Nemotron-IMO-Bench 2026

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 $m,n\geq 2$ be integers and let $f\in \mathbb{R}[X_1,...,X_n]$ be a polynomial such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor,\, \forall\, x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

Prove that for all positive real numbers $a,b,c,d$ $$\frac{a+b}{1+c}\cdot \frac{b+c}{1+d}\cdot \frac{c+d}{1+a}\cdot \frac{d+a}{1+b}\geq \frac{16abcd}{(1+\sqrt[4]{abcd})^4}.$$

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 exist distinct block-similar polynomials of degree $n+1$.

(b) Prove that there do not exist distinct block-similar polynomials of degree $n$.

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 $n\geq 1$.

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 $n,k$.

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 $a+b+c+d = 100$.

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 $1<\sum_{i=1}^n 1/b_i\leq C$.

Find all functions $f: \{1,2,...\}\to \mathbb{Z}$ such that for all positive integers $m,n$ we have $$\lfloor f(mn)/n\rfloor=f(m).$$

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 $f(x)^2\leq f(y)\leq f(x)^3$.

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 $n$?

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.

Prove that for all real numbers $x,y,z$ $$(x+y+z)^2+\sum \frac{(y+x)(y+z)}{1+|x-z|}\geq xy+yz+zx.$$

Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, $$\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.$$

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 derivatives consist of positive numbers. Prove that if $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ are good sequences, then so is the sequence $(a_nb_n)_{n\geq 1}$.

Prove that for all positive real numbers $a,b,c$ $$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}+\frac{45}{a+b+c}\geq \frac{7}{a}+\frac{7}{b}+\frac{7}{c}.$$

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.

Find the largest real number $k$ such that for all positive real numbers $a,b,c,d$ with $ab+bc+cd+da=1$ $$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}-4\geq k(a^2+b^2+c^2+d^2-4).$$

Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be sequences of real numbers for which $a_1 > b_1$ and $$a_{n+1}= a_n^2 - 2b_n,\, b_{n+1}= b_n^2 - 2a_n$$ 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 $k > N$.

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 $$\sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.$$

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 $Q(-1/2)$?

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.

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.

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 $n$ 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 $1\leq k\leq K_n$.

Let $n$ be an even positive integer and let $z$ be a complex number such that $z^{2^{n+1}-1}=1$. Evaluate $\prod_{i=0}^n ({\rm Re}(z^{2^i})-\frac{1}{2})$.

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 $a_{n+1}=a_1$.

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 $\sum_{i=0}^n \frac{1}{x_i}=n+\frac{1}{a}$?

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 $P(Q(X))$.

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 $$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$

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.

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 $\mathbf{Z}_{>0}$ 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 $m,n\in \mathbb{Z}_{>0}$.

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 $n\geq 1$.

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 $P(x + y)$.

Let $n>1$ be an integer. Find the largest positive integer $k$ such that for all real numbers $a_1\geq a_2\geq\ldots\geq a_n\geq 0$ $$\left(\frac{a_1+\ldots+a_k}{k}\right)^2\geq \frac{a_1^2+\ldots+a_n^2}{n}.$$

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 $x_1+...+x_n=0$.

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 $0\leq x_1\leq x_2\leq\ldots\leq x_n\leq 1$?

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 $$\frac{1}{ab+3}+\frac{1}{bc+3}+\frac{1}{cd+3}+\frac{1}{da+3}+\frac{1}{ac+3}+\frac{1}{bd+3}\geq \frac{3}{2}.$$

Find all functions $f: (0,\infty)\to (0,\infty)$ such that for all $x,y>0$ $$f(x+y)\geq f(x)+yf(f(x)).$$

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 $f_1,\ldots, f_n$ is linear (i.e. of the form $x\mapsto cx$ for some constant $c$).

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 $p$ 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 $\mathbb{F}_p[X]$.

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 $$a(1-b)+b(1-c)+c(1-d)+d(1-a)\geq 0.$$

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 $$\sum_{P\in C\cap M} f(P)=0.$$

Let $n$ be a positive integer. Prove that $$\sum_{k=1}^n (-1)^{\lfloor k (\sqrt{2} - 1) \rfloor} \geq 0.$$

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 $0\leq k\leq n-1$.

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 $a$?

For which integers $M$ can the polynomial $MX$ be written as the sum of cubes of some polynomials with integer coefficients?

Prove that for all $a_1,\ldots, a_n\in (-1,1)$ we have $$\prod_{i=1}^n \prod_{j=1}^n \frac{1+a_i a_j}{1-a_ia_j}\geq 1.$$

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 the winner of the tournament.

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 $e$.

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.

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 denotes composition. Prove that for sufficiently large $n$ $$g(n)=\max _{\tau \in T_n} \operatorname{ord}(\tau)<\max _{\tau \in S_n} \operatorname{ord}(\tau)+n^{0.501}.$$

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 for all $M>k$ $$N_M\leq a_k+\frac{M}{k+1}.$$

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 $x_1,\ldots, x_n$ which are relatively prime to $p_1\ldots p_s$ and let $$M=M^{(1)}-M^{(2)}+\ldots+(-1)^{n-1}M^{(n)}.$$ For $1\leq i\leq s$ and $1\leq j\leq n$ let $N_i^{(j)}$ be the number of sums of $j$ numbers among $x_1,\ldots, x_n$ which are relatively prime to $p_i$, but not to $P_i$ and let $$N_i=N_i^{(1)}-N_i^{(2)}+\ldots+(-1)^{n-1}N_i^{(n)}.$$ Prove that $$p_1\ldots p_s\mid M+\sum_{i=1}^n P_i P_i'N_i.$$

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 $$|S|\leq 2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.$$

Let $\sigma(x)$ be the sum of positive divisors of the positive integer $x$. Prove that for all integers $n>1$ $$\sum_{k=0}^{n-1} (-1)^k (2k+1)\sigma (\frac{n^2+n}{2}-\frac{k^2+k}{2})=(-1)^{n-1}\frac{n(n+1)(2n+1)}{6}.$$

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 at least $n-1$ indicators have equal or higher values than in the previous report. Prove that the government can publish annual reports in an infinitely long sequence.

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 $$\sum_{i=1}^n |p_i-q_i|=1.$$

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 trusting one of his accomplices. Prove that if less than $2^n$ policemen are involved, then the thief can always make sure that by the $n$th evening no policeman shadows exactly those $n$ accomplices who are still trusted by the thief.

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.

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 $$p<\frac{3-\sqrt{5}}{2}\cdot 2^n.$$

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. What is the most negative integer which can appear in a list?

Are there two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that for all $x\in (0,1)$ $$\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}?$$

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 $$|a_i+b_j-ij|\leq m.$$

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 $p$.

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 $K_n$?

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 $S\geq x^n +x-1$.

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 $n$ 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 $S$ 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 $\sum_{i=1}^k m_i^2\leq n^3-n^2$.

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 $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$

A triangle-free graph on $n$ vertices and having maximal degree $\leq d$ has an independent set of size at least $\frac{\log_2 d}{8d}n$.

Prove that if $A_1,\ldots, A_m$ are subsets of $\{1,\ldots, n\}$, then $$\sum_{i,j=1}^m |A_i|\cdot |A_i\cap A_j|\geq \frac{1}{mn} (\sum_{i=1}^m |A_i|)^3.$$

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 $$g(a, b, c)\ge \sqrt{2abc}.$$

$100$ 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.

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 $S_1-S_2\leq \binom{k}{k/2}$.

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.

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 $f(v)$ is at most the number of neighbors of $v$ in $V\setminus A$.

How many ordered $64$-tuples $(x_0,x_1,\dots,x_{63})$ consisting of pairwise distinct elements of $\{1,2,\dots,2017\}$ satisfy $$2017\mid x_0+x_1+2x_2+3x_3+\cdots+63x_{63}?$$

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 always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour?

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 $|S|\geq (n/4)^{2/3}$.

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 ratings that his followers had just before midnight. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days.

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 $n>1$ 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 $a_1,\ldots, a_n$ such that $a_1+a_2+\ldots+a_n<n\cdot 2^n$.\ b) Prove that $$\min (a_1,\ldots, a_n)\geq 2^{n-1}.$$

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 $\sum_{x\in A} a_x=0$?

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 symmetric difference of two sets as the set of objects belonging to exactly one of the two sets.

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 $x+y\equiv z\pmod p$.

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.

Prove that for all positive integers $a,b$ $$\sum_{i,j \ge 0} \binom{i+j}{i}^2 \binom{(a-i)+(b-j)}{a-i}^2=\frac{1}{2} \binom{(2a+1)+(2b+1)}{2a+1}.$$

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 $$S\leq \frac{\sqrt{n-1}}{2}\cdot \left(x_1^2+x_2^2+\dots+x_n^2\right).$$

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 $p$ 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?