Nemotron Labs IMO 2026
Collection
Checkpoints, training data and benchmark from 'An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics' (IMO 2026). • 6 items • Updated • 5
id string | problem string | proof string | subject string |
|---|---|---|---|
A1 | Prove that for all positive integers $n$
$$\sum_{k=0}^n \frac{2^{2^k+k+1}}{2^{2^k}+3^{2^k}}<4.$$ | Let $x_k=(3/2)^{2^k}-1$, so that $$x_k(x_k+2)=(x_k+1)^2-1=x_{k+1}.$$
It follows that
$$\frac{2^{2^k+k+1}}{2^{2^k}+3^{2^k}}=\frac{2^{k+1}}{(3/2)^{2^k}+1}=\frac{2^{k+1}}{x_k+2}=$$
$$\frac{2^{k+1}}{x_k}-\frac{2^{k+2}}{x_k(x_k+2)}=\frac{2^{k+1}}{x_k}-\frac{2^{k+2}}{x_{k+1}}.$$
Thus
$$\sum_{k=0}^n \frac{2^{2^k+k+1}}{2... | algebra |
A2 | Let $m,n\geq 2$ be integers and let $f\in \mathbb{R}[X_1,...,X_n]$ be a polynomial such that
$$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor,\,\, \quad \forall\,\, x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$
Prove that the total degree of $f$ is at least $n$. | We start by considering the Lagrange interpolation polynomial $g$ for which $g(x)=\lfloor x/m\rfloor$ for $x=0,1,...,n(m-1)$, i.e. the unique polynomial of degree not exceeding $n(m-1)$ satisfying the previous equalities. Thus $f(x_1,...,x_n)=g(x_1+...+x_n)$ for
$x_1,...,x_n\in \{0,1,...,m-1\}$. Since $g(0)=0=g(1)$... | algebra |
A3 | Prove that for all positive real numbers $a,b,c,d$
$$\frac{a+b}{1+c}\cdot \frac{b+c}{1+d}\cdot \frac{c+d}{1+a}\cdot \frac{d+a}{1+b}\geq \frac{16abcd}{(1+\sqrt[4]{abcd})^4}.$$ | The key observation is that
$$\frac{x+y}{(1+x)(1+y)}\geq \frac{2\sqrt{xy}}{(1+\sqrt{xy})^2}$$
for all $x,y>0$. Indeed, writing this in the form
$$\frac{x+y}{2\sqrt{xy}}-1\geq \frac{1+x+y+xy}{(1+\sqrt{xy})^2}-1$$
it becomes equivalent to
$$\frac{(\sqrt{x}-\sqrt{y})^2}{2\sqrt{xy}}\geq \frac{(\sqrt{x}-\sqrt{y})^2}{... | algebra |
A4 | Let $n>1$ be an integer. Two polynomials $P,Q\in \mathbb{R}[X]$ are called block-similar if for each $i \in \{1, 2, \ldots, n\}$ the sequences
$$P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) \,\, \text{and}\,\,
Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014)$$
are permutations of each other.
(a) Prove that there ex... | For simplicity set $k=2015$. Part a) is fairly easy, since one easily checks that setting $$P(X)=X(X-k)(X-2k)...(X-nk),\,\, Q(X)=P(X-1)$$
yields distinct block-similar polynomials of degree $n+1$. Part b) is much harder and will require some preparation. Suppose that
$P\ne Q$ are block-similar, of degree $n$. If $... | algebra |
A5 | Find all periodic sequences $a_1,a_2,...$ of real numbers such that
$|a_{n+1}-a_n|\leq 1$ and $$a_{n+2}+a_n^2=a_n+a_{n+1}^2$$
for all $n\geq 1$. | Let $x_n=a_n+a_{n+1}$, so that the recurrence relation becomes $$x_{n+1}=x_n(a_{n+1}-a_n+1).$$
We will discuss two cases. If there is some $k$ with $x_k=0$ then $x_n=0$ for $n\geq k$ and since
$(x_n)$ is periodic (because $(a_n)$ is so), it follows that $x_n=0$ for all $n$ and so the sequence $(a_n)$ is of the form ... | algebra |
A6 | Prove that for any polynomial $f\in \mathbb{R}[X]$ of degree greater than $1$ there are infinitely many positive integers $m$ for which the equation $$f(n+1)+f(n+2)+...+f(n+k)=m$$
has no solutions in positive integers $n,k$. | Let $d=\deg(f)\geq 2$ and let $a$ be the leading coefficient of $f$. If $a<0$ there is $N$ such that $f(x)<0$ for $x>N$. If
$m,n,k$ are positive integers with $f(n+1)+...+f(n+k)=m$ then $n<N$ and $m\leq f(n+1)+...+f(N)$, thus we are done. Thus we may assume that $a>0$. Choose $C>0$ such that $f(x)\geq \frac{a}{2}x^d$... | algebra |
A7 | Find the maximal value of
\[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. | We will prove that \[ S \le \frac{8}{\sqrt[3]{7}} \]. Note that we have equality for $(a,b,c,d)=(49, 1, 49, 1)$.
The shortest and trickiest proof is to observe that \[\sqrt[3]{\frac{7a}{64(b+7)}}\leq \frac{1}{3}\left(\frac{a+7}{64}+\frac{a}{a+7}+\frac{7}{b+7}\right)\]
Writing down similar inequalities for the other ter... | algebra |
A8 | Find the smallest real number $C>1$ with the following property: for any integer $n>1$ and any non-integers
$a_1,...,a_n>0$ with $\sum_{i=1}^n 1/a_i=1$ there are positive integers $b_i\in \{\lfloor a_i\rfloor, \lfloor a_i\rfloor+1\}$ such that
$1<\sum_{i=1}^n 1/b_i\leq C$. | Take a large positive integer $n$ and set $a_1=2-\frac{1}{2n-1}$ and $a_i=2n-1-\frac{1}{2}$ for $2\leq i\leq n$. We must have
$b_1\in \{1,2\}$, $b_i\in \{2n-2, 2n-1\}$ for $2\leq i\leq n$. We can't have $b_1=2$, since otherwise $\sum_{i=1}^n 1/b_i$ would not exceed
$1/2+\frac{n-1}{2n-2}=1$, thus $b_1=1$ and
$$C\g... | algebra |
A9 | Find all functions $f: \{1,2,...\}\to \mathbb{Z}$ such that for all positive integers $m,n$ we have
$$\lfloor f(mn)/n\rfloor=f(m).$$ | One easily checks (using the identity $\lfloor \frac{\lfloor x\rfloor}{n}\rfloor=\lfloor \frac{x}{n}\rfloor$) that all functions of the form
$f(n)=\lfloor nx\rfloor$ or $f(n)=\lceil nx\rceil-1$ are solutions of the problem. We will prove that these are all solutions. By assumption
$$\lfloor f(n!(n+1))/(n+1)\rfloo... | algebra |
A10 | Find all functions $f: (1,\infty)\to (1,\infty)$ such that whenever $x,y>1$ satisfy $x^2\leq y\leq x^3$ we have
$f(x)^2\leq f(y)\leq f(x)^3$. | It is clear that for all $k>0$ the function $f(x)=x^k$ is a solution, and we will prove that these are the only solutions.
Define $g: (0,\infty)\to (0,\infty)$ by $g(x)=\ln f(e^x)$. Then $2x\leq y\leq 3x$ (with $x,y>0$) forces $2g(x)\leq g(y)\leq 3g(x)$.
In particular $g(2x)\geq 2g(x)$ and $g(3x)\leq 3g(x)$ for all ... | algebra |
A11 | Is there a sequence of pairs of real numbers $(x_n, y_n)_{n\geq 1}$ such that for any sequence
$(b_n)_{n\geq 1}$ of real numbers there is a polynomial $f\in \mathbb{R}[X,Y]$ with $f(x_n, y_n)=b_n$ for all $n$? | Suppose that such a sequence exists and note that it is unbounded, for otherwise the sequence $(f(x_n, y_n))_{n\geq 1}$ would be bounded for any $f\in \mathbb{R}[X,Y]$. Let $z_n=1+\max(|x_n|, |y_n|)$. By assumption there is $f\in \mathbb{R}[X,Y]$ with
$f(x_n, y_n)=2^{z_n}$ for all $n$. Let $d=\deg(f)$ and observe tha... | algebra |
A12 | Find all functions $f: \mathbb{Q}[X]\to \mathbb{Q}[X]$ such that
$f(P)+f(Q)=f(P+Q)$ for all $P,Q\in \mathbb{Q}[X]$ and such that
$\gcd(P, f(P))=1$ if and only if $P$ is squarefree. | It is easy to see that any function of the form $f(P)=AP+bP'$ with $b\in \mathbb{Q}$ nonzero and
$A\in \mathbb{Q}[X]$ is a solution of the problem. We will prove that these are the only solutions. Let
$f$ be a solution and note that $f$ is actually $\mathbb{Q}$-linear. Let $P_n=f(X^n)\in \mathbb{Q}[X]$. By line... | algebra |
A13 | Prove that for all real numbers $x,y,z$
$$(x+y+z)^2+\sum \frac{(y+x)(y+z)}{1+|x-z|}\geq xy+yz+zx.$$ | With the substitutions $y+z=2a, z+x=2b, x+y=2c$ the inequality is equivalent to
$$a^2+b^2+c^2+2\sum \frac{ab}{1+2|a-b|}\geq 0$$
for all real numbers $a,b,c$. We will actually prove that
the second sum is already nonnegative. Renaming $2a, 2b, 2c$ into $a,b,c$, it suffices to check that
$$\sum \frac{ab}{1... | algebra |
A14 | Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, \[\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.\] | We use AM-GM in a slightly tricky way: for $i<j$ we have
$$a_i^2 \cdot \frac{1}{2^{j-i-1}}+a_j^2 \cdot \frac{1}{2^{i-j}}\ge 2a_ia_j\cdot \sqrt{2}.$$
Adding these inequalities yields
$$\frac{1}{2\sqrt{2}}\cdot \sum_{i=1}^{n} a_i^2\cdot \sum_{j=1}^{n-1} \frac{1}{2^{j-i}}\ge \sum_{i<j} a_ia_j.$$
Since $\sum_{j=1}^{n-1} ... | algebra |
A15 | The first derivative of an infinite sequence $(a_n)_{n\geq 1}$ is the sequence $(a'_n)_{n\geq 1}$ with $a'_n=a_{n+1}-a_n$ for $n\geq 1$. Define the $k$th derivative as the first derivative of the $(k-1)$th derivative for $k\geq 2$. Say the sequence $(a_n)_{n\geq 1}$ is good if the sequence itself and all of its derivat... | Write $a^{(k)}$ for the $k$th derivative of the sequence $a$ (with the convention that $a^{(0)}=a$). Since
$$a_{i+1}b_{i+1}-a_ib_i=(a_{i+1}-a_i)(b_{i+1}-b_i)+a_i(b_{i+1}-b_i)+b_i(a_{i+1}-a_i),$$
we have $(ab)'=a'b'+a'b+ab'$ for any sequences $a,b$. It is also clear that taking derivatives is a linear operations.
... | algebra |
A16 | Prove that for all positive real numbers $a,b,c$ $$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}+\frac{45}{a+b+c}\geq \frac{7}{a}+\frac{7}{b}+\frac{7}{c}.$$ | We may assume that $a+b+c=1$. The inequality is equivalent to
$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+45\geq \frac{8}{a}+\frac{8}{b}+\frac{8}{c},$$
or $$\sum a^2b^2-abc+abc(1-3\sum ab)\geq 5abc(\sum ab-9abc).$$ Using the identities
$$\sum a^2b^2-abc=\sum a^2b^2-abc(a+b+c)=\sum \frac{a^2}{2}(b-c)^2,$$
$$1-3\sum ab=... | algebra |
A17 | Let $n\geq 2$, and let $a_1,...,a_n$ be positive real numbers such that there are real numbers $c,d$ and $b_1,...,b_n$ with $$\sum_{i=1}^n \lfloor a_ix+b_i\rfloor=\lfloor cx+d\rfloor$$ for all $x\in \mathbb{R}$. Prove that $a_1,...,a_n$ are not pairwise distinct. | Since the left-hand side is unbounded, we must have $c\ne 0$. Replacing $x$ by $\frac{x-d}{c}$ and the
$a_i$'s by $a_i/c$ (and $b_i$ by $b_i-a_id/c$), we may assume that $c=1$ and $d=0$.
Let $f_i(x)=\lfloor a_ix+b_i\rfloor$. Then $f_i$ is non-degreasing and increases by $1$ if $x$ belongs to an arithmetic progression... | algebra |
A18 | Find the largest real number $k$ such that for all positive real numbers $a,b,c,d$ with $ab+bc+cd+da=1$ $$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}-4\geq k(a^2+b^2+c^2+d^2-4).$$ | If $k$ has this property, by taking $b=d=x$, $c=1$ and $a=(2-x)/x$ with $x>0$ very small yields $k\leq 1/2$. Next, we show that
$k=1/2$ has the required property. Let $a,b,c,d$ be as in the statement and write $$a+c=2x,\, \, b+d=2y,$$ where without loss of generality
$x\ge y$. Let $$p=ac\,\, \text{and}\,\, q=bd$$ and ... | algebra |
A19 | Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be sequences of real numbers for which $a_1 > b_1$ and $$a_{n+1}= a_n^2 - 2b_n,\,\, \quad b_{n+1}= b_n^2 - 2a_n$$ for all positive integers $n$. Prove that $a_1, a_2, \dots$ is eventually increasing, i.e. there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $... | Choose complex numbers $x,y,z$ such that $$xyz=1, \quad a_1=x+y+z,\quad b_1=1/x+1/y+1/z,$$ namely the roots of the polynomial
$$p(t)=t^3-a_1t^2+b_1t-1.$$ A simple induction then shows that $$a_n=x^{2^{n-1}}+y^{2^{n-1}}+z^{2^{n-1}}$$ and
$$b_n=x^{-2^{n-1}}+y^{-2^{n-1}}+z^{-2^{n-1}}$$ for all $n$. Note that since $p(1)... | algebra |
A20 | Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ for all but finitely many $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Big... | Set
$$\|f\|_{\infty} = \max_{i\in \mathbb{Z}}|f(i)|$$
and for $p<\infty$ set
$$\|f\|_{p} = \left( \sum_{i\in \mathbb{Z}} |f(i)|^p \right)^{\frac{1}{p}}.$$
A first key remark is that
$$\sum_{n\in\mathbb{Z}}h(n) \geq \|f\|_{\infty} \|g\|_1\,\, \text{and}\,\, \sum_{n\in\mathbb{Z}}h(n) \geq \|g\|_{\infty} \|1\|_1.$$
Ind... | algebra |
A21 | Let $P,Q\in \mathbb{R}[X]$ be polynomials of degree not exceeding $n$ such that $$P(X)X^{n+1}+Q(X)(X+1)^{n+1}=1.$$ What are the possible values of $Q(-1/2)$? | Note that such $P,Q$ do exist since $X^{n+1}$ and $(X+1)^{n+1}$ are relatively prime. Moreover, they are unique, for if $P_1,Q_1$ is another such pair then $$X^{n+1}(P-P_1)=(X+1)^{n+1}(Q_1-Q),$$ thus
$X^{n+1}$ divides $Q_1-Q$ and for degree reasons we must have $Q_1=Q$ and then $P_1=P$. Next, replacing $X$ by $-1-X$ y... | algebra |
A22 | Let $\{x\}$ denote the fractional part of $x$. If $\alpha$ is a real number such that there are only finitely many distinct numbers in the sequence $\{\alpha\}, \{\alpha^2\}, \{\alpha^3\}, \cdots $, prove that $\alpha$ is an integer. | Let $x_n=\{\alpha^n\}$. Since there are only finitely many distinct terms in the sequence $x_1,x_2,...$, there are only finitely many distinct pairs of numbers of the form $(x_n, x_{n+1})$, thus we can find $i<j$ with $$x_i=x_j\,\, \text{and}\,\, x_{i+1}=x_{j+1}.$$ It follows that $$\alpha^{j}-\alpha^i\,\,\text{and}\,\... | algebra |
A23 | Find all sets $A,B\subset\mathbb{R}$ with the following property: if $f: A\times B\to \mathbb{R}$ is a function such that $b\mapsto f(a,b)$ is a polynomial function on $B$ for all $a\in A$ and $a\mapsto f(a,b)$ is a polynomial function on $A$ for all $b\in B$, then $f$ is a polynomial function. | The answer is: at least one of $A,B$ is finite or at least one of $A,B$ is uncountable. First assume that $A$ is finite, say
$A=\{a_1,...,a_n\}$. By assumption there are polynomials $P_i$ such that $f(a_i, b)=P_i(b)$ for $b\in B$. By Lagrange interpolation we can pick polynomials
$Q_j$ such that $Q_j(a_i)=1_{i=j}$. T... | algebra |
A24 | Evaluate the product $$\sin \frac{\pi}{42} \sin \frac{5\pi}{42}\sin \frac{13\pi}{42}\sin \frac{17\pi}{42}\sin \frac{19\pi}{42}\sin \frac{31\pi}{42}.$$ | Let $P$ be the given product and note that
$$P=\cos \frac{\pi}{21} \cos \frac{2\pi}{21}\cos \frac{4\pi}{21}\cos \frac{5\pi}{21}\cos \frac{8\pi}{21}\cos \frac{10\pi}{21},$$
since we easily see that
$$\sin \frac{\pi}{42}=\cos \frac{10\pi}{21},\, \sin \frac{5\pi}{42}=\cos \frac{8\pi}{21},\, \sin \frac{13\pi}{42}=\cos \... | algebra |
A25 | Let $n$ be a positive integer. Find the largest positive integer $K_n$ for which there is a set $S=\{z_1,z_2,\dots,z_n\}$ of $n$ complex numbers, stable under complex conjugation, not containing $0$ and such that $z_1^k+...+z_n^k\leq 0$ for all $1\leq k\leq K_n$. | The answer is $K_n=2n-1$. It is not difficult to see that the set $S$ of solutions of the equation $z^n=-1$ has the desired properties for $K_n=2n-1$. Suppose we can find a set $S$ as in the problem such that $p_k\leq 0$ for $1\leq k\leq 2n$, where
$p_k=z_1^k+...+z_n^k$. Note that since $S$ is stable under complex con... | algebra |
A26 | Let $n$ be an even positive integer and let $z$ be a complex number such that $z^{2^{n+1}-1}=1$. Evaluate $\prod_{i=0}^n ({\rm Re}(z^{2^i})-\frac{1}{2})$. | We will prove that the desired product $P$ is always equal to $\frac{1}{2^{n+1}}$.
This is obvious if $z=1$, so assume that this is not the case.
Observe that
$$2^{n+1}(z+z^{-1}+1)P=(z+z^{-1}+1)\prod_{i=0}^n (z^{2^i}+z^{-2^i}-1)$$
$$=(z^2+z^{-2}+1)\prod_{i=1}^n (z^{2^i}+z^{-2^i}-1)=(z^4+z^{-4}+1)\prod_{i=2}^n (z^{2^i... | algebra |
A27 | Let $n\geq 4$ and let $a_1\geq a_2\geq\ldots\geq a_n$ be real numbers such that $a_1=a_2$ and $a_{n-1}=a_n$. Prove that $$n\sum_{i=1}^n a_ia_{i+1}\geq (\sum_{i=1}^n a_i)^2,$$ where $a_{n+1}=a_1$. | We may assume that $n>4$, since for $n=4$ we actually always have equality. Let
$$x=\frac{a_2+a_{n-1}}{2},\,\, y=\frac{a_3+\ldots+a_{n-2}}{n-4}.$$
Since $a_1=a_2$ and $a_{n-1}=a_n$, the inequality is equivalent to
$$n(4x^2-a_2a_{n-1}+a_2a_3+\ldots+a_{n-2}a_{n-1})\geq (4x+(n-4)y)^2.$$
Chebyshev's inequality yields
$$a... | algebra |
A28 | Let $n$ be a positive integer and let $a\in (0,1)$. What is the minimal value of $x_0^2+...+x_n^2$ among all tuples $(x_0,...,x_n)$ of positive real numbers with sum $n+a$ and such that $\sum_{i=0}^n \frac{1}{x_i}=n+\frac{1}{a}$? | We first prove that $x_j\in [a, 1/a]$ for all $j$. By symmetry we may assume that $j=0$. Note that
$$(n+a-x_0)(n+\frac{1}{a}-x_0)=(x_1+...+x_n)(1/x_1+...+1/x_n)\geq n^2,$$
by Cauchy-Schwarz. Expanding and simplifying yields
$$n(a+1/a)+2\geq n(x_0+1/x_0)+a/x_0+x_0/a\geq n(x_0+1/x_0)+2,$$
which immediately yields $x_0\i... | algebra |
A29 | Let $P\in \mathbb{Q}[X]$ be a nonconstant irreducible polynomial of degree $n$. Prove that there are at most $n$ polynomials $Q\in \mathbb{Q}[X]$ of degree less than $n$ such that $P(X)$ divides $P(Q(X))$. | Since $P$ is nonconstant and irreducible, its roots $z_1,...,z_n$ are pairwise distinct. If $P(X)$ divides
$P(Q(X))$ then $P(Q(z_1))=0$, thus $Q(z_1)=z_i$ for some $i$. Suppose that $Q_1,...,Q_{n+1}$ have rational coefficients, degree less than $n$ and $P(X)$ divides $P(Q_i(X))$. Let $Q_i(z_1)=z_{k_i}$ for $1\leq i\le... | algebra |
A30 | Let $m,n$ be nonnegative integers and let $a_0,\ldots, a_m, b_0,\ldots, b_n$ nonnegative real numbers. Let $c_k:=\max_{i+j=k}a_ib_j.$ Prove that $$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$ | We will induct on $m+n$, the cases $m=0$ or $n=0$ being immediate. We may assume that $$\sum_{i=0}^m a_i=\sum_{j=0}^n b_j=1,$$ by homogeneity. Note that if
$\sum_{i=0}^m a_i=0$ or $\sum_{j=0}^n b_j=0$ the inequality is clear. Applying the induction hypothesis to $a_1,...,a_m$ and $b_0,...,b_n$ we obtain
$$\frac{1}{m+... | algebra |
A31 | Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$, as well as all values of $(a,b,c,d)$ realizing the minimum. | We will prove that the minimum is $-1/8$, achieved for the tuple
$$(a,b,c,d)=(\frac{1+\sqrt{3}}{4}, -\frac{1+\sqrt{3}}{4}, \frac{1-\sqrt{3}}{4}, -\frac{1-\sqrt{3}}{4}),$$
the tuples obtained by cyclic shifting as well the tuples obtained from these four ones by performing the operation
$(a,b,c,d)\mapsto (-a,-b,-c,-d)... | algebra |
A32 | Prove that if real numbers $a,b,c,d$ add up to $2$, then
$$ \frac{a}{a^2-a+1}+ \frac{b}{b^2-b+1}+ \frac{c}{c^2-c+1}+ \frac{d}{d^2-d+1}\leq \frac{8}{3}.$$ | Let $$x=a-1/2,\,\, y=b-1/2, \,\, z=c-1/2, \,\, t=d-1/2,$$ so that $$x+y+z+t=0.$$ We need to prove that
$$\sum \frac{x+2}{4x^2+3}\leq \frac{8}{3}.$$
This is equivalent to
$$\sum \frac{(2x-1)^2}{4x^2+3}\geq \frac{4}{3}.$$
Since
$$4x^2=3x^2+(y+z+t)^2\leq 3(x^2+y^2+z^2+t^2),$$
we have
$$\sum \frac{(2x-1)^2}{4x^2+3}\geq ... | algebra |
A33 | Let $\mathbf{Z}_{>0}$ be the set of positive integers. Find all functions $f:\mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(m)\geq m$ and $$f(m+n) \mid f(m)+f(n)$$ for all $m,n\in \mathbb{Z}_{>0}$. | We will prove below that $L:=\lim_{n\to\infty} f(n)/n$ exists, is finite and satisfies $f(n)\geq nL$ for all
$n$. Pick $k$ such that $|f(n)-nL|<n/2$ for $n\geq k$. If $m,n\geq k$ then $f(m+n), f(m), f(n)$ are at distance less than
$\frac{m+n}{2}, \frac{m}{2}, \frac{n}{2}$ from $L(m+n), Lm, Ln$ respectively. Thus
$$|f... | algebra |
A34 | A sequence $(x_n)_{n\geq 1}$ of real numbers has the property that $x_1>0$ and that for all $n\geq 1$
$$x_{n+1}\geq (n+2)x_n-\sum_{k=1}^{n-1} kx_k.$$
Prove that $x_n\geq n!x_1$ for all $n\geq 1$. | We prove first by induction that $$x_{n+1}>\sum_{k=1}^n kx_k$$ for all $n$. This is trivial for $n=1$ since $x_2\geq 3x_1$ by assumption. Assume that it holds for $n$, then
$$x_{n+2}\geq (n+3)x_{n+1}-\sum_{k=1}^n kx_k=(n+1)x_{n+1}+2x_{n+1}-\sum_{k=1}^n kx_k>$$
$$>(n+1)x_{n+1}+2\sum_{k=1}^n kx_k-\sum_{k=1}^n kx_k=\sum_... | algebra |
A35 | Determine all polynomials $P$ with real coefficients satisfying the following condition: whenever $x$ and $y$ are real numbers such that $P(x)$ and $P(y)$ are both rational, so is $P(x + y)$. | The answer is: constant polynomials and linear polynomials with rational constant terms. It is clear that these are solutions, and it is easy to see that among polynomials of degree at most one these are the only solutions. Suppose we found a solution $P$ of degree $d\geq 2$. Since $P$ is not constant, there is
some n... | algebra |
A36 | Let $n>1$ be an integer. Find the largest positive integer $k$ such that for all
real numbers $a_1\geq a_2\geq\ldots\geq a_n\geq 0$ $$\left(\frac{a_1+\ldots+a_k}{k}\right)^2\geq \frac{a_1^2+\ldots+a_n^2}{n}.$$ | Taking $a_1=1$ and $a_2=\ldots=a_n=0$ we obtain $k\leq \lfloor \sqrt{n}\rfloor$. We will prove that
the inequality holds for $k=\lfloor \sqrt{n}\rfloor$, thus the answer is $\lfloor \sqrt{n}\rfloor$. Write
$n=k^2+i$ with $0\leq i\leq 2k$ and rewrite the inequality in the form
$$(k^2+i)(a_1+\ldots+a_k)^2\geq k^2 \sum_... | algebra |
A37 | Let $n$ be an integer greater than $1$. Find the minimum value of $$\left( \sum_{i=1}^n x_i^2\right) \left( \sum_{i=1}^n x_i^{-2} \right)$$ over all families of nonzero real numbers $x_1,...,x_n$ for which $x_1+...+x_n=0$. | By symmetry, we may assume that $x_1,...,x_k$ are positive and $x_{k+1},...,x_n$ are negative, for some $1\leq k<n$. Let $m=n-k$ and $y_i=-x_{i+k}$ for $1\leq i\leq n-k$, so that $y_1,...,y_m$ are positive and $S:=x_1+...+x_k=y_1+...+y_m$. Note that if $z_1,...,z_c$ are positive then
$$\sum_{i=1}^c \frac{1}{z_i^2}\geq... | algebra |
A38 | Let $f: [0,1]\to \mathbf{R}$ be a non-decreasing function. What is the maximal possible value of the expression
$$\sum_{k=1}^n f(|x_k-\frac{2k-1}{2n}|)$$ over all possible $n$-tuples $(x_1,\ldots, x_n)$ such that $0\leq x_1\leq x_2\leq\ldots\leq x_n\leq 1$? | The answer is $\sum_{k=1}^n f(a_k)$, with $a_k=\frac{2k-1}{2n}$, attained for instance when all $x_i$'s are zero or when all are equal to $1$. Let us prove that indeed $$\sum_{k=1}^n f(|x_k-\frac{2k-1}{2n}|)\leq \sum_{k=1}^n f(a_k)$$
for all $0\leq x_1\leq x_2\leq\ldots\leq x_n\leq 1$. Let $A: \{1,\ldots, n\}\to \{1,\l... | algebra |
A39 | Let $a,b,c,d$ be positive real numbers such that $ab+bc+cd+da=4$ and at most one of $a,b,c,d$ is smaller than $1$. Prove that $$\frac{1}{ab+3}+\frac{1}{bc+3}+\frac{1}{cd+3}+\frac{1}{da+3}+\frac{1}{ac+3}+\frac{1}{bd+3}\geq \frac{3}{2}.$$ | We may assume that $a\leq 1$ and $b,c,d\geq 1$. Write $b+d=2x$ with $x\geq 1$, so that $a+c=2/x$. Then
$$\frac{1}{ab+3}+\frac{1}{bc+3}+\frac{1}{cd+3}+\frac{1}{da+3}+\frac{1}{ac+3}+\frac{1}{bd+3}=\left(\frac{1}{ab+3}+\frac{1}{ad+3}\right)+$$ $$+\left(\frac{1}{bc+3}+\frac{1}{cd+3}\right)+\frac{1}{ac+3}+\frac{1}{bd+3}\ge... | algebra |
A40 | Find all functions $f: (0,\infty)\to (0,\infty)$ such that for all $x,y>0$
$$f(x+y)\geq f(x)+yf(f(x)).$$ | We will prove that there is no such function. Suppose that $f$ is a solution of the problem, and let $a=f(f(1))$. Setting
$x=1$ yields $f(y+1)>ya$, thus $$\lim_{y\to\infty} f(y)=\infty.$$ Setting $y=1$ yields $$f(x+1)-f(x)\geq f(f(x)),$$ thus also
$$\lim_{x\to\infty} f(x+1)-f(x)=\infty.$$ Since $f(x+y)>f(x)$ for all ... | algebra |
A41 | A function $f: \mathbf{R}\to \mathbf{R}$ is called additive if $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbf{R}$. Let $a$ be a real number, $n$ a positive integer and let $f_1,\ldots, f_n$ be additive functions such that $f_1(x)f_2(x)\ldots f_n(x)=ax^n$ for all $x\in \mathbf{R}$. Prove that at least one of the functions $... | Suppose first that $a\ne 0$ and take $x\in \mathbf{R}$. For an integer $k$ we have
$$a(1+kx)^n=\prod_{i=1}^n f_i(1+kx)=\prod_{i=1}^n (f_i(1)+kf_i(x)).$$
This is a polynomial equality in the variable $k$, which holds for infinitely many values of $k$, thus it is a polynomial identity. Since the left-hand side is a powe... | algebra |
A42 | Prove that for all non-degenerate subintervals $ I_{1},\dots, I_{k} $ of the interval $ [0, 1]$
we have \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2},\] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$. | Let $S$ be the set of pairs $(i,j)$ for which $I_i\cap I_j\neq \emptyset$.
We first claim that for all $(i,j)\in S$ we have
$$\dfrac{1}{|I_i \cup I_j|} \ge \dfrac{|I_i \cap I_j|}{|I_i| \cdot |I_j|}.$$
Indeed, we have $|I_i\cup I_j|+|I_i\cap I_j|=|I_i|+|I_j|$ and
$|I_i\cap I_j|\leq \min (|I_i|, |I_j|)$. But if $a,b,c,... | algebra |
A43 | Let $p$ be an odd prime number, and let $\mathbb{F}_p[X]$ be the ring of polynomials over the field $\mathbb{F}_p$ of integers modulo $p$. Let $q\in \mathbb{F}_p[X]$ be given by $$q(X) = \sum_{k=1}^{p-1} k^{\frac{p-1}{2}} X^k.$$ Find the greatest nonnegative integer $n$ such that $(X-1)^n$ divides $q$ in $\mathbb{F}_p[... | The condition $(X-1)^n\mid q$ is equivalent to saying that $$q^{(m)}(1)=0, \,\, \forall \,\, 0\leq m<n,$$ where $q^{(m)}$ is the $k$th (formal) derivative of
$q$. We have
$$q^{(m)}(1)=\sum_{k=1}^{p-1} k^{\frac{p-1}{2}}k(k-1)...(k-m+1).$$
Let $$P_m(X)=X^{\frac{p-1}{2}}X(X-1)...(X-m+1),$$ thus we want to find the large... | algebra |
A44 | Let $a,b,c,d$ be positive real numbers such that $$\frac{1-c}{a}+\frac{1-d}{b}+\frac{1-a}{c}+\frac{1-b}{d}\geq 0.$$ Prove that $$a(1-b)+b(1-c)+c(1-d)+d(1-a)\geq 0.$$ | The inequality we want to prove is equivalent to
$$\frac{1}{a+c}+\frac{1}{b+d}\geq 1$$
and it is clear if $a+c\leq 1$ or $b+d\leq 1$, so assume that $a+c>1$ and $b+d>1$. By assumption
$$0\leq \frac{1-c}{a}+\frac{1-a}{c}+\frac{1-d}{b}+\frac{1-d}{b}=\frac{a+c-a^2-c^2}{ac}+\frac{b+d-b^2-d^2}{bd}$$
$$=\frac{(a+c)(1-a-c)}... | algebra |
A45 | Let $n>3$ be an integer and let $M$ be a set of $n$ points in the plane, no three being collinear. Suppose that there is no circle that contains all points of $M$. Find all functions $f: M\to \mathbf{R}$ such that for any circle $C$ containing at least three points of $M$ we have $$\sum_{P\in C\cap M} f(P)=0.$$ | We will prove that the zero function is the only solution. Suppose that $f$ is a nonzero solution. Let $A\ne B\in M$ and let $C_{A,B}$ be the set of circles passing through $A,B$ and some other point of $M$. By assumption
$k:=|C_{A,B}|\geq 2$ and also for all $C\in C_{A,B}$ we have $\sum_{P\in C\cap M} f(P)=0$. It fol... | algebra |
A46 | Let $n$ be a positive integer. Prove that
$$\sum_{k=1}^n (-1)^{\lfloor k (\sqrt{2} - 1) \rfloor} \geq 0.$$ | Let $a_k=\lfloor k (\sqrt{2} - 1) \rfloor$. The sequence $(-1)^{a_k}$ starts as follows: $1,1,1, -1, -1, 1,1,1,...$. It consists of groups of $1$ and $-1$, whose lengths are denoted
$c_0=3, c_1=2, c_2=3,...$. For each $n$ the sum $c_0+c_1+...+c_n$ counts the number of indices
$k$ with $a_k<n+1$, or equivalently $k<(n... | algebra |
A47 | A polynomial $P=a_0+a_1X+...+a_nX^n$ with complex coefficients and degree $n$ has the property that
$|P(z)|\leq 1$ for all $|z|=1$. Prove that $|a_k|\leq 1-|a_n|^2$ for all $0\leq k\leq n-1$. | By a standard argument, if $Q=b_dX^d+...+b_0$ is a polynomial and $z_1,...,z_N$ are the $N$th roots of unity with $N>d$ then
$$\frac{1}{N}\sum_{i=1}^N|Q(z_i)|^2=|b_0|^2+\cdots+|b_d|^2.$$
Let $t$ be a complex number and apply the above observation to $$Q=P(1+t X^k)=a_0+...+(a_n+ta_{n-k})X^n+...+ta_n X^{n+k}$$
and to th... | algebra |
A48 | Is there an irreducible monic polynomial $f\in \mathbf{Z}[X]$ such that $\deg(f)>1$ and $f(X^2+aX)$ is reducible in $\mathbf{Q}[X]$ for infinitely many integers $a$? | The answer is negative. Suppose that $f$ is such a polynomial and let $d=\deg(f)$. Let $a$ be an integer for which
$f(X^2+aX)$ is reducible and let $u$ be a root of $f(X^2+aX)$. Then $z=u^2+au\in \mathbf{Q}(u)$ is a root of $f$ and since
$f$ is irreducible we must have $[\mathbf{Q}(z): \mathbf{Q}]=d$. On the other ha... | algebra |
A49 | For which integers $M$ can the polynomial $MX$ be written as the sum of cubes of some polynomials with integer coefficients? | Note that
$$6X=(X+1)^3+(X-1)^3+(-X)^3+(-X)^3,$$
thus any multiple of $6$ is a solution of the problem. We will prove that these are the only solutions. Let $M$ be an integer and suppose that
we can write $$MX=f_1(X)^3+\ldots+f_n(X)^3$$
for some polynomials $f_1,\ldots, f_n$ with integer coefficients. Taking the deriv... | algebra |
A50 | Prove that for all $a_1,\ldots, a_n\in (-1,1)$ we have
$$\prod_{i=1}^n \prod_{j=1}^n \frac{1+a_i a_j}{1-a_ia_j}\geq 1.$$ | Let
$$S_k=a_1^{2k-1}+\ldots+a_n^{2k-1}$$
and note that $$|S_k|\leq n (\max_{i} |a_i|)^{2k-1}$$ by assumption. It follows that the series
$\sum_{k\geq 1} \frac{1}{2k-1} S_k^2$ converges and we will prove that
$$\ln \prod_{i=1}^n \prod_{j=1}^n \frac{1+a_i a_j}{1-a_ia_j}=2\sum_{k\geq 1} \frac{1}{2k-1} S_k^2,$$
which wil... | algebra |
C1 | In a chess tournament with $n\geq 5$ players, each player played all other players. One gets a point for a win, half a point for a draw, and zero points for a loss. At the end of the tournament, each player had a different number of points. Prove that the second and third ranked players had together more points than th... | Let $x_i$ be the score of player ranked $i$. By assumption $x_1,\ldots, x_n$ are pairwise distinct, thus
$x_i\geq x_{i+1}+1/2$ for all $i$, in particular $$x_{k+3}\leq \min(x_2-\frac{k+1}{2}, x_3-\frac{k}{2})\leq \frac{x_2+x_3}{2}-\frac{2k+1}{2}$$
for $1\leq k\leq n-3$. Adding the previous inequalities yields
... | combinatorics |
C2 | A graph $G$ has $n>1$ vertices. Prove that
$$\sum_{e} \frac{c(e)}{c(e)-1}\leq \frac{n^2}{2},$$
the sum being taken over all edges of $G$ and $c(e)$ being the number of vertices of the largest complete subgraph of $G$ containing $e$. | We induct on $n$, the case $n=2$ being trivial. We may assume that $G$ has at least one edge. Let
$k$ be the maximal size of a clique (i.e. complete subgraph) of $G$ and pick a $k$-clique $K$ in $G$. There are $\binom{k}{2}$ edges $e$ within
$K$, and for each of them $c(e)=k$ by maximality of $K$. Thus the contributi... | combinatorics |
C3 | Let $T$ be a finite set of integers greater than $1$.
A subset $S$ of $T$ is good
if for any $t\in T$ one can find $s\in S$ such that $\gcd(s,t)>1$.
Prove that the number
of good subsets of $T$ is odd. | We will count in two ways pairs $(A,B)$, where any element of $A$ is relatively prime to any element of $B$.
If $S$ is a subset of $T$, let $\bar{S}$ be the set of $t\in T$ which are relatively prime to all elements of $S$.
Given $A$, there are $2^{|\bar{A}|}$ possibilities for $B$, since it can (and should) be ... | combinatorics |
C4 | Let $n$ be a positive integer and let
$S_n$ (respectively $T_n$) be the set of all bijections (respectively all maps) from $[n]:=\{1,\ldots, n\}$ to $[n]$. For $\tau\in T_n$ let ${\rm ord}(\tau)$ be the
number of distinct maps in the set $\{\tau, \tau \circ \tau, \tau \circ \tau \circ \tau, \ldots\}$, where o denot... | Set $$
f(n)=\max _{\tau \in S_n} \operatorname{ord}(\tau)$$
and
$$g(n)=\max _{\tau \in T_n} \operatorname{ord}(\tau) .$$
For large enough $n$ we have $\pi(n^{0, 501})> n^{0, 5009}$, where
$\pi(x)$ is the number of primes not exceeding $x$. This follows from a weak version of the prime number theorem, such as a Che... | combinatorics |
C5 | Let $0<a_1<a_2<\ldots$ be an increasing sequence of positive integers such that for any
$n>1$ and any $\varepsilon_1,\ldots, \varepsilon_{n-1}\in \{0,1\}$
$$a_n\ne \varepsilon_1a_1+\ldots+\varepsilon_{n-1} a_{n-1}.$$ Let $N_M$ be the number of terms of the sequence $(a_n)_{n\geq 1}$ not exceeding $M$. Prove that ... | We will prove the stronger inequality
$$N_M\leq \frac{M}{k+1}+\frac{a_1+\ldots+a_k}{k+1}+\frac{k}{2}.$$
This is indeed stronger, since $a_i\leq a_k-(k-i)$ for $i\leq k$, thus
$$\frac{a_1+\ldots+a_k}{k+1}+\frac{k}{2}\leq \frac{1}{k+1}\sum_{i=1}^k (a_k-k+i)+\frac{k}{2}=\frac{ka_k+k}{k+1}\leq a_k.$$
In order to ... | combinatorics |
C6 | Let $p_1,\ldots, p_s$ be pairwise distinct primes and let
$x_1,\ldots, x_n$ be integers, with $n\geq \max p_i$. Let $P_i=\frac{p_1\ldots p_s}{p_i}$ and let
$P'_i$ be an integer such that $P_iP'_i\equiv 1\pmod {p_i}$. For $1\leq j\leq n$ let $M^{(j)}$ be the number of sums of $j$ numbers among $x_1,\ldots, x_n$ whic... | Since $p_1,\ldots, p_s$ are pairwise distinct, it suffices to prove that $X:=M+\sum_{i=1}^n P_i P_i'N_i$ is divisible by
$p_i$ for each $1\leq i\leq s$. Fix such $i$ and observe that by definition $X\equiv M+N_i\pmod {p_i}$, thus it suffices to prove that $M\equiv -N_i\pmod {p_i}$.
For each subset $A$ of $\{1,\ldot... | combinatorics |
C7 | Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that
\[|S|\leq 2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\] | There is one very tricky observation which instantly solved the problem, namely
the identity
\[2^{-1} + 2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} + 2^{-10} = \frac{2018}{2048}.\] Letting $A = \{1, 2, 3, 4, 5, 6, 10\}$, it follows that
$$\left(\frac{2018}{2048}\right)^{2018}= (2^{-1} + 2^{-2} + 2^{-3} + 2^{-4} + 2^{-5... | combinatorics |
C8 | Let $\sigma(x)$ be the sum of positive divisors of the positive integer $x$. Prove that for all integers $n>1$
$$\sum_{k=0}^{n-1} (-1)^k (2k+1)\sigma (\frac{n^2+n}{2}-\frac{k^2+k}{2})=(-1)^{n-1}\frac{n(n+1)(2n+1)}{6}.$$ | We will use generating functions. Let
$$P=\prod_{n\geq 1} (1-X^n).$$
Note that
$$\sum_{n\geq 1} \sigma(n)X^n=\sum_{n\geq 1} \sum_{d\mid n} dX^n=\sum_{d\geq 1} \frac{dX^d}{1-X^d}=-\frac{XP'}{P}.$$
On the other hand Jacobi's triple product identity yields
$$P^3=\sum_{n\geq 0} (-1)^n (2n+1)X^{n(n+1)/2}.$$
Thus
$... | combinatorics |
C9 | Let $a_1,\ldots, a_n$ be positive integers such that
$$\frac{1}{a_1}+\ldots+\frac{1}{a_n}\leq \frac{1}{2}.$$
The government of Optimistica publishes each year an annual report, with $n$ economic indicators, the values of the
$i$th indicator being in the set $\{1,\ldots, a_i\}$. We say the report is optimistic if a... | Let $k_i$ be nonnegative integers such that $2^{k_i}\leq a_i<2^{k_i+1}$ and note that by assumption
$$\sum_{i=1}^n 2^{-k_i}\leq 1.$$ We may assume that $k_1\leq\ldots\leq k_n$.
We will prove below that for each $1\leq i\leq n$ there is a residue class $A_i$ modulo $2^{k_i}$ such that
$A_1,\ldots, A_n$ are pairwise... | combinatorics |
C10 | Prove that for any $n\geq 1$ there is a subset $S$ of $\mathbf{Z}^n$ such that no two points in $S$ are neighbors, and any point in $\mathbf{Z}^n\setminus S$ has exactly one neighbor in $S$. Here
two points $p=(p_1,\ldots, p_n)$ and $q=(q_1,\ldots, q_n)$ in $\mathbf{Z}^n$ are called neighbors if
$$\sum_{i=1}^n |p_i-q... | Let $$f(p_1,\ldots, p_n)=p_1+2p_2+\ldots+np_n\pmod {2n+1}$$
and let $S$ be the set of points $p$ with $f(p)=0$. Note that if $p,q$ are neighbors
there is a unique $1\leq i\leq n$ with $p_k=q_k$ for $k\ne i$ and $p_i-q_i=\pm 1$, thus
$$f(p)-f(q)=\sum_{k=1}^n k (p_k-q_k)=\pm i.$$
In particular we cannot have $f(p)=... | combinatorics |
C11 | Let $n$ be a positive integer. A thief has $2n$ accomplices. Several policemen try to catch him, by placing his accomplices under surveillance. In the beginning no accomplice is followed by the policemen. Every morning each policeman places under his surveillance one of the accomplices and every evening the thief stops... | Call a policeman useless if he shadows someone not trusted by the thief. Note that such a policeman will never catch the thief (we say that the thief is caught by a policeman $P$ if by the $n$th evening $P$ shadows exactly those
$n$ accomplices still trusted by the thief). We will prove that the thief can dismiss eve... | combinatorics |
C12 | Let $n>2^k$ and let $1\leq a_1<a_2<\ldots <a_n$ be integers. Prove that
$$N:=\prod_{1\leq i<j\leq n} (a_i+a_j)$$
has at least $k+1$ pairwise distinct prime factors. | Suppose that it has less, so there are $k$ pairwise distinct primes $p_1,\ldots, p_k$ such that all prime factors of $N$ are among the $p_i$'s. Consider the complete graph $K_n$ and the subgraph $G_i$ of $K_n$ with the same vertex set (namely the set $[n]=\{1,\ldots,n\}$), two vertices $x,y$ being adjacent if
$$v_{p_i... | combinatorics |
C13 | Prove that if $S_1,\ldots, S_p, T_1,\ldots, T_p$ are pairwise distinct subsets of
$\{1,\ldots, n\}$ such that $S_i\cap T_j$ is nonempty for all $i$ and $j$, then
$$p<\frac{3-\sqrt{5}}{2}\cdot 2^n.$$ | Let $\mathcal{S}$ be the set of subsets $X$ of $[n]:=\{1,\ldots, n\}$ which contain $S_i$ for some
$1\leq i\leq p$ and define similarly $\mathcal{T}$. The hypothesis of the problem implies that whenever
$X\in \mathcal{S}$ we have $X^c:=[n]\setminus X\notin \mathcal{T}$. This shows that there is an injection of
$\mat... | combinatorics |
C14 | Starting with a finite list of distinct positive integers, we may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish. W... | We will look for an invariant of the form $\sum_{n\in L} x^n$, where
$L$ is the list of integers. To have an invariant we should at least ask that
$x^n+x^{n+1}=x^{n-2}$
and $x^{n}+x^{n+4}=x^{n-1}$ for all $n$.
This looks like a lot to ask for, but it reduces to $x^2+x^3=1$ and $x^5+x=1$.
Is there such $x$? Start ... | combinatorics |
C15 | Are there two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that for all $x\in (0,1)$
\[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}?\] | The answer is positive. The first step consists in finding a simple criterion that ensures the desired inequality.
Suppose that we have two different sets of at most $2011^2$ positive integers
$A,B$ such that the polynomial
$P(x)=\sum_{a \in A} x^a - \sum_{b \in B} x^b$ is divisible by $(1-x)^{2012}$.
We claim ... | combinatorics |
C16 | Let $m\geq n\geq 2022$ be integers and let $a_1,\ldots, a_n$ and
$b_1,\ldots, b_n$ be real numbers. Prove that there are at most $3n\sqrt{m\log n}$ pairs
$(i,j)$ with $1\leq i,j\leq n$ and
$$|a_i+b_j-ij|\leq m.$$ | Consider the bipartite graph $G$ with two classes $A=\{1,\ldots, n\}$ and $B=\{1,\ldots, n\}$ of $n$ vertices each, where we declare
$i\in A$ adjacent to $j\in B$ if $|a_i+b_j-ij|\leq m$. Let us double count triples $(x,y,z)$ with
$x\in A$ and $y\ne z\in B$ such that $xy$ and $xz$ are edges in $G$. Any $x\in A$ contr... | combinatorics |
C17 | Let $p$ be a prime number. A flea is at point $0$ of the real line. At each minute,
the flea has three possibilities: to stay at its position, or to move by $1$ to the left or to the right.
After $p-1$ minutes, it wants to be at $0$ again. If $f(p)$ is number of its strategies to do this, find $f(p)$ modulo $p$. | Note that $f(p)$ is nothing but the coefficient of $x^0$ in $(x+1+x^{-1})^{p-1}$, or equivalently the coefficient of $x^{p-1}$ in $(x^2+x+1)^{p-1}=\sum_{k=0}^{p-1} \binom{p-1}{k} x^{2k} (x+1)^{p-1-k}$. This is $\sum_{2k+j=p-1} \binom{p-1}{k} \binom{p-1-k}{j}=\sum_{k=0}^{(p-1)/2} \frac{(p-1)!}{k!^2 (p-1-2k)!}$. Alternat... | combinatorics |
C18 | An operation on a graph consists in choosing a $4$-cycle (if there is one) and deleting an edge (the one you like)
from this cycle. Let $n\geq 4$. What is the least number of edges you can get by applying operations to $K_n$? | When you delete an edge from a cycle in a connected graph, you don't disconnect the graph, so the final graph is connected and in particular it has at least $n-1$ edges.
The key point is that this is not optimal: we will show that the final graph is not bipartite, in particular it cannot have $n-1$ edges (as then it w... | combinatorics |
C19 | A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a positive integer $x$ into a number $S > x^n + 1$ for some positive integer $n$. Prove that $S\geq x^n +x-1$. | We exclude the obvious case $x=1$. The key point is to study how the remainder of the numbers on the board changes modulo $x^2+x+1$. Playing a bit seems to show that there are only four possibilities: $\pm x, \pm x^2$ for these remainders.
To show this consider the minimal counterexample, so the calculator gives us ... | combinatorics |
C20 | Let $n>1$ be an integer. Let $A,B$ be subsets of the set
$D$ of positive divisors of $n$ such that $a\nmid b$ and $b\nmid a$ for any $a\in A$ and $b\in B$. Prove that
$$\sqrt{|A|}+\sqrt{|B|}\leq \sqrt{|D|}.$$ | Let $A^-$ be the set of positive divisors of some element of $A$ and let $A^+$ be the set of positive divisors of
$n$ which are multiples of some element of $A$. By assumption $A^+\cap B^-=\emptyset$ and
$A^-\cap B^+=\emptyset$, so that
$$|A^+|+|B^-|\leq |D|,\,\, |A^-|+|B^+|\leq |D|.$$
Using Cauchy-Schwarz
$$\sqrt{... | combinatorics |
C21 | Let $n$ be a positive integer. On each edge of $K_{2^n+1}$ we put an integer in $\{0,1,...,2^{n-1}-1\}$ such that for any triangle
the number on some edge of the triangle is the sum of the numbers on the other two edges. Prove that there is a triangle having only
zeros on its edges. | Let $V$ be the set of vertices of $K_{2^n+1}$. Let $f(e)$ be the number written on the edge $e$. We induct on $n$, the case $n=1$ being clear.
In order to apply the inductive hypothesis, it suffices to find a subset $V'$ of $V$ with $|V'|\geq 2^{n-1}+1$ such that $f(e)$ is even
for all edges $e$ connecting two vertic... | combinatorics |
C22 | Let $S$ be a set of $n$ points in the plane such that no four points are collinear.
Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$.
Let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$.
Prove that $\sum_{i=1}^k... | We will count ordered triples $(P, Q, R)$, where $P,Q,R$ are distinct elements of $S$ such that $PQ=PR$. First, for any ordered pair $(Q,R)$ of elements of $S$ there are at most $3$ triples of the form $(P,Q,R)$. Indeed, the
points $P$ such that $PQ=PR$ all lie on a line, and no four points of $S$ are collinear. So the... | combinatorics |
C23 | Let $1<a_1<a_2<\ldots < a_n < 2a_1$ be integers and let
$m$ be the number of distinct prime factors of $a_1a_2\cdots a_n$. Prove that
$$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$ | Fix a prime divisor $p$ of $a_1a_2\ldots a_n$ and write
$$a_i = p^{k_i} \cdot b_i,$$
with $k_i\geq 0$ and $b_i$ relatively prime to $p$. If $b_i=b_j$ for some $i\ne j$, then by symmetry we may assume that
$k_i>k_j$ and we have $pa_j\mid a_i$, but then
$$a_n\geq a_i\geq pa_j\geq pa_1\geq 2a_1,$$
a contradiction. Thu... | combinatorics |
C24 | A triangle-free graph on $n$ vertices and having maximal degree $\leq d$ has an independent set of size
at least $\frac{\log_2 d}{8d}n$. | For $d\leq 15$ the bound $n/(1+d)$ given by the Caro-Wei theorem is strongly enough to conclude, so assume that
$d>15$. Pick a random independent
set $W$ (for the uniform distribution).
For a vertex $v\in V(G)$ consider the random variable $X_v$ defined by
$$X_v(W)=d|\{v\}\cap W|+|N(v)\cap W|.$$
We will prove th... | combinatorics |
C25 | Prove that if $A_1,\ldots, A_m$ are subsets of $\{1,\ldots, n\}$, then
$$\sum_{i,j=1}^m |A_i|\cdot |A_i\cap A_j|\geq \frac{1}{mn} (\sum_{i=1}^m |A_i|)^3.$$ | For $x\in [n]:=\{1,\ldots, n\}$ let $n_x$ be the number of $i$'s with $x\in A_i$. Clearly
$$\sum_{x=1}^n n_x=\sum_{i=1}^m |A_i|.$$
Rewrite the inequality as
$$\sum_{i=1}^m |A_i|(\sum_{j=1}^m |A_i\cap A_j|)\geq \frac{1}{mn}(\sum_{x=1}^n n_x)^3.$$
Fix $1\leq i\leq m$ and double count pairs $(x,j)$ with $x\in A_i\cap A_... | combinatorics |
C26 | The positive integers $a,\ b,\ c$ are pairwise relatively prime. Let $g(a, b, c)$ be the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that
\[g(a, b, c)\ge \sqrt{2abc}.\] | Let $A=\sqrt{2abc}$ and let $X$ be the set of integers of the form
$xa+yb+zc$ with $x,y,z\in \mathbb{Z}_{>0}$. Thus $g(a,b,c)=\max(\{1,2,...\}\setminus X)$. Suppose that
$g(a,b,c)<A$, thus $X$ contains all integers greater than $A$.
Choose $c$ consecutive integers greater than $A$. Each such integer
$n$ is of the ... | combinatorics |
C27 | $100$ circles of radius one in the plane have the property that the triangle formed by the centres of any three given circles has area at most $2017$. Prove that there is a line intersecting at least three of the circles. | We can replace $100$ by $n$ and $2017$ by some $A>0$ and prove the existence of a line intersecting at least $\frac {n}{\sqrt A+1}$ of the circles.
We will prove below that there is a line $\ell$ such that when the centers of the $n$ circles are projected onto the line, the $n$ points lie in some interval with length ... | combinatorics |
C28 | Let $k$ be an even positive integer, and let
$p_1,...,p_k$ be pairwise distinct prime numbers. Finally, let
$N=p_1...p_k$, $a,b\in \{1,2,...,N\}$ and let $S_1$ (resp. $S_2$)
be the number of divisors $d$ of $N$ belonging to $[a,b]$ and having an even (resp. odd) number of prime
factors. Prove that $S_1-S_2\leq \bin... | Call a divisor $d$ of $n$ even-nice (resp. odd-nice) if it belongs to $[a,b]$ and has an even (resp. odd) number of prime factors.
Decompose the set of positive divisors of $N=p_1...p_k$ into symmetric chains. There are
$\binom{k}{k/2}$ such chains.
Fix such a chain, say $C=\{d_1,...,d_l\}$, where $d_{i+1}/d_i$... | combinatorics |
C29 | Label the edges of a regular icosahedron $1,2,\ldots, 30$. Find the number of ways to paint each edge red, white or blue such that each of the $20$ triangular faces of the icosahedron has two edges of some color and a third of a different color. | We will prove that the answer is $12^{10}$. Replace colors by $0,1,2$ and work in
the field with $3$ elements $k=\mathbf{F}_3$. Let $E, F$ be the sets of edges and faces of the icosahedron.
The set ${\rm Col}_E$ of colorings $c: E\to k$ has a natural structure of
$k$-vector space of dimension $|E|=30$. Consider the l... | combinatorics |
C30 | Let $G$ be a graph with $n$ vertices, each of positive degree and let $E$ be the set of edges of $G$. Prove that there are at least $n$ and at most $n!$ of assigning an integer $f(v)$ to each each vertex $v$ such that $\sum_{v} f(v)=|E|$ and for any nonempty subset
$A$ of $V$ there is a vertex $v\in A$ such that $f(v)... | Let $f$ be such an assignment. Taking $A=V$ we obtain a vertex $v_1$ with $f(v_1)\leq 0$. Taking $A=V\setminus \{v_1\}$ we obtain
$v_2\ne v_1$ such that $f(v_2)$ is at most the number of neighbors of $v_2$ in $\{v_1\}$ (which is thus $0$ or $1$). Continuing like this (at the next step apply the hypothesis to $A=V\setm... | combinatorics |
C31 | How many ordered $64$-tuples $(x_0,x_1,\dots,x_{63})$ consisting of pairwise distinct elements of
$\{1,2,\dots,2017\}$ satisfy
\[2017\mid x_0+x_1+2x_2+3x_3+\cdots+63x_{63}?\] | Note that $p:=2017$ is a prime number and $p=1+1+2+3+...+63$. We will work in $\mathbb{F}_p$. Since
$p=1+1+2+...+63$, if $(x_0,x_1,...,x_{63})$ is a solution of the congruence $x_0+x_1+...+63x_{63}\equiv 0\pmod p$ with
$x_0,...,x_{63}$ pairwise distinct, then so are $(x_0+t, x_1+t,..., x_{63}+t)$ for all $t\in \mathb... | combinatorics |
C32 | An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions and flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna a... | Unsurprisingly the answer is positive.
Consider the bipartite graph $G$ with bipartition $A,B$, where vertices in $A$ (resp $B$) are the flights of Anna (resp. Bella). Connect $b\in B$ and $a\in A$ if $p(b)\geq p(a)$, where $p(?)$ is the price of flight $?$.
We will show that $G$ has a perfect matching, so
there is a ... | combinatorics |
C33 | Let $n\geq 2$ be an integer and let $C_n$ be the set of points in the plane with coordinates in $\{1,2,\ldots, n\}$. Let
$S$ be a subset of $C_n$ such that any point of $C_n$ lies on a line determined by two points of $S$. Prove that
$|S|\geq (n/4)^{2/3}$. | Consider the slope of a line passing through two points of $C_n$. This slope is $0,\infty$ or of the form
$\pm a/b$ with $a,b\in \{1,2,\ldots, n\}$ relatively prime. If the slope is $0$ or $\infty$ the line contains $n$ points of $C_n$. Suppose that the slope is
$\pm a/b$, with $a,b$ as above, thus the equation of th... | combinatorics |
C34 | Let $m$ be a positive integer and consider a social network with a fixed finite set of users, each user having a fixed set of followers (among the other users), as well as an initial rating, a positive integer which may not be the same for all users. Every midnight, the rating of every user increases by the sum of the ... | Say there are $n$ users, called $1,\ldots, n$.
Let $a_{ij}$ be $1$ if user $j$ follows user $i$ and let $A=(a_{ij})$ the resulting binary matrix. Each day we can record the user's ratings taken modulo $m$ by a vector. Note that by assumption if the vector recording the ratings is
$v$ before midnight, it will be $Av$ ... | combinatorics |
C35 | A graph $G$ contains no $K_4$ and its chromatic number is $\geq 4$. Prove that there is a partition $V(G)=A\cup B$ such that the graphs induced by $A$ and $B$ have chromatic numbers $\geq 3$ and $\geq 2$. | Let $\chi(X)$ be the chromatic number of $X$. Suppose that this is not the case, i.e. for any partition $V=A\cup B$ of the vertices,
the induced subgraphs on $A$ and $B$ satisfy $\chi(A)\leq 2$ or $\chi(B)\leq 1$.
Pick an odd cycle $C$ of minimal length (it exists since $G$ is not bipartite). Since $\chi(C)=3$, the g... | combinatorics |
C36 | Let $n>1$ be an integer and let $a_1,\ldots, a_n$ be positive integers. Suppose that we have an unlimited supply of coins, each with value in $\{a_1,\ldots, a_n\}$ and that the sum $a_1+a_2+\ldots+a_n$ can be collected in only one way, by taking one coin of each value.\\
a) Give an example of such integers $a_1,\ldots,... | a) We will show that setting $a_i=2^n-2^{n-i}$ yields a solution. Note that
$$a_1+\ldots+a_n=(n-1)2^n+1<n\cdot 2^n.$$
Suppose that we could collect the sum $S=a_1+\ldots+a_n$ in the form
$$S=a_{i_1}+\ldots+a_{i_p}.$$
We will prove that $j_k:=n-i_k$ are necessarily given by $j_k=k-1$, so we have uniqueness. Note that ... | combinatorics |
C37 | Let $a_1,\ldots, a_n$ be nonzero real numbers, not necessarily distinct. What is the largest possible number of subsets $A\subset \{1,\ldots, n\}$ such that $\sum_{x\in A} a_x=0$? | Taking the first $\lfloor n/2\rfloor$ $a_i$'s equal to $1$ and the others to $-1$ we obtain
$N=\binom{n}{\lfloor n/2\rfloor}$ such subsets. To prove that this is maximal set
$a_A=\sum_{x\in A} a_x$ and say $A$ is good if $a_A=0$. For each good set
$A$ let $X_A$ be the set of those
$i\in \{1,\ldots, n\}$ for which e... | combinatorics |
C38 | Let $n$ be a positive integer and let $A_1,\ldots, A_{2^n+1}$ be finite sets. We color these sets red and blue such that there is at least one red and at least one blue set. Prove that there are at least $2^n$ different sets which can be obtained as the symmetric difference of a red set and a blue set. Here the symmetr... | Let $d$ be the size of the union of the $A_i$'s and say $x_1,\ldots, x_d$ are the elements of $\cup_{i} A_i$. For each subset
$S$ of $\{x_1,\ldots, x_d\}$ let $v_S\in\mathbf{F}_2^d$ be the vector whose $j$th coordinate is $1$ if and only if $x_j\in S$.
Note that
$$v_{A_i\Delta A_j}=v_{A_i}+v_{A_j}.$$ Thus we are red... | combinatorics |
C39 | Let $p>3$ be a prime and color each of the numbers $1,2,\ldots, p-1$ in red, blue or green. Assume that all three colors are used. Prove that there are three numbers $x,y,z$ with pairwise distinct colors such that $x+y\equiv z\pmod p$. | Let $A,B,C$ be the sets of numbers colored red, blue or green respectively and suppose that the congruence $x+y\equiv z\pmod p$ has no solution with $x,y,z$ belonging to three different subsets among $A,B,C$. We work in $\mathbf{F}_p$ from now on. For $1\leq r\leq p-1$ the sets
$rA, rB, rC$ still form a partition of $... | combinatorics |
C40 | A connected graph has $1998$ vertices and each vertex has degree $3$. Prove that we can delete $200$ vertices, no two of them joined by an edge, such that the resulting graph is connected. | Let $G_0=G$ be the original graph. We will construct
graphs $G_1=G_0-v_1$, $G_2=G_1-v_2=G-\{v_1,v_2\}$,..., $G_{i}=G-\{v_1,...,v_{i}\}$
such that $d_{G_{i-1}}(v_i)=3$ and all $G_i$ are connected for $1\leq i\leq 200$.
The condition $d_{G_{i-1}}(v_i)=3$ ensures that all $3$ neighbors of $v_i$ in $G=G_0$ are in $G_{i-1... | combinatorics |
C41 | Prove that for all positive integers $a,b$
\[\sum_{i,j \ge 0} \binom{i+j}{i}^2 \binom{(a-i)+(b-j)}{a-i}^2=\frac{1}{2} \binom{(2a+1)+(2b+1)}{2a+1}.\] | Set $$S=\sum_{i,j \ge 0} \binom{i+j}{i}^2 \binom{(a-i)+(b-j)}{a-i}^2.$$
Letting $s=i+j$, we observe that
$$S = \sum_{s\geq 0}\sum_{i=0}^s \binom{s}{i}^2 \binom{a+b-s}{a-i}^2.$$ The generating function
$$F(x,y) = \sum_{s,i} \binom{s}{i}^2 x^i y^s = (1-2y+y^2-2xy-2xy^2+x^2y^2)^{-1/2}$$ has the property that $S$ is the c... | combinatorics |
C42 | Real numbers $x_1,...,x_n$ are attached to the $n\geq 2$ vertices of a tree (one number per vertex). For each edge of the tree consider the product of the numbers associated to its endpoints. Prove that the sum $S$ of all products over the edges of the tree satisfies $$S\leq \frac{\sqrt{n-1}}{2}\cdot \left(x_1^2+x_2^2+... | We will induct on $n$, the base case being clear. Suppose that it holds for $n-1$. Note that we may assume that the $x_i\geq 0$.
Pick a leaf $a$ and let $b$ be its unique neighbor. Delete $a$ and the edge $ab$ to get a new tree
$T'=(V',E')$. We need to show that
$$\frac{2}{\sqrt{n-1}} x_ax_b+\frac{2}{\sqrt{n-1}} \su... | combinatorics |
C43 | A graph $G$ on $n$ vertices is not complete, has no isolated vertex (i.e. of degree $0$) and each dominating set has size
$\geq k\geq 1$. Prove that the chromatic number of $G$
is $\leq n-k$. | Let $\Delta$ be the maximal degree of a vertex of $G$.For any vertex $v$ the set $S=V\setminus N(v)$ is dominating, so $n-d(v)\geq k$ and so
$\Delta\leq n-k$.
If $\Delta\leq n-k-1$ we conclude thanks to the inequality
$\chi(G)\leq \Delta+1$.
Assume now that $\Delta=n-k$ and pick a vertex $v$ of degree $n-k$.
Let $... | combinatorics |
C44 | Let $p$ be an odd prime. What is the largest positive integer positive integer $n$ for which there are $n$ lattice points
$A_1,...,A_n$ in the plane, no three collinear and such that $p$ does not divide twice the area of any non-degenerate triangle with vertices among the $A_i$'s? | The answer is $n=p+1$. Let us prove first that we cannot find $p+2$ such points. Indeed, since no three points are collinear, for each $x$-coordinate $x$ of one of the points there is at most one other point with the same $x$-coordinate. Since $p+2$ is odd, it follows that there is some point
$A_{i_0}$ whose $x$-coord... | combinatorics |
C45 | Let $v_1,v_2,...,v_n$ be $n$ vectors in the plane, whose coordinates are integers of absolute value less that $\frac{1}{100}\sqrt{\frac{2^n}{n}}$. Prove that there are disjoint subsets $I,J$ of $\{1,2,...,n\}$ such that $\sum_{i\in I} v_i=\sum_{j\in J} v_j$. | It is enough to find two distinct such subsets $I,J$,
for then it is enough to take out of each the common elements of $I,J$.
Assume that this is not the case and write $v_i=(x_i,y_i)$.
Consider a random binary sequence $(a_1,a_2,...,a_n)$, each $a_i$ being $0$ or $1$ with probability
$1/2$ (all these events being ind... | combinatorics |
C46 | Prove that for any integer $n>1$ any set $M$ of positive rational numbers with size $2n^2-3n+2$ has a subset $A$ of size $n$ such that
for any $k\in \{2,\ldots, n\}$ the sum of any $k$ (not necessarily distinct) numbers from $A$ is not in $A$. | Multiplying all elements of $M$ by a suitable positive integer, we may assume that the elements of $M$ are positive integers. Take a prime
$p>\max M$ and identify $M$ with its image in $\mathbf{F}_p=\{0,1,\ldots, p-1\}$ (note that two distinct elements of $M$ must give distinct remainders when divided by $p$).
Conside... | combinatorics |
C47 | Let $n$ be a positive integer. A permutation $\sigma$ of $\{1,\ldots, n\}$ is called special if $k^4+\sigma(k)^4$ is a prime number for $k=1,2,\ldots, n$. Prove that the number of special permutations is a perfect square. | Let
$\sigma$ be a special permutation, thus $k^4+\sigma(k)^4=p_k$ for $1\leq k\leq n$, with $p_1,\ldots, p_n$ prime numbers. Then
$$p_1+\ldots+p_n=2(1^4+\ldots+n^4)$$
is even. On the other hand for $k>1$ we have $p_k>1$, thus $p_k$ is odd. It follows that $p_1+n-1$ must be even. In particular, if
$n$ is even then $p... | combinatorics |
C48 | Let $\varepsilon>0$. Prove that for all but finitely many positive integers
$n$, a graph with $n$ vertices and at least $(1+\varepsilon)n$ edges
has two different cycles of equal length. | Suppose that $G$ is a graph on $n$ vertices and at least $(1+\varepsilon)n$ edges. For simplicity,
suppose that $G$ is connected and pick a spanning tree $T$.
For each edge $e$ of $G$ not in $T$, adding $e$ produces a unique cycle
$C_e$ and these $C_e$ have pairwise distinct lengths by assumption.
Double count pairs... | combinatorics |
C49 | In a club with $42$ members it is known that among any $31$ members one can find a boy and a girl that know each other.
Prove that we can find $12$ pairs of boys and girls, such that the boy and the girl in each pair know each other. | Say there are $b$ boys and $g$ girls in the club, so that $b+g=42$.
Pick any $k\in \{1,2,...,b\}$ boys and let $r$ be the number of girls known by at least one of these $k$ boys.
Then $k+(g-r)\leq 30$, otherwise these $k$ boys and the $g-r$ girls they don't know would form a group of at least $31$ persons, contradicti... | combinatorics |
C50 | Prove that there are $c,n_0>0$ such that for any set
$A$ consisting of $n>n_0$ integers we have
$$|A-A|-|A+A|\leq n^2-cn^{8/5},$$
where $A\pm A=\{a\pm b|\, a,b\in A\}$. | For each $x\in A-A$ let $n_x$ be the number of pairs
$(a,b)\in A\times A$ such that $a-b=x$. Define similarly
$m_y$ for $y\in A+A$. Double counting pairs $(a,b)\in A\times A$ yields
$$\sum_{x\in A-A} n_x=n^2= \sum_{y\in A+A} m_y$$
The first key observation is that
there is another relation between the $n_x$ and th... | combinatorics |
End of preview. Expand in Data Studio
Nemotron-IMO-Bench
Dataset Description:
Nemotron-IMO-Bench is an English-language evaluation benchmark containing 200 challenging, proof-oriented mathematics problems. The problems and reference proofs were created in collaboration with Professor Titu Andreescu, a mathematics educator and olympiad problem author; they were written for this benchmark and have not been published before. Each problem is paired with a reference proof. The benchmark is evenly divided among algebra, combinatorics, geometry, and number theory, with 50 problems in each subject. The benchmark is intended to measure a model's ability to construct solutions to difficult olympiad-style problems.
This dataset is ready for commercial or non-commercial uses.
Dataset Owner(s)
NVIDIA Corporation
Dataset Creation Date
Created on: 03/29/2026
Versioning
Version: 1.0
Previous versions: No previous public version.
License/Terms of Use
Nemotron-IMO-Bench is licensed under the Creative Commons Attribution 4.0 International license.
Intended Usage
Nemotron-IMO-Bench is intended for researchers and developers evaluating the mathematical reasoning capabilities of language models. Potential uses include:
- evaluating olympiad-level problem solving and proof generation;
- comparing mathematical reasoning systems under a common benchmark;
- analyzing proof correctness, completeness, and reasoning quality; and
- studying failure modes on difficult, multi-step mathematical problems.
The benchmark is intended for evaluation rather than model training. Users should avoid incorporating its problems or reference solutions into training corpora, because doing so would compromise the validity of subsequent benchmark results.
Dataset Characterization
Data Collection Method
- Manually collected and human-curated in collaboration with Professor Titu Andreescu.
Labeling Method
- Manually curated. Every problem is paired with a human-curated reference proof.
Dataset Format
The dataset is distributed as UTF-8 encoded JSON Lines (.jsonl) text. Each line represents one mathematics problem.
The JSONL schema contains four fields:
| Field | Type | Coverage | Description |
|---|---|---|---|
id |
string | 200/200 | Unique problem identifier. |
problem |
string | 200/200 | Mathematical problem statement. |
proof |
string | 200/200 | Reference proof. |
subject |
string | 200/200 | One of algebra, combinatorics, geometry, or number theory. |
The dataset uses a test split.
Dataset Quantification
- Record count: 200 mathematics problems and 200 corresponding reference proofs.
- Subject distribution: 50 algebra, 50 combinatorics, 50 geometry, and 50 number theory problems.
- Feature count: Four fields.
- Uniqueness: All 200 problem statements, reference proofs, and identifiers are unique.
- Total storage: 315 KB
References
- An Open Recipe for IMO Gold: Training Nemotron for Olympiad Mathematics (technical report)
- NeMo-Skills
recipes/nemotron-imo-tts(inference pipeline, script assembling the 30-problem development set, submitted proofs)
Ethical Considerations
NVIDIA believes Trustworthy AI is a shared responsibility and has established policies and practices to enable development for a wide array of AI applications. Developers should work with their internal teams to ensure this dataset meets requirements for the relevant industry and use case and addresses unforeseen product misuse.
Because benchmark contamination can inflate reported performance, users should disclose any possibility that these problems or their solutions were included in model training data. Users should also document prompting, inference-time compute, external tools, and grading methodology when reporting results.
Please report quality, risk, security vulnerabilities, or NVIDIA AI concerns here.
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