RU-ALLRUSSIAN 2021

On a circle there're $1000$ marked points, each colored in one of $k$ colors. It's known that among any $5$ pairwise intersecting segments, endpoints of which are $10$ distinct marked points, there're at least $3$ segments, each of which has its endpoints colored in different colors. Determine the smallest possible value of $k$ for which it's possible.

Let $n$ be a natural number. An integer $a>2$ is called $n$ -decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$ , where $d\neq n$ is a natural divisor of $n$ . Find all composite $n\in \mathbb{N}$ , for which there's an $n$ -decomposable number.

On a line $n+1$ segments are marked such that one of the points of the line is contained in all of them. Prove that one can find $2$ distinct segments $I, J$ which intersect at a segment of length at least $\frac{n-1}{n}d$ , where $d$ is the length of the segment $I$ .

Given an acute triangle $ABC$ , point $D$ is chosen on the side $AB$ and a point $E$ is chosen on the extension of $BC$ beyond $C$ . It became known that the line through $E$ parallel to $AB$ is tangent to the circumcircle of $\triangle ADC$ . Prove that one of the tangents from $E$ to the circumcircle of $\triangle BCD$ cuts the angle $\angle ABE$ in such a way that a triangle similar to $\triangle ABC$ is formed.

The reals $b>0$ and $a$ are such that the quadratic $x^2+ax+b$ has two distinct real roots, exactly one of which lies in the interval $[-1;1]$ . Prove that one of the roots lies in the interval $(-b;b)$ .

Given is a non-isosceles triangle $ABC$ with $\angle ABC=60^{\circ}$ , and in its interior, a point $T$ is selected such that $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$ . Let $M$ the intersection point of the medians in $ABC$ . Let $TM$ intersect $(ATC)$ at $K$ . Find $TM/MK$ .

One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set { $1,2,3$ }, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly $200$ ; else it is successful. Prove that the sages can create a strategy, by which they can win the game.

On the side $BC$ of the parallelogram $ABCD$ , points $E$ and $F$ are given ( $E$ lies between $B$ and $F$ ) and the diagonals $AC, BD$ meet at $O$ . If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$ , prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

Find all sets of positive integers $\{x_1, x_2, \dots, x_{20}\}$ such that $$x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1})$$ for $i=1, 2, \dots, 20$ where $x_0=x_{20}, x_{21}=x_1, x_{22}=x_2$ .

In the country there're $N$ cities and some pairs of cities are connected by two-way airlines (each pair with no more than one). Every airline belongs to one of $k$ companies. It turns out that it's possible to get to any city from any other, but it fails when we delete all airlines belonging to any one of the companies. What is the maximum possible number of airlines in the country ?

Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$ . On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$ . Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$ .

Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.

Given is a cyclic pentagon $ABCDE$ , inscribed in a circle $k$ . The line $CD$ intersects $AB$ and $AE$ in $X$ and $Y$ respectively. Segments $EX$ and $BY$ intersect again at $P$ , and they intersect $k$ in $Q$ and $R$ , respectively. Point $A'$ is reflection of $A$ across $CD$ . The circles $(PQR)$ and $(A'XY)$ intersect at $M$ and $N$ . Prove that $CM$ and $DN$ intersect on $(PQR)$ .

For some positive integer $n>m$ , it turns out that it is representable as sum of $2021$ non-negative integer powers of $m$ , and that it is representable as sum of $2021$ non-negative integer powers of $m+1$ . Find the maximal value of the positive integer $m$ .

Let $P(x)$ be a nonzero polynomial of degree $n>1$ with nonnegative coefficients such that function $y=P(x)$ is odd. Is that possible thet for some pairwise distinct points $A_{1}, A_{2}, \dots A_{n}$ on the graph $G: y = P(x)$ the following conditions hold: tangent to $G$ at $A_{1}$ passes through $A_{2}$ , tangent to $G$ at $A_{2}$ passes through $A_{3}$ , $\dots$ , tangent to $G$ at $A_{n}$ passes through $A_{1}$ ?

Some language has only three letters - $A, B$ and $C$ . A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all $A$ . What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?

In triangle $ABC$ angle bisectors $AA_{1}$ and $CC_{1}$ intersect at $I$ . Line through $B$ parallel to $AC$ intersects rays $AA_{1}$ and $CC_{1}$ at points $A_{2}$ and $C_{2}$ respectively. Let $O_{a}$ and $O_{c}$ be the circumcenters of triangles $AC_{1}C_{2}$ and $CA_{1}A_{2}$ respectively. Prove that $\angle{O_{a}BO_{c}} = \angle{AIC}$

In tetrahedron $ABCS$ no two edges have equal length. Point $A'$ in plane $BCS$ is symmetric to $S$ with respect to the perpendicular bisector of $BC$ . Points $B'$ and $C'$ are defined analagously. Prove that planes $ABC, AB'C', A'BC'$ abd $A'B'C$ share a common point.

Find all permutations $(a_1, a_2,...,a_{2021})$ of $(1,2,...,2021)$ , such that for every two positive integers $m$ and $n$ with difference bigger than $20^{21}$ , the following inequality holds: $GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|$ .

Each girl among $100$ girls has $100$ balls; there are in total $10000$ balls in $100$ colors, from each color there are $100$ balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has $100$ balls, colored in the $100$ distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has $100$ balls, colored in the $100$ colors.