$ is ( ).\n(A) 0\n(B) 1\n(C) 2\n(D) 4"
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"value": "<think>We need to find the number of real solutions to |x^2 - 3x + 2| + |x^2 + 2x - 3| =... | [
"We need to find the number of real solutions to |x^2 - 3x + 2| + |x^2 + 2x - 3| = 11.",
"Let's define f1(x) = x^2 - 3x + 2, f2(x) = x^2 + 2x - 3. Both are quadratic. The sum of absolute values equals 11.",
"We need to consider intervals where each quadratic is non-negative or negative. Find roots of each quadr... | [
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"value": "## Problem Statement\n\nCalculate the limit of the numerical sequence:\n\n$\\lim _{n \\rightarrow \\infty} \\frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$"
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"value": "<think>\n\nWe need to calculate:\n\nlim_{n→∞} ((n+1)^4 - (n-1)^4) / ((n+1)... | [
{
"from": "user",
"value": "## Problem Statement\n\nCalculate the limit of the numerical sequence:\n\n$\\lim _{n \\rightarrow \\infty} \\frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$"
},
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"from": "assistant",
"value": "<think>We need to calculate:\n\nlim_{n→∞} ((n+1)^4 - (n-1)^4) / ((n+1)^3 +... | [
"We need to calculate:",
"lim_{n→∞} ((n+1)^4 - (n-1)^4) / ((n+1)^3 + (n-1)^3).",
"We can compute directly by expanding or factoring. It's a rational function of n.",
"Let’s compute numerator: (n+1)^4 - (n-1)^4. This is a difference of two squares? Actually we can expand or use identity: a^4 - b^4 = (a^2 - b^2... | [
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"value": "Condition of the problem\n\nFind the derivative.\n\n$$\ny=\\frac{1}{24}\\left(x^{2}+8\\right) \\sqrt{x^{2}-4}+\\frac{x^{2}}{16} \\arcsin \\frac{2}{x}, x>0\n$$"
},
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"value": "<think>\n\nWe are asked to find the derivative of y with respect to x. T... | [
{
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"value": "Condition of the problem\n\nFind the derivative.\n\n$$\ny=\\frac{1}{24}\\left(x^{2}+8\\right) \\sqrt{x^{2}-4}+\\frac{x^{2}}{16} \\arcsin \\frac{2}{x}, x>0\n$$"
},
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"value": "<think>We are asked to find the derivative of y with respect to x. The f... | [
"We are asked to find the derivative of y with respect to x. The function:",
"y = (1/24) (x^2 + 8) sqrt(x^2 - 4) + (x^2/16) arcsin(2/x), for x>0.",
"We need to compute dy/dx.",
"We'll compute derivative term by term.",
"Let’s denote first term: A = (1/24)(x^2+8) sqrt(x^2-4).",
"Second term: B = (x^2/16) a... | [
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"value": "\\section*{Problem 5 - 071225}\n\nAll ordered pairs of real numbers \\((x, y)\\) are to be determined for which the system of equations\n\n\\[\n\\begin{aligned}\nx \\cdot\\left(a x^{2}+b y^{2}-a\\right) & =0 \\\\\ny \\cdot\\left(a x^{2}+b y^{2}-b\\right) & =0\n\\end{aligned}\n\\]... | [
{
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"value": "\\section*{Problem 5 - 071225}\n\nAll ordered pairs of real numbers \\((x, y)\\) are to be determined for which the system of equations\n\n\\[\n\\begin{aligned}\nx \\cdot\\left(a x^{2}+b y^{2}-a\\right) & =0 \\\\\ny \\cdot\\left(a x^{2}+b y^{2}-b\\right) & =0\n\\end{aligned}\n\\]... | [
"We are given problem: \"All ordered pairs of real numbers (x, y) are to be determined for which the system of equations",
"\\[\n\\begin{aligned}\nx \\cdot\\left(a x^{2}+b y^{2}-a\\right) & =0 \\\\\ny \\cdot\\left(a x^{2}+b y^{2}-b\\right) & =0\n\\end{aligned}\n\\]",
"is satisfied. Here, a and b are real number... | [
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"value": "6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,\n$$\n\\begin{array}{c}\nS=\\left[\\frac{1}{1}\\right]+\\left[\\frac{2}{1}\\right]+\\left[\\frac{1}{2}\\right]+\\left[\\frac{2}{2}\\right]+\\left[\\frac{3}{2}\\right]+ \\\\\n{\\left[\\frac{4}{2}\\right]+\\... | [
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"value": "6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,\n$$\n\\begin{array}{c}\nS=\\left[\\frac{1}{1}\\right]+\\left[\\frac{2}{1}\\right]+\\left[\\frac{1}{2}\\right]+\\left[\\frac{2}{2}\\right]+\\left[\\frac{3}{2}\\right]+ \\\\\n{\\left[\\frac{4}{2}\\right]+\\... | [
"We need to compute S defined as:",
"S = [1/1] + [2/1] + [1/2] + [2/2] + [3/2] + [4/2] + [1/3] + [2/3] + [3/3] + [4/3] + [5/3] + [6/3] + ... up to 2016 terms.",
"The pattern: For denominator k (starting from k=1?), the segment has up to 2k terms: [1/k], [2/k], ..., [2k/k]. But the description says: \"for a segm... | [
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"value": "40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers."
},
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"value": "<think>\n\nWe need to find triples of three consecutive odd numbers such that their sum equals th... | [
{
"from": "user",
"value": "40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers."
},
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"from": "assistant",
"value": "<think>We need to find triples of three consecutive odd numbers such that their sum equals the fo... | [
"We need to find triples of three consecutive odd numbers such that their sum equals the fourth power of a single-digit number. Let the odd numbers be (2n-1), (2n+1), (2n+3) for some integer n (since consecutive odd numbers differ by 2). Their sum = (2n-1)+(2n+1)+(2n+3) = 6n+3 = 3(2n+1). So the sum is three times t... | [
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"value": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$(\\mathrm {A}) \\ 1 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 10 \\qquad (\\mathrm {E})\\ 20$"
},
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"from": "assistant",
"value": "<think>\n\nWe are given: \"Two is 1... | [
{
"from": "user",
"value": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$(\\mathrm {A}) \\ 1 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 10 \\qquad (\\mathrm {E})\\ 20$"
},
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"from": "assistant",
"value": "<think>We are given: \"Two is 10% o... | [
"We are given: \"Two is 10% of x and 20% of y.\" So: 2 = 0.10 * x => x = 2 / 0.10 = 20. Also 2 = 0.20 * y => y = 2 / 0.20 = 10. Then x - y = 20 - 10 = 10. So answer D.",
"But let's confirm: 10% of 20 is 2, yes. 20% of 10 is 2, yes. So answer is 10.",
"Thus answer D."
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"value": "19. Given $m \\in\\{11,13,15,17,19\\}$, $n \\in\\{1999,2000, \\cdots, 2018\\}$.\nThen the probability that the unit digit of $m^{n}$ is 1 is ( ).\n(A) $\\frac{1}{5}$\n(B) $\\frac{1}{4}$\n(C) $\\frac{3}{10}$\n(D) $\\frac{7}{20}$\n(E) $\\frac{2}{5}$"
},
{
"from": "assistant... | [
{
"from": "user",
"value": "19. Given $m \\in\\{11,13,15,17,19\\}$, $n \\in\\{1999,2000, \\cdots, 2018\\}$.\nThen the probability that the unit digit of $m^{n}$ is 1 is ( ).\n(A) $\\frac{1}{5}$\n(B) $\\frac{1}{4}$\n(C) $\\frac{3}{10}$\n(D) $\\frac{7}{20}$\n(E) $\\frac{2}{5}$"
},
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"from": "assistant... | [
"We are given: m ∈ {11,13,15,17,19} (so five odd numbers, specifically odd numbers from 11 to 19 inclusive, odd only? Actually 11,13,15,17,19 all odd). n ∈ {1999,2000,...,2018} (that's 20 numbers? Let's check: from 1999 to 2018 inclusive: count = 2018-1999+1 = 20). So total possible (m,n) pairs = 5 * 20 = 100. We n... | [
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"value": "1. (6 points) Today is January 31, 2015, welcome to the 2015 \"Spring Cup\" Final. The calculation result of the expression $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$ is"
},
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"from": "assistant",
"value": "<think>\n\nWe need to evaluate the expression: ( (2... | [
{
"from": "user",
"value": "1. (6 points) Today is January 31, 2015, welcome to the 2015 \"Spring Cup\" Final. The calculation result of the expression $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$ is"
},
{
"from": "assistant",
"value": "<think>We need to evaluate the expression: ( (2015/... | [
"We need to evaluate the expression: ( (2015/1) + (2015/0.31) ) / (1 + 0.31). Actually the expression is written as:",
"$\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$",
"Simplify: numerator: 2015/1 = 2015. plus 2015/0.31. So numerator = 2015 + 2015/0.31.",
"Denominator: 1 + 0.31 = 1.31.",
"We can comp... | [
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"value": "Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?"
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"value": "<think>\n\nWe need to find the ratio of the area inside the incircle to the area of the hexagon. The hexagon... | [
{
"from": "user",
"value": "Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?"
},
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"from": "assistant",
"value": "<think>We need to find the ratio of the area inside the incircle to the area of the hexagon. The hexagon is ... | [
"We need to find the ratio of the area inside the incircle to the area of the hexagon. The hexagon is regular, and it has an incircle (a circle tangent to all its sides). For a regular polygon, the incircle is the circle inscribed in the polygon, touching each side at its midpoint. The question: \"Consider a regula... | [
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"value": "Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\\cdot}AG + BP{\\cdot}BG + CP{\\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$."
},
{
"from": "... | [
{
"from": "user",
"value": "Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\\cdot}AG + BP{\\cdot}BG + CP{\\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$."
},
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"from": "... | [
"We need to solve: Let ABC be a triangle with centroid G. For any point P in the plane, consider the sum S(P) = AP * AG + BP * BG + CP * CG. Find P that minimizes S(P), and find the minimal value in terms of side lengths a=BC, b=CA, c=AB.",
"First, note that AG, BG, CG are lengths from vertices to centroid. They ... | [
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"value": "1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is"
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"value": "<think>\n\nWe are given: \"Four points $A, B, C, D$ in... | [
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"value": "1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is"
},
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"value": "<think>We are given: \"Four points $A, B, C, D$ in spa... | [
"We are given: \"Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is\". It seems incomplete? Probably the question: Four points A, B, C, D in space are pairwise 1 unit apart (so it's a regular t... | [
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"value": "The function $f$ maps the set of positive integers into itself, and satisfies the equation\n\n$$\nf(f(n))+f(n)=2 n+6\n$$\n\nWhat could this function be?"
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"value": "<think>\n\nWe are asked to find a function f: positive integers -> positive in... | [
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"value": "The function $f$ maps the set of positive integers into itself, and satisfies the equation\n\n$$\nf(f(n))+f(n)=2 n+6\n$$\n\nWhat could this function be?"
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"value": "<think>We are asked to find a function f: positive integers -> positive intege... | [
"We are asked to find a function f: positive integers -> positive integers (i.e., f: Z+ → Z+) satisfying functional equation:",
"f(f(n)) + f(n) = 2n + 6 for all positive integers n.",
"We need to determine what the function could be. Probably there is a unique solution (or maybe a family). Let's try to solve.... | [
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"value": "30. Find the remainder when the 2018-digit number $\\underbrace{\\overline{55 \\cdots}}_{2018 \\text { 555 }}$ is divided by 13."
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"value": "<think>\n\nWe need to find remainder when a 2018-digit number consisting of all 5's (i.e., 555...5 wit... | [
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"value": "30. Find the remainder when the 2018-digit number $\\underbrace{\\overline{55 \\cdots}}_{2018 \\text { 555 }}$ is divided by 13."
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"value": "<think>We need to find remainder when a 2018-digit number consisting of all 5's (i.e., 555...5 with 20... | [
"We need to find remainder when a 2018-digit number consisting of all 5's (i.e., 555...5 with 2018 digits) is divided by 13. That is number N = 5 * (10^{2018} - 1)/9. Actually careful: number with n digits each 5 is: 5*(10^{n-1}+10^{n-2}+...+10^0) = 5*(10^n -1)/9. So N = 5*(10^{2018} - 1)/9. We need N mod 13.",
"... | [
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"value": "1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).\nA. 1\nB. 2\nC. 3\nD. 4"
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"value": "<think>\n\nWe need to determine the maximum number of intersection points between a line and a hyperbola.\n\nA... | [
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"value": "1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).\nA. 1\nB. 2\nC. 3\nD. 4"
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"value": "<think>We need to determine the maximum number of intersection points between a line and a hyperbola.\n\nA hyp... | [
"We need to determine the maximum number of intersection points between a line and a hyperbola.",
"A hyperbola is a conic section, defined as the set of points such that the absolute difference of distances to two foci is constant, or algebraically as a second-degree equation of the form:",
"Ax^2 + Bxy + Cy^2 +... | [
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"value": "1. Let the universal set be the set of real numbers. If $A=\\{x \\mid \\sqrt{x-2} \\leqslant 0\\}, B=\\left\\{x \\mid 10^{x^{2}-2}=10^{x}\\right\\}$, then $A \\cap \\bar{B}$ is\nA. $\\{2\\}$\nB. $\\{-1\\}$\nC. $\\{x \\mid x \\leqslant 2\\}$\nD. $\\varnothing$"
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"from": "user",
"value": "1. Let the universal set be the set of real numbers. If $A=\\{x \\mid \\sqrt{x-2} \\leqslant 0\\}, B=\\left\\{x \\mid 10^{x^{2}-2}=10^{x}\\right\\}$, then $A \\cap \\bar{B}$ is\nA. $\\{2\\}$\nB. $\\{-1\\}$\nC. $\\{x \\mid x \\leqslant 2\\}$\nD. $\\varnothing$"
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"We are given:",
"Universal set = real numbers.",
"A = { x | sqrt(x-2) ≤ 0 }",
"Recall sqrt of a real number: sqrt(x-2) is defined only for x-2 ≥ 0, i.e., x ≥ 2. sqrt(x-2) ≥ 0 always. So sqrt(x-2) ≤ 0 means sqrt(x-2) = 0 (since it's nonnegative). So sqrt(x-2) = 0 => x-2 = 0 => x = 2. Also we must have domain:... | [
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"value": "4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \\perp A M$, then $\\angle B A C$ equals ( ).\n... | [
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"value": "4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \\perp A M$, then $\\angle B A C$ equals ( ).\n... | [
"We are given a right triangular prism ABC-A1B1 C1. Typically a right triangular prism means the lateral edges (like AA1, BB1, CC1) are perpendicular to the base. But here they say \"right triangular prism\", which probably means it's a prism with the lateral edges perpendicular to the base (so it's a right prism).... | [
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"value": "7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$."
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"value": "<think>\n\nWe need to solve: \n(16 * 5^(2x-1) - 2 * 5^(x-1) - 0.048) * lg(x^3 + 2x + 1) = 0.\n\nHere lg is presumably base 10 log... | [
{
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"value": "7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$."
},
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"from": "assistant",
"value": "<think>We need to solve: \n(16 * 5^(2x-1) - 2 * 5^(x-1) - 0.048) * lg(x^3 + 2x + 1) = 0.\n\nHere lg is presumably base 10 logarit... | [
"We need to solve: \n(16 * 5^(2x-1) - 2 * 5^(x-1) - 0.048) * lg(x^3 + 2x + 1) = 0.",
"Here lg is presumably base 10 logarithm? Usually in these contexts, \"lg\" denotes base 10. But could be natural? However typical notation: lg = log10, ln = natural. So we'll assume log10.",
"Equation is product equals zero, s... | [
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"value": "In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?\n$\\text{(A)}\\ 40 \\qquad \\text{(B)}\\ 200 \\qquad \\text{(C)}\\ 400 \\qquad \\t... | [
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"value": "In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?\n$\\text{(A)}\\ 40 \\qquad \\text{(B)}\\ 200 \\qquad \\text{(C)}\\ 400 \\qquad \\t... | [
"We need to find area of rectangular backyard given conditions. Let length = L, width = W. Units in meters.",
"Condition 1: walking the length 25 times equals 1000 meters? Actually: \"he must walk the length 25 times or walk its perimeter 10 times.\" Means that to walk a kilometer (1000 m) in his backyard, he cou... | [
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"value": "11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base."
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"value": "<think>\n\nWe need to solve: \"11.005. The plane angle at th... | [
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"value": "11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base."
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"value": "<think>We need to solve: \"11.005. The plane angle at the ve... | [
"We need to solve: \"11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base.\"",
"Interpretation: \"Regular triangular pyramid\" means a tetrahedron with all edges equal? Actually, a regular triangul... | [
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"value": "The knights in a certain kingdom come in two colors. $\\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What frac... | [
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"value": "The knights in a certain kingdom come in two colors. $\\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What frac... | [
"We need to find the fraction of red knights that are magical.",
"Let total number of knights = T (or use fractions, but can use numbers for convenience).",
"Given: 2/7 are red. So red = (2/7)T. Blue = (5/7)T.",
"Given: 1/6 of all knights are magical. So magical total = (1/6)T.",
"Let fraction of blue knigh... | [
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"value": "A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to t... | [
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"value": "A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to t... | [
"We need to find the number of children such that the property distribution pattern described leads to each child receiving the same amount. Let's denote total property as P (yuan). The process:",
"For child i (i starting from 1), they get: i * 1000 yuan (i.e., 1000, 2000, 3000, ...) plus one-tenth of the remaini... | [
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"value": "Example 11 Let $x>0, y>0, \\sqrt{x}(\\sqrt{x}+2 \\sqrt{y})$ $=\\sqrt{y}(6 \\sqrt{x}+5 \\sqrt{y})$. Find the value of $\\frac{x+\\sqrt{x y}-y}{2 x+\\sqrt{x y}+3 y}$."
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"value": "<think>\n\nWe are given: \"Example 11 Let $x>0, y>0, \\sqrt{x}(\\s... | [
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"from": "user",
"value": "Example 11 Let $x>0, y>0, \\sqrt{x}(\\sqrt{x}+2 \\sqrt{y})$ $=\\sqrt{y}(6 \\sqrt{x}+5 \\sqrt{y})$. Find the value of $\\frac{x+\\sqrt{x y}-y}{2 x+\\sqrt{x y}+3 y}$."
},
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"value": "<think>We are given: \"Example 11 Let $x>0, y>0, \\sqrt{x}(\\sqrt{... | [
"We are given: \"Example 11 Let $x>0, y>0, \\sqrt{x}(\\sqrt{x}+2 \\sqrt{y})$ $=\\sqrt{y}(6 \\sqrt{x}+5 \\sqrt{y})$. Find the value of $\\frac{x+\\sqrt{x y}-y}{2 x+\\sqrt{x y}+3 y}$.\" So we need to solve for the ratio given the equation. Let's rewrite the given equation:",
"\\sqrt{x}(\\sqrt{x}+2\\sqrt{y}) = \\sqr... | [
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"value": "9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?"
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"value": "<think>\n\nWe are given a regular tetrahedron of volum... | [
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"value": "9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?"
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"value": "<think>We are given a regular tetrahedron of volume 1.... | [
"We are given a regular tetrahedron of volume 1. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?",
"Interpretation: We have a regular tetrahedron T. Its center (the centroid) is the same as its circumcenter, incenter, etc for a regul... | [
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"value": "Let's determine all the triples of numbers $(x, y, m)$ for which\n\n$$\n-2 x + 3 y = 2 m, \\quad x - 5 y = -11\n$$\n\nand $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number."
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"value": "<think>\n\nWe have a system:\... | [
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"value": "Let's determine all the triples of numbers $(x, y, m)$ for which\n\n$$\n-2 x + 3 y = 2 m, \\quad x - 5 y = -11\n$$\n\nand $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number."
},
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"value": "<think>We have a system:\n\n-... | [
"We have a system:",
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"We can solve the system for x and y in terms of m, or solve for x and y integers first and then find m."... | [
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"value": "The equation of line $\\ell_1$ is $24x-7y = 319$ and the equation of line $\\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of ... | [
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"value": "The equation of line $\\ell_1$ is $24x-7y = 319$ and the equation of line $\\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of ... | [
"We need to check the problem statement and solution: The problem says: \"The equation of line ℓ1 is 24x-7y = 319 and the equation of line ℓ2 is 12x-5y = 125. Let a be the number of positive integer values n less than 2023 such that for both ℓ1 and ℓ2 there exists a lattice point on that line that is a distance of ... | [
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"value": "Let's determine $m$ such that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$.\n\n---\n\nDetermine $m$ so that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$."
},
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"value": "<... | [
{
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"value": "Let's determine $m$ such that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$.\n\n---\n\nDetermine $m$ so that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$."
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"We need to consider cases:",
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"value": "2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$."
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"value": "<think>\n\nWe are asked: \"How many integers b exist such that... | [
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"value": "2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$."
},
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"value": "<think>We are asked: \"How many integers b exist such that the... | [
"We are asked: \"How many integers b exist such that the equation x^2 + b x - 9600 = 0 has an integer solution that is a multiple of both 10 and 12? Specify the largest possible b.\"",
"Interpretation: We have quadratic equation x^2 + b x - 9600 = 0, with integer b. We require that there exists an integer solutio... | [
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"value": "Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \\leqslant x \\leqslant 0)$ is $(\\quad)$.\n(A) $(-\\infty, 4]$\n(B) $[3,12]$\n(C) $[-12,4]$\n(D) $[4,12]$"
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"value": "<think>\n\nWe are asked: \"Example 1 The range of the function $y=-x^... | [
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"value": "Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \\leqslant x \\leqslant 0)$ is $(\\quad)$.\n(A) $(-\\infty, 4]$\n(B) $[3,12]$\n(C) $[-12,4]$\n(D) $[4,12]$"
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"value": "<think>We are asked: \"Example 1 The range of the function $y=-x^{2}-... | [
"We are asked: \"Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \\leqslant x \\leqslant 0)$ is $(\\quad)$.\" Options: (A) $(-\\infty, 4]$ (B) $[3,12]$ (C) $[-12,4]$ (D) $[4,12]$. So we need to compute the range of y = -x^2 -2x +3 on the domain x ∈ [-5, 0].",
"Let's find the quadratic. Write y = -x^2 -2x +... | [
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"value": "4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,\n$$\\begin{array}{l}\n\\vec{f}_{1}=\\overrightarrow{O A}=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n\\vec{f}_{2}=\\overrightarrow{O B}=2\\left[\\cos \\left(-30... | [
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"value": "4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,\n$$\\begin{array}{l}\n\\vec{f}_{1}=\\overrightarrow{O A}=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n\\vec{f}_{2}=\\overrightarrow{O B}=2\\left[\\cos \\left(-30... | [
"We need to parse the problem: There are two forces f1 and f2 acting on origin O. They are given as vectors in complex plane representation: \nf1 = OA = sqrt(2) (cos45° + i sin45°)\nf2 = OB = 2 [cos(-30°) + i sin(-30°)]",
"So f1 and f2 are force vectors. (1) Find magnitude and direction of resultant force (sum f1... | [
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"value": "6. Given that $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. If $t \\in[0,1]$, then\n$$\n|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right|... | [
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"value": "6. Given that $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. If $t \\in[0,1]$, then\n$$\n|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right|... | [
"We need to find the minimum value of the expression:",
"S(t) = | t * AB - AO | + | (5/12) BO - (1-t) BA |, where t ∈ [0,1].",
"Given that OA ⟂ OB, and |OA| = |OB| = 24.",
"We need to interpret vectors relative to origin O. We have points A and B such that OA and OB are perpendicular and of length 24.",
"We... | [
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"value": "All three vertices of $\\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? \n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)... | [
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"value": "All three vertices of $\\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? \n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(C)... | [
"We are given: All three vertices of triangle ABC lie on the parabola y = x^2, with A at the origin and BC parallel to the x-axis. The area of the triangle is 64. What is the length of BC?",
"We need to find the length BC. Let's set up coordinates: A = (0,0) since A at origin, and on parabola: (0,0) is indeed on ... | [
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"value": "1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$."
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"value": "<think>\n\nWe are given non-zero real numbers a, b... | [
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"value": "1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$."
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"value": "<think>We are given non-zero real numbers a, b, c ... | [
"We are given non-zero real numbers a, b, c satisfying:",
"(1) a^2 + a = b^2\n(2) b^2 + b = c^2\n(3) c^2 + c = a^2",
"We need to find (a - b)(b - c)(c - a).",
"We need to find possible values given that a,b,c are non-zero reals.",
"Let’s try to solve the system. The equations are cyclic but not symmetric? A... | [
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"value": "2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).\n(A) $\\frac{\\sqrt{5}+1}{2}$\n(B) $\\frac{\\sqrt{5}-1}{2}$\n(C) 1\n(D) 2"
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"v... | [
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"value": "2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).\n(A) $\\frac{\\sqrt{5}+1}{2}$\n(B) $\\frac{\\sqrt{5}-1}{2}$\n(C) 1\n(D) 2"
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"We are given: Rhombus ABCD, side length = a. O is a point on diagonal AC, with OA = a, OB = OC = OD = 1. Then a equals? Options: (A) (√5+1)/2, (B) (√5-1)/2, (C) 1, (D) 2.",
"We need to find a.",
"Interpret geometry: Rhombus ABCD, all sides equal to a. Usually vertices order: A, B, C, D. Diagonal AC. O is on AC... | [
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"value": "V-2 If one side of the rectangle is reduced by $3 \\mathrm{~cm}$, and the other side is reduced by $2 \\mathrm{~cm}$, we get a square whose area is $21 \\mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.\n\n![](https://cdn.mathpix.co... | [
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"value": "V-2 If one side of the rectangle is reduced by $3 \\mathrm{~cm}$, and the other side is reduced by $2 \\mathrm{~cm}$, we get a square whose area is $21 \\mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.\n\n![](https://cdn.mathpix.co... | [
"We are given: \"V-2 If one side of the rectangle is reduced by 3 cm, and the other side is reduced by 2 cm, we get a square whose area is 21 cm^2 less than the area of the rectangle. Calculate the dimensions of the rectangle.\"",
"Interpretation: There is a rectangle. Let its sides be length and width. One side ... | [
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"value": "1. Given $a, b>0, a \\neq 1$, and $a^{b}=\\log _{a} b$, then the value of $a^{a^{b}}-\\log _{a} \\log _{a} b^{a}$ is"
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"value": "<think>\n\nWe are given: a, b > 0, a ≠ 1, and a^b = log_a b. We need to find the value of a^{a^b} - log_a (log_a (... | [
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"value": "1. Given $a, b>0, a \\neq 1$, and $a^{b}=\\log _{a} b$, then the value of $a^{a^{b}}-\\log _{a} \\log _{a} b^{a}$ is"
},
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"value": "<think>We are given: a, b > 0, a ≠ 1, and a^b = log_a b. We need to find the value of a^{a^b} - log_a (log_a (b^a)... | [
"We are given: a, b > 0, a ≠ 1, and a^b = log_a b. We need to find the value of a^{a^b} - log_a (log_a (b^a)). Let's denote something.",
"We have equation: a^b = log_a b.",
"Let’s write log_a b = ln b / ln a. So equation: a^b = (ln b)/(ln a). Not sure if helpful.",
"We need to compute expression: E = a^{a^b} ... | [
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"value": "4. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \\geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2... | [
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"value": "4. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \\geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2... | [
"We are given a sequence {a_n} with a1 = 0, a2 = 1, and for n ≥ 3, a_n is the smallest positive integer greater than a_{n-1} such that there is no subsequence of a_1, a_2, ..., a_n that forms an arithmetic sequence. So we need to find a_2014.",
"Interpretation: The condition: \"there is no subsequence of a_1, a_2... | [
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"value": "In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?"
},
{
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"value"... | [
{
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"value": "In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?"
},
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"value"... | [
"We have a problem: In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?",
"Interpretation: There are 10 distinct mailboxes. First di... | [
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"value": "18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues tow... | [
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"value": "18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues tow... | [
"We are given: Li Shuang rides a bike at a speed of 320 meters per minute from location A to location B. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from B to repair the bike. After 15 minutes, he continues towards B at 1.5 times his original riding sp... | [
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"value": "53. How many four-digit numbers contain at least one even digit?"
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"value": "<think>\n\nWe need to count how many four-digit numbers contain at least one even digit. The total number of four-digit numbers is from 1000 to 9999 inclusive: 9999 -... | [
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"value": "53. How many four-digit numbers contain at least one even digit?"
},
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"value": "<think>We need to count how many four-digit numbers contain at least one even digit. The total number of four-digit numbers is from 1000 to 9999 inclusive: 9999 - 100... | [
"We need to count how many four-digit numbers contain at least one even digit. The total number of four-digit numbers is from 1000 to 9999 inclusive: 9999 - 1000 + 1 = 9000. Count the complement: numbers that contain no even digits, i.e., all digits are odd. Digits are from 0-9 but for four-digit numbers, the thous... | [
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"value": "1. The range of the function $f(x)=\\sin x+\\cos x+\\tan x+$ $\\arcsin x+\\arccos x+\\arctan x$ is $\\qquad$ ."
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"value": "<think>\n\nWe are asked: \"The range of the function f(x) = sin x + cos x + tan x + arcsin x + arccos x + arctan x is __... | [
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"value": "1. The range of the function $f(x)=\\sin x+\\cos x+\\tan x+$ $\\arcsin x+\\arccos x+\\arctan x$ is $\\qquad$ ."
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"value": "<think>We are asked: \"The range of the function f(x) = sin x + cos x + tan x + arcsin x + arccos x + arctan x is ______... | [
"We are asked: \"The range of the function f(x) = sin x + cos x + tan x + arcsin x + arccos x + arctan x is ______.\" We need to find the range of this function. But careful: The domain of f? The function involves sin, cos, tan of x, which are defined for all real x except tan has domain all real x except x ≠ π/2 +... | [
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"value": "10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid."
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"value": "<think>\n\nWe are given: \"10.319. The diagonals of an isosceles trapezoid are perpend... | [
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"value": "10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid."
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"value": "<think>We are given: \"10.319. The diagonals of an isosceles trapezoid are perpendicul... | [
"We are given: \"10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid.\"",
"Interpretation: We have an isosceles trapezoid with diagonals perpendicular. Area = a^2. Find its height in terms of a? Possibly height = something... | [
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"value": "8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result directly."
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"value": "<think>\n\nWe need to translat... | [
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"value": "8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result directly."
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"value": "<think>We need to translate th... | [
"We need to translate the given text: \"8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\" The context: It's a math problem. Probably from some contest. The phrase \"100 sevens\" suggests there are 100 sevens in a tower? Actually \"7^{7^{-7}}\" might be misinterpreted; maybe it's 7^{7^{7^{...}}} with 100 ... | [
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"value": "1. Let $S=\\{1,2, \\cdots, n\\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find the ... | [
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"value": "1. Let $S=\\{1,2, \\cdots, n\\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find the ... | [
"We need to find the number of arithmetic sequences A (with at least two terms) consisting of elements from S = {1,2,...,n}, with positive common difference d, such that adding any other element x in S (x not already in A) would destroy the property that the new set is an arithmetic sequence with the same common di... | [
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"value": "9.171. $0.6^{\\lg ^{2}(-x)+3} \\leq\\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}$."
},
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"value": "<think>\n\nWe need to solve inequality:\n\n$$0.6^{\\lg^{2}(-x)+3} \\leq \\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}$$\n\n<latent><latent_1></latent>\n\n... | [
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"value": "9.171. $0.6^{\\lg ^{2}(-x)+3} \\leq\\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}$."
},
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"value": "<think>We need to solve inequality:\n\n$$0.6^{\\lg^{2}(-x)+3} \\leq \\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}$$\n\nFirst, note domain: We have $\\lg(-... | [
"We need to solve inequality:",
"$$0.6^{\\lg^{2}(-x)+3} \\leq \\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}$$",
"First, note domain: We have $\\lg(-x)$ meaning logarithm base 10? Usually $\\lg$ denotes base 10. So argument of $\\lg$ must be positive: $-x > 0 \\implies x < 0$. Also we have $\\lg x^{2}$; for $\\lg(x... | [
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"value": "2. Find all integer solutions of the inequality\n\n$$\nx^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \\leq 0\n$$"
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"value": "<think>\n\nWe are given an inequality:\n\n<latent><latent_1></latent>\n\nWe need to find all int... | [
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"value": "2. Find all integer solutions of the inequality\n\n$$\nx^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \\leq 0\n$$"
},
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"value": "<think>We are given an inequality:\n\n$$\nx^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x... | [
"We are given an inequality:",
"$$\nx^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \\leq 0\n$$",
"We need to find all integer solutions (x, y, z integers). Possibly we can factor or rewrite as sum of squares plus something. Let's attempt to rearrange.",
"We have terms: x^2 y^2, y^2 z^2, x... | [
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"value": "Solve the triangle whose area $t=357.18 \\mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$."
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"value": "<think>\n\nWe are given: area t = 357.18 cm^2, sides are in ratio a:b:c = 4:5:6. Need to solve the triangle: find sides,... | [
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"value": "Solve the triangle whose area $t=357.18 \\mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$."
},
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"value": "<think>We are given: area t = 357.18 cm^2, sides are in ratio a:b:c = 4:5:6. Need to solve the triangle: find sides, ang... | [
"We are given: area t = 357.18 cm^2, sides are in ratio a:b:c = 4:5:6. Need to solve the triangle: find sides, angles, etc.",
"Let sides: a = 4k, b = 5k, c = 6k for some k>0.",
"We have area. Use Heron's formula: t = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 = (4k+5k+6k)/2 = (15k)/2 = 7.5k.",
"Then s-a = 7.5... | [
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"value": "Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.\n\\[ 2^m p^2 + 1 = q^5 \\]"
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"value": "<think>\n\nWe need to find all triples (m, p, q) with m positive integer, p and q primes, satisfying:\n\n<latent><lat... | [
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"value": "Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.\n\\[ 2^m p^2 + 1 = q^5 \\]"
},
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"value": "<think>We need to find all triples (m, p, q) with m positive integer, p and q primes, satisfying:\n\n2^m p^2 + 1 = q^... | [
"We need to find all triples (m, p, q) with m positive integer, p and q primes, satisfying:",
"2^m p^2 + 1 = q^5.",
"We need to find all solutions. This is an exponential Diophantine equation.",
"We have q^5 - 1 = (q - 1)(q^4 + q^3 + q^2 + q + 1). And it equals 2^m p^2.",
"Thus we have factorization: 2^m p^... | [
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"value": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two... | [
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"value": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two... | [
"We need to parse the problem. There's a square ABCD area 196 cm², so side length = sqrt(196) = 14 cm. It contains two partially overlapping smaller squares. The larger of the two smaller squares has area 4 times the area of the smaller one. Their overlapping area is 1 cm². We need area of the shaded part. Probably... | [
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"value": "25. Anna, Bridgit and Carol run in a $100 \\mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \\mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \\mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna fin... | [
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"value": "25. Anna, Bridgit and Carol run in a $100 \\mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \\mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \\mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna fin... | [
"We need to solve: In a 100m race, Anna, Bridgit, Carol run at constant speeds. When Anna finishes (at 100m), Bridgit is 16m behind her, i.e., Bridgit has run 84m. When Bridgit finishes (at 100m), Carol is 25m behind her, i.e., Carol has run 75m. We need: How far behind was Carol when Anna finished? That is, when A... | [
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"value": "Given the complex number $z$ has a modulus of 1. Find\n$$\nu=\\frac{(z+4)^{2}-(\\bar{z}+4)^{2}}{4 i} \\text {. }\n$$\n\nthe maximum value."
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"value": "<think>\n\nWe need to find the maximum value of u, where\n\nu = ((z+4)^2 - (\\bar{z}+4)^2) /... | [
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"value": "Given the complex number $z$ has a modulus of 1. Find\n$$\nu=\\frac{(z+4)^{2}-(\\bar{z}+4)^{2}}{4 i} \\text {. }\n$$\n\nthe maximum value."
},
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"value": "<think>We need to find the maximum value of u, where\n\nu = ((z+4)^2 - (\\bar{z}+4)^2) / (4i... | [
"We need to find the maximum value of u, where",
"u = ((z+4)^2 - (\\bar{z}+4)^2) / (4i)",
"given that |z| = 1.",
"We can simplify expression.",
"Let z be a complex number on unit circle: z = e^{iθ} or more generally z = cosθ + i sinθ. But we can also treat algebraically.",
"Compute (z+4)^2 - (\\bar{z}+4)^... | [
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"value": "## Problem 1\n\nPerform the calculations:\n\na) $7 \\cdot 147 - 7 \\cdot 47$ (1p)\n\nb) $(2+4+6+8+\\cdots+50)-(1+3+5+7+\\cdots+49)$ (2p)\n\nc) $10 \\cdot 9^{2} : 3^{2} - 3^{4} \\quad(2 \\text{p})$\n\nd) $(\\overline{a b} + \\overline{b c} + \\overline{c a}) : (a + b + c) \\quad$ ... | [
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"value": "## Problem 1\n\nPerform the calculations:\n\na) $7 \\cdot 147 - 7 \\cdot 47$ (1p)\n\nb) $(2+4+6+8+\\cdots+50)-(1+3+5+7+\\cdots+49)$ (2p)\n\nc) $10 \\cdot 9^{2} : 3^{2} - 3^{4} \\quad(2 \\text{p})$\n\nd) $(\\overline{a b} + \\overline{b c} + \\overline{c a}) : (a + b + c) \\quad$ ... | [
"We need to solve each part. The problem statements:",
"a) 7·147 - 7·47 (1p)",
"b) (2+4+6+8+...+50) - (1+3+5+7+...+49) (2p)",
"c) 10·9^{2} : 3^{2} - 3^{4} (2p) Note: likely means 10 * 9^2 / 3^2 - 3^4. Possibly careful order of operations: multiplication and division left to right, then subtraction. Or maybe... | [
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"value": "1. The curve $(x+2 y+a)\\left(x^{2}-y^{2}\\right)=0$ represents three straight lines intersecting at one point on the plane if and only if\nA. $a=0$\nB. $a=1$\nC. $a=-1$\nD. $a \\in \\mathbf{R}$"
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"value": "<think>\n\nWe are given: The curve $... | [
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"value": "1. The curve $(x+2 y+a)\\left(x^{2}-y^{2}\\right)=0$ represents three straight lines intersecting at one point on the plane if and only if\nA. $a=0$\nB. $a=1$\nC. $a=-1$\nD. $a \\in \\mathbf{R}$"
},
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"value": "<think>We are given: The curve $(x+2... | [
"We are given: The curve $(x+2 y+a)\\left(x^{2}-y^{2}\\right)=0$ represents three straight lines intersecting at one point on the plane if and only if",
"Options: A. a=0, B. a=1, C. a=-1, D. a∈R",
"We need to determine for which value(s) of a the equation factorizes into three lines that all intersect at a sing... | [
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"value": "9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?"
},
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"value": "<think>\n\nWe need to read the problem: \"In the metro t... | [
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"from": "user",
"value": "9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?"
},
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"from": "assistant",
"value": "<think>We need to read the problem: \"In the metro train... | [
"We need to read the problem: \"In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?\"",
"Interpretation: There are 100 passengers who board at the initial stop. The train will make 16 subsequent s... | [
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"value": "2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).\n(A) the I quadrant\n(B) the II quadrant\n(C) the III quadrant\n(D) the IV quadra... | [
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"value": "2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).\n(A) the I quadrant\n(B) the II quadrant\n(C) the III quadrant\n(D) the IV quadra... | [
"We are given that there are 7 points corresponding to the 7 roots of the equation $x^7 = -1 + \\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ) which quadrant? So we need to determine in which quadrant exactly one of the 7 roots lies.",
"We have to solve $x^7 = -1 + ... | [
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"value": "Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? \n\n$ \\tex... | [
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"value": "Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? \n\n$ \\tex... | [
"We need to read the problem carefully. There are 36 students total. Preferences: chocolate = 12, apple = 8, blueberry = 6. So far we have accounted for 12+8+6 = 26 students. Remaining students = 36 - 26 = 10. Half of these (i.e., 5) prefer cherry, and half (5) prefer lemon. So cherry = 5, lemon = 5.",
"We need t... | [
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"value": "Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.\n\nHow many games did father play with uncle?\n\n(M. Volfová)... | [
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"value": "Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.\n\nHow many games did father play with uncle?\n\n(M. Volfová)... | [
"We need to solve the problem.",
"Let’s restate: There are two players: father and uncle. They play some number of games. For each game: if someone wins, winner receives 8 crowns from opponent. If it's a draw, no exchange. Uncle won four times. There were five draws. At the end, father earned 24 crowns. How many ... | [
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"value": "## Problem 4\n\nGiven the numbers $1,2,3, \\ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.\n\nSelected problems by Prof. C... | [
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"value": "## Problem 4\n\nGiven the numbers $1,2,3, \\ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.\n\nSelected problems by Prof. C... | [
"We need to solve: Given numbers 1,2,...,1000. Find the largest m such that for any removal of m numbers from the set, among the remaining (1000-m) numbers there exist two such that one divides the other. In other words, the complement of any m-subset is guaranteed to contain a divisible pair (a divides b, a<b). We... | [
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"value": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $x\\... | [
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"value": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $x\\... | [
"We are given a functional equation: f: ℝ → ℝ such that f(x+1) = 2 f(x) for all x ∈ ℝ. Also, we know f(x) = x(x-1) for all x ∈ (0,1]. We need to find the greatest real number m for which the inequality f(x) ≥ -8/9 holds for all x ∈ (-∞, m].",
"So essentially, we need to understand the behavior of f on all reals g... | [
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"value": "4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \\neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$."
... | [
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"value": "4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \\neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$."
... | [
"We are given: A and B are n-digit numbers (n is odd) which give the same remainder r ≠ 0 when divided by k. So A ≡ r (mod k), B ≡ r (mod k). We need to find at least one number k, which does not depend on n, such that the number C, obtained by appending the digits of A and B, is divisible by k. That is, if we writ... | [
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"value": "7. Given that $z$ is a complex number, and $|z|=1$. When $\\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\\qquad$ or . $\\qquad$"
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"value": "<think>\n\nWe are asked: \"Given that $z$ is a complex number, ... | [
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"value": "7. Given that $z$ is a complex number, and $|z|=1$. When $\\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\\qquad$ or . $\\qquad$"
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"value": "<think>We are asked: \"Given that $z$ is a complex number, and ... | [
"We are asked: \"Given that $z$ is a complex number, and $|z|=1$. When $\\mid 1+z+ 3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\\qquad$ or . $\\qquad$\". So we need to find z on the unit circle such that the absolute value of the polynomial P(z) = 1 + z + 3z^2 + z^3 + z^4 is minimized. ... | [
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"value": "Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \\sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\\prime}$ such that $B^{\\prime} C: C B$ $=1: 2... | [
"We have a problem about a frustum of a right circular cone. The radii: upper base radius? Wait: \"Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \\sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix... | [
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"value": "5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is"
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"value": "<think>\n\nWe are given: \"5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^... | [
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"value": "5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is"
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"value": "<think>We are given: \"5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}... | [
"We are given: \"5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is\". Probably: \"Given positive real numbers a and b satisfy a + b = 1, then M = sqrt(1 + a^2) + sqrt(1 + 2b). The integer part is?\" We need to find the integer part of M (the gr... | [
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"value": "1. Arrange the consecutive natural numbers from 1 to 99 in sequence to form a large number:\n1234567891011…979899,\nBy extracting four consecutive digits, you can get a four-digit number, such as 5678, 1011, etc. Among the four-digit numbers obtained by this method, the largest i... | [
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"value": "1. Arrange the consecutive natural numbers from 1 to 99 in sequence to form a large number:\n1234567891011…979899,\nBy extracting four consecutive digits, you can get a four-digit number, such as 5678, 1011, etc. Among the four-digit numbers obtained by this method, the largest i... | [
"We need to form a number by writing numbers 1 to 99 in order: 1 2 3 ... 10 11 ... 99. Then consider all possible windows of four consecutive digits (i.e., sliding a window of length 4 over the digit string). Among those windows, we get four-digit numbers (some may have leading zeros? Typically when we talk about a... | [
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"value": "1. Simplify\n$$\n\\sqrt{1+2 \\sin \\alpha \\cdot \\cos \\alpha}+\\sqrt{1-2 \\sin \\alpha \\cdot \\cos \\alpha}\n$$\n$\\left(0^{\\circ}<\\alpha \\leqslant 90^{\\circ}\\right)$ The result is $\\qquad$ ."
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"value": "<think>\n\nWe need to simplify... | [
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"value": "1. Simplify\n$$\n\\sqrt{1+2 \\sin \\alpha \\cdot \\cos \\alpha}+\\sqrt{1-2 \\sin \\alpha \\cdot \\cos \\alpha}\n$$\n$\\left(0^{\\circ}<\\alpha \\leqslant 90^{\\circ}\\right)$ The result is $\\qquad$ ."
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"value": "<think>We need to simplify the... | [
"We need to simplify the expression: √(1+2 sin α cos α) + √(1-2 sin α cos α) for 0° < α ≤ 90°. The result is something like 2 sin α or 2 cos α or something like that. Let's compute.",
"Recall that 1 ± 2 sin α cos α = sin^2 α + cos^2 α ± 2 sin α cos α = (sin α ± cos α)^2. Indeed, sin^2 α + cos^2 α = 1, and 2 sin α... | [
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"value": "8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1."
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"value": "<think>\n\nWe are asked: \"If n is a natural number less t... | [
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"value": "8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1."
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"value": "<think>We are asked: \"If n is a natural number less than ... | [
"We are asked: \"If n is a natural number less than 50, find all values of n such that the values of the algebraic expressions 4n+5 and 7n+6 have a common divisor greater than 1.\"",
"Interpretation: For a given n (natural number, i.e., positive integer, less than 50), consider the two numbers A = 4n+5 and B = 7n... | [
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"value": "6. Let set $A=\\{1,2,3,4,5,6\\}$, and a one-to-one mapping $f: A \\rightarrow A$ satisfies that for any $x \\in A$, $f(f(f(x)))$ $=x$. Then the number of mappings $f$ that satisfy the above condition is ( ).\n(A) 40\n(B) 41\n(C) 80\n(D) 81"
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"... | [
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"value": "6. Let set $A=\\{1,2,3,4,5,6\\}$, and a one-to-one mapping $f: A \\rightarrow A$ satisfies that for any $x \\in A$, $f(f(f(x)))$ $=x$. Then the number of mappings $f$ that satisfy the above condition is ( ).\n(A) 40\n(B) 41\n(C) 80\n(D) 81"
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"... | [
"We need to count number of permutations (bijections) f on set A = {1,...,6} such that f^3 = identity permutation. That is, each element is of order dividing 3, so cycles can be of lengths 1 or 3 only (since permutation order dividing 3; also cycles of length dividing 3; possible cycle lengths: 1 (fixed points) and... | [
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"value": "9. (16 points) Let the real number $t \\in [0, \\pi]$. If the equation $\\cos (x+t)=1-\\cos x$ has a solution for $x$, find the range of values for $t$."
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"value": "<think>\n\nWe need to find t in [0,π] such that the equation cos(x+t)=1-cos x ... | [
{
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"value": "9. (16 points) Let the real number $t \\in [0, \\pi]$. If the equation $\\cos (x+t)=1-\\cos x$ has a solution for $x$, find the range of values for $t$."
},
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"value": "<think>We need to find t in [0,π] such that the equation cos(x+t)=1-cos x has ... | [
"We need to find t in [0,π] such that the equation cos(x+t)=1-cos x has a solution for x (real x). So we need to determine for which t there exists x satisfying this.",
"We have cos(x+t) = 1 - cos x.",
"We can write left side as cos(x+t) = cos x cos t - sin x sin t.",
"So equation: cos x cos t - sin x sin t =... | [
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"value": "5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7."
},
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"value": "<think>\n\nWe need to find the largest natural number such that:\n\n- All digits are different (i.e., no digit ... | [
{
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"value": "5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7."
},
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"value": "<think>We need to find the largest natural number such that:\n\n- All digits are different (i.e., no digit repe... | [
"We need to find the largest natural number such that:",
"- All digits are different (i.e., no digit repeats).\n- Any two adjacent digits differ by either 6 or 7.",
"We need the largest possible natural number (presumably in decimal) satisfying these constraints.",
"Interpretation: \"largest natural number\" ... | [
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"value": "5. For a convex $n$-sided polygon, if circles are constructed with each side as the diameter, the convex $n$-sided polygon must be covered by these $n$ circles. Then the maximum value of $n$ is: ( ).\n(A) 3 .\n(B) 4 .\n(C) 5 .\n(D) Greater than 5 ."
},
{
"from": "assistan... | [
{
"from": "user",
"value": "5. For a convex $n$-sided polygon, if circles are constructed with each side as the diameter, the convex $n$-sided polygon must be covered by these $n$ circles. Then the maximum value of $n$ is: ( ).\n(A) 3 .\n(B) 4 .\n(C) 5 .\n(D) Greater than 5 ."
},
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"from": "assistan... | [
"We are given: \"For a convex n-sided polygon, if circles are constructed with each side as the diameter, the convex n-sided polygon must be covered by these n circles. Then the maximum value of n is: ( ). (A) 3. (B) 4. (C) 5. (D) Greater than 5.\"",
"Interpretation: We have a convex polygon with n sides. For eac... | [
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"value": "15. Lucia and Carla are playing bingo against each other (with no other opponents). Each has a card with 15 numbers; the two cards have exactly 4 numbers in common. What is the probability that, after 89 numbers have been drawn, neither of them has yet completed a bingo?\n(A) $2 ... | [
{
"from": "user",
"value": "15. Lucia and Carla are playing bingo against each other (with no other opponents). Each has a card with 15 numbers; the two cards have exactly 4 numbers in common. What is the probability that, after 89 numbers have been drawn, neither of them has yet completed a bingo?\n(A) $2 ... | [
"We are told: Lucia and Carla are playing bingo against each other (with no other opponents). Each has a card with 15 numbers; the two cards have exactly 4 numbers in common. We need: probability that after 89 numbers have been drawn, neither of them has yet completed a bingo.",
"Interpretation: Bingo is a game w... | [
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"value": "## Task 2 - 080522\n\nIn a warehouse, there are three times as many kilograms of wheat as in a second one. After removing $85000 \\mathrm{~kg}$ from the first and $5000 \\mathrm{~kg}$ from the second, the stocks were equal.\n\nHow many tons of wheat were in the first and how many... | [
{
"from": "user",
"value": "## Task 2 - 080522\n\nIn a warehouse, there are three times as many kilograms of wheat as in a second one. After removing $85000 \\mathrm{~kg}$ from the first and $5000 \\mathrm{~kg}$ from the second, the stocks were equal.\n\nHow many tons of wheat were in the first and how many... | [
"We are given: \"In a warehouse, there are three times as many kilograms of wheat as in a second one. After removing 85000 kg from the first and 5000 kg from the second, the stocks were equal. How many tons of wheat were in the first and how many in the second warehouse before the removal?\" So we need to find init... | [
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"value": "$17 \\cdot 27$ As shown in the figure, in $\\triangle A B C$, $\\angle A=42^{\\circ}$. If the trisectors of $\\angle B$ and $\\angle C$ intersect at $D, E$. Then the degree measure of $\\angle B D C$ is\n(A) $67^{\\circ}$.\n(B) $84^{\\circ}$.\n(C) $88^{\\circ}$.\n(D) $110^{\\circ... | [
{
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"value": "$17 \\cdot 27$ As shown in the figure, in $\\triangle A B C$, $\\angle A=42^{\\circ}$. If the trisectors of $\\angle B$ and $\\angle C$ intersect at $D, E$. Then the degree measure of $\\angle B D C$ is\n(A) $67^{\\circ}$.\n(B) $84^{\\circ}$.\n(C) $88^{\\circ}$.\n(D) $110^{\\circ... | [
"We are given a triangle ABC with angle A = 42°. The trisectors of angle B and angle C intersect at D and E? The phrasing: \"If the trisectors of ∠B and ∠C intersect at D, E. Then the degree measure of ∠BDC is ...\" Wait, read carefully: \"As shown in the figure, in triangle ABC, angle A = 42°. If the trisectors of... | [
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"value": "Example 5. Expand the function\n\n$$\nf(z)=\\frac{z}{z^{2}-2 z-3}\n$$\n\ninto a Taylor series in the neighborhood of the point $z_{0}=0$ using expansion (12), and find the radius of convergence of the series."
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"value": "<think>\n\nWe need to ... | [
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"value": "Example 5. Expand the function\n\n$$\nf(z)=\\frac{z}{z^{2}-2 z-3}\n$$\n\ninto a Taylor series in the neighborhood of the point $z_{0}=0$ using expansion (12), and find the radius of convergence of the series."
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"value": "<think>We need to expa... | [
"We need to expand f(z) = z/(z^2 - 2z - 3) into Taylor series about z0 = 0. Use expansion (12) presumably referencing some known series? Probably they want to do partial fractions and use geometric series.",
"First, factor denominator: z^2 - 2z - 3 = (z - 3)(z + 1). So f(z) = z / ((z-3)(z+1)). We can decompose in... | [
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"value": "7. The set \n$\\left\\{x \\left\\lvert\\,-1 \\leqslant \\log _{\\frac{1}{x}} 10<-\\frac{1}{2}\\right., x \\in \\mathbf{N}\\right\\}$ has $\\qquad$ proper subsets."
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"value": "<think>\n\nWe need to solve the inequality: -1 ≤ log_{1/x}(10) < -1/... | [
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"value": "7. The set \n$\\left\\{x \\left\\lvert\\,-1 \\leqslant \\log _{\\frac{1}{x}} 10<-\\frac{1}{2}\\right., x \\in \\mathbf{N}\\right\\}$ has $\\qquad$ proper subsets."
},
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"value": "<think>We need to solve the inequality: -1 ≤ log_{1/x}(10) < -1/2, w... | [
"We need to solve the inequality: -1 ≤ log_{1/x}(10) < -1/2, with x natural number (positive integer, presumably N = {1,2,3,...} but careful: base of logarithm is 1/x, and we require base positive and not equal to 1. So 1/x > 0, and 1/x ≠ 1 => x > 0 and x ≠ 1. Since x ∈ N (natural numbers), likely N = {1,2,3,...}. ... | [
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"value": "2. Given $0<x<1$. Simplify\n$$\n=\\quad \\sqrt{\\left(x-\\frac{1}{x}\\right)^{2}+4}-\\sqrt{\\left(x+\\frac{1}{x}\\right)^{2}-4}\n$$"
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"value": "<think>\n\nWe are given: $0 < x < 1$. Simplify\n\n$$\nE = \\sqrt{\\left(x - \\frac{1}{x}\\right)^2 ... | [
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"value": "2. Given $0<x<1$. Simplify\n$$\n=\\quad \\sqrt{\\left(x-\\frac{1}{x}\\right)^{2}+4}-\\sqrt{\\left(x+\\frac{1}{x}\\right)^{2}-4}\n$$"
},
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"value": "<think>We are given: $0 < x < 1$. Simplify\n\n$$\nE = \\sqrt{\\left(x - \\frac{1}{x}\\right)^2 + 4}... | [
"We are given: $0 < x < 1$. Simplify",
"$$\nE = \\sqrt{\\left(x - \\frac{1}{x}\\right)^2 + 4} - \\sqrt{\\left(x + \\frac{1}{x}\\right)^2 - 4}.\n$$",
"We need to simplify this expression.",
"First, note that $\\left(x - \\frac{1}{x}\\right)^2 + 4 = x^2 - 2 + \\frac{1}{x^2} + 4 = x^2 + 2 + \\frac{1}{x^2} = \\le... | [
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"value": "Find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are not equal, but the difference between the larger and the smaller of them is as small as possible.\n\n#"
},
{
"from": "assistant",
"value": "<think>\n\nW... | [
{
"from": "user",
"value": "Find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are not equal, but the difference between the larger and the smaller of them is as small as possible.\n\n#"
},
{
"from": "assistant",
"value": "<think>We ne... | [
"We need to find two fractions: one with denominator 8, i.e., a/8 where a is integer? The problem says \"common fractions\" typically meaning fractions in simplest form? But not necessarily? It says \"find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are... | [
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"value": "1. As shown in Figure 1, in the Cartesian coordinate system, the graph of the quadratic function $y=a x^{2}+m c(a \\neq$ $0)$ passes through three vertices $A$, $B$, and $C$ of the square $A B O C$, and $a c=-2$. Then the value of $m$ is ( ).\n(A) 1\n(B) -1\n(C) 2\n(D) -2"
},
... | [
{
"from": "user",
"value": "1. As shown in Figure 1, in the Cartesian coordinate system, the graph of the quadratic function $y=a x^{2}+m c(a \\neq$ $0)$ passes through three vertices $A$, $B$, and $C$ of the square $A B O C$, and $a c=-2$. Then the value of $m$ is ( ).\n(A) 1\n(B) -1\n(C) 2\n(D) -2"
},
... | [
"We are given a problem: \"As shown in Figure 1, in the Cartesian coordinate system, the graph of the quadratic function $y = a x^2 + m c (a \\neq 0)$ passes through three vertices $A$, $B$, and $C$ of the square $A B O C$, and $a c = -2$. Then the value of $m$ is ( ).\" Options: 1, -1, 2, -2.",
"We need to inter... | [
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