Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.

From the first 64 positive integers are chosen two subsets with 16 numbers in each. The first subset contains only odd numbers while the second one contains only even numbers. Total sums of both subsets are the same. Prove that among all the chosen numbers there are two whose sum equals 65. [i](3 points)[/i]

Prove that if $f$ is a non-constant real-valued function such that for all real $x$, $f(x+1) + f(x-1) = \sqrt{3} f(x)$, then $f$ is periodic. What is the smallest $p$, $p > 0$ such that $f(x+p) = f(x)$ for all $x$?

$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$. [i]proposed by Mohammadmahdi Yazdi[/i]

Prove that there exist infinitely many pairs $ (a, b)$ of natural numbers not equal to $ 1$ such that $ b^b \plus{}a$ is divisible by $ a^a \plus{}b$.

A permutation of a finite set $S$ is a one-to-one function from $S$ to $S$. A permutation $P$ of the set $\{ 1, 2, 3, 4, 5 \}$ is called a W-permutation if $P(1) > P(2) < P(3) > P(4) < P(5)$. A permutation of the set $\{1, 2, 3, 4, 5 \}$ is selected at random. Compute the probability that it is a W-permutation.

Let $n \in \mathbb{N}_{\geq 2}$ and $a_1,a_2, \dots , a_n \in (1,\infty).$ Prove that $f:[0,\infty) \to \mathbb{R}$ with $$f(x)=(a_1a_2...a_n)^x-a_1^x-a_2^x-...-a_n^x$$ is a strictly increasing function.

In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

A tourist going to visit the [i]Complant[/i], found that: a) in this country $1024$ cities, numbered by integers from $0$ to $1023$ , b) two cities with numbers $m$ and $n$ are connected by a straight line if and only if the binary entries of numbers $m$ and $n$ they differ exactly in one digit, c) during the stay of a tourist in that country $8$ roads will be closed for scheduled repairs. Prove that a tourist can make a closed route along the existing roads of [i]Complant[/i], passing through each of its cities exactly once.

Find the smallest real number $M$, for which $\{a\}+\{b\}+\{c\}\leq M$ for any real positive numbers $a, b, c$ with $abc = 2024$. Here $\{a\}$ denotes the fractional part of number $a$. [i]Proposed by Fedir Yudin, Anton Trygub[/i]

There are $2014$ students from high schools nationwide communications sit around a round table in arbitrary manner. Then the organizers want to rearrange students from the same school sit next to each other by performing the following swapping: permutation view of two adjacent groups of students (see illustration). Find the smallest $k$ number so that a result can be obtained results as desired by the organizers with no more than $k$ swapping permits. Permission to change places like after $...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$ Vu The Khoi, Institute of Mathematics, Vietnam Academy of Science and Technology, Cau Giay, Hanoi.

Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$

Let $u_1=1,u_2=1$ and for all $k \geq 1$'s $$u_{k+2}=u_{k+1}+u_{k}$$ Prove that for all $m \geq 1$'s $5$ divides $u_{5m}$

Let $a$, $b$ and $c$ be complex numbers such that $abc = 1$. Find the value of the cubic root of \begin{tabular}{|ccc|} $b + n^3c$ & $n(c - b)$ & $n^2(b - c)$\\ $n^2(c - a)$ & $c + n^3a$ & $n(a - c)$\\ $n(b - a)$ & $n^2(a - b)$ & $a + n^3b$ \end{tabular}

Let $ABC$ be an acute-angled triangle with $A=60^{\circ}$. Let $E,F$ be the feet of altitudes through $B,C$ respectively. Prove that $CE-BF=\tfrac{3}{2}(AC-AB)$ [i]Proposed by Fatemeh Sajadi[/i]

A polygon with $2n+1$ vertices is given. Show that it is possible to assign numbers $1,2,\ldots ,4n+2$ to the vertices and midpoints of the sides of the polygon so that for each side the sum of the three numbers assigned to it is the same.

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .

Let $P$ be a point outside a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the symmetric of $B$ with respect to $P$. Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$. Line $EB$ cuts back to $\omega _1 $ in $F$. Prove that $CF = AB$.

Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.

Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$

Jar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green marbles, and one red marble. Jar #3 contains five green marbles, three red marbles, and one blue marble. You randomly select one marble from each jar. Given that you select one marble of each color, the probability that the red marble came from jar #1, the blue marble came from jar #2, and the green marble came from jar #3 can be expressed as $\frac{m}{n}$, where m and n are relatively prime positive integers. Find m + n.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.