There were $n{}$ positive integers. For each pair of those integers Boris wrote their arithmetic mean onto a blackboard and their geometric mean onto a whiteboard. It so happened that for each pair at least one of those means was integer. Prove that on at least one of the boards all the numbers are integer. Boris Frenkin

Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} + bx^{n-2}+ \dots$ has $n$ positive integer roots then there exist $a$ lines in the plane such that they have exactly $b$ intersection points. Is the baron’s theorem true?

Two circles $\alpha{}$ and $\beta{}$ with centers $A{}$ and $B{}$ respectively intersect at points $C{}$ and $D{}$. The segment $AB{}$ intersects $\alpha{}$ and $\beta{}$ at points $K{}$ and $L{}$ respectively. The ray $DK$ intersects the circle $\beta{}$ for the second time at the point $N{}$, and the ray $DL$ intersects the circle $\alpha{}$ for the second time at the point $M{}$. Prove that the intersection point of the diagonals of the quadrangle $KLMN$ coincides with the incenter of the triangle $ABC$. Konstantin Knop

There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does? Mikhail Svyatlovskiy

Does there exist a rectangle which can be cut into a hundred rectangles such that all of them are similar to the original one but no two are congruent? Mikhail Murashkin

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? Andrey Arzhantsev

A white bug sits in one corner square of a $1000$ × $n$ chessboard, where $n$ is an odd positive integer and $n > 2020$. In the two nearest corner squares there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach the opposite corner square by never visiting a square occupied or attacked by a bishop, and visiting every other square exactly once. Show that the number of ways for the bug to attain its goal does not depend on $n$.

Let us say that a circle intersects a quadrilateral properly if it intersects each of the quadrilateral’s sides at two distinct interior points. Is it true that for each convex quadrilateral there exists a circle which intersects it properly? Alexandr Perepechko

Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.) Boris Frenkin

Alice and Bob are playing the following game. Each turn Alice suggests an integer and Bob writes down either that number or the sum of that number with all previously written numbers. Is it always possible for Alice to ensure that at some moment among the written numbers there are at least a hundred copies of number 5? at least a hundred copies of number 10? Andrey Arzhantsev

The $X{}$ pentomino consists of five $1\times1$ squares where four squares are all adjacent to the fifth one. Is it possible to cut nine such pentominoes from an $8\times 8$ chessboard, not necessarily cutting along grid lines? (The picture shows how to cut three such $X{}$ pentominoes.) Alexandr Gribalko

Do there exist 100 positive distinct integers such that a cube of one of them equals the sum of the cubes of all the others? Mikhail Evdokimov

There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that: one of its angles is not greater than $60^\circ{}$; one of its angles is at least $120^\circ$. Maxim Didin

Each of the quadratic polynomials $P(x), Q(x)$ and $P(x)+Q(x)$ with real coefficients has a repeated root. Is it guaranteed that those roots coincide? Boris Frenkin

There were ten points $X_1, \ldots , X_{10}$ on a line in this particular order. Pete constructed an isosceles triangle on each segment $X_1X_2, X_2X_3,\ldots, X_9X_{10}$ as a base with the angle $\alpha{}$ at its apex. It so happened that all the apexes of those triangles lie on a common semicircle with diameter $X_1X_{10}$. Find $\alpha{}$. Egor Bakaev

A positive integer number $N{}$ is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number $N{}$ have? Sergey Tokarev

The sides of a triangle are divided by the angle bisectors into two segments each. Is it always possible to form two triangles from the obtained six segments? Lev Emelyanov

There are 101 coins in a circle, each weights 10g or 11g. Prove that there exists a coin such that the total weight of the $k{}$ coins to its left is equal to the total weight of the $k{}$ coins to its right where a) $k = 50$ and b) $k = 49$. Alexandr Gribalko

Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points? Sergey Dvoryaninov

A group of 8 players played several tennis tournaments between themselves using the single-elimination system, that is, the players are randomly split into pairs, the winners split into two pairs that play in semifinals, the winners of semifinals play in the final round. It so happened that after several tournaments each player had played with each other exactly once. Prove that each player participated in semifinals more than once; each player participated in at least one final. Boris Frenkin

There are $n{}$ stones in a heap. Two players play the game by alternatively taking either 1 stone from the heap or a prime number of stones which divides the current number of stones in the heap. The player who takes the last stone wins. For which $n{}$ does the first player have a strategy so that he wins no matter how the other player plays? Fedor Ivlev

There is an equilateral triangle with side $d{}$ and a point $P{}$ such that the distances from $P{}$ to the vertices of the triangle are positive numbers $a, b, c$. Prove that there exist a point $Q{}$ and an equilateral triangle with side $a{}$, such that the distances from $Q{}$ to the vertices of this triangle are $b, c, d$. Alexandr Evnin

The director of a Zoo has bought eight elephants numbered by $1, 2, \ldots , 8$. He has forgotten their masses but he remembers that each elephant starting with the third one has the mass equal to the sum of the masses of two preceding ones. Suddenly the director hears a rumor that one of the elephants has lost his mass. How can the director perform two weightings on balancing scales without weights to either find this elephant or make sure that this was just a rumor? (It is known that no elephant gained mass and no more than one elephant lost mass.) Alexandr Gribalko

In a room there are several children and a pile of 1000 sweets. The children come to the pile one after another in some order. Upon reaching the pile each of them divides the current number of sweets in the pile by the number of children in the room, rounds the result if it is not integer, takes the resulting number of sweets from the pile and leaves the room. All the boys round upwards and all the girls round downwards. The process continues until everyone leaves the room. Prove that the total number of sweets received by the boys does not depend on the order in which the children reach the pile. Maxim Didin

Does there exist a positive integer $n{}$ such that for any real $x{}$ and $y{}$ there exist real numbers $a_1, \ldots , a_n$ satisfying \[x=a_1+\cdots+a_n\text{ and }y=\frac{1}{a_1}+\cdots+\frac{1}{a_n}?\]Artemiy Sokolov

Let $M{}$ be the midpoint of the side $BC$ of the triangle $ABC$. The circle $\omega$ passes through $A{}$, touches the line $BC$ at $M{}$, intersects the side $AB$ at the point $D{}$ and the side $AC$ at the point $E{}$. Let $X{}$ and $Y{}$ be the midpoints of $BE$ and $CD$ respectively. Prove that the circumcircle of the triangle $MXY$ touches $\omega$. Alexey Doledenok

There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays? Ivan Mitrofanov

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. Fyodor Ivlev

Find at least one real number $A{}$ such that for any positive integer $n{}$ the distance between $\lceil A^n\rceil$ and the nearest square of an integer is equal to two. Dmitry Krekov

An integer $n > 2$ is given. Peter wants to draw $n{}$ arcs of length $\alpha{}$ of great circles on a unit sphere so that they do not intersect each other. Prove that for all $\alpha<\pi+2\pi/n$ it is possible; for all $\alpha>\pi+2\pi/n$ it is impossible; Ilya Bogdanov

The number $2021 = 43 \cdot 47$ is composite. Prove that if we insert any number of digits “8” between 20 and 21 then the number remains composite. Mikhail Evdikomov

There is an equilateral triangle $ABC$. Let $E, F$ and $K$ be points such that $E{}$ lies on side $AB$, $F{}$ lies on the side $AC$, $K{}$ lies on the extension of side $AB$ and $AE = CF = BK$. Let $P{}$ be the midpoint of the segment $EF$. Prove that the angle $KPC$ is right. Vladimir Rastorguev

A traveler arrived to an island where 50 natives lived. All the natives stood in a circle and each announced firstly the age of his left neighbour, then the age of his right neighbour. Each native is either a knight who told both numbers correctly or a knave who increased one of the numbers by 1 and decreased the other by 1 (on his choice). Is it always possible after that to establish which of the natives are knights and which are knaves? Alexandr Gribalko

In the center of each cell of a checkered rectangle $M{}$ there is a point-like light bulb. All the light bulbs are initially switched off. In one turn it is allowed to choose a straight line not intersecting any light bulbs such that on one side of it all the bulbs are switched off, and to switch all of them on. In each turn at least one bulb should be switched on. The task is to switch on all the light bulbs using the largest possible number of turns. What is the maximum number of turns if: $M$ is a square of size $21 \times 21$; $M$ is a rectangle of size $20 \times 21$? Alexandr Shapovalov

Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$. Nikolay Belukhov

A convex pentagon is partitioned into three triangles by nonintersecting diagonals. Is it possible for centroids of these triangles to lie on a common straight line? The same question for a non-convex pentagon. Alexandr Gribalko

Maria has a balance scale that can indicate which of its pans is heavier or whether they have equal weight. She also has 4 weights that look the same but have masses of 1001, 1002, 1004 and 1005g. Can Maria determine the mass of each weight in 4 weightings? The weights for a new weighing may be picked when the result of the previous ones is known. The Jury (For the senior paper) The same question when the left pan of the scale is lighter by 1g than the right one, so the scale indicates equality when the mass on the left pan is heavier by 1g than the mass on the right pan. Alexey Tolpygo

For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? Boris Frenkin

It is well-known that a quadratic equation has no more than 2 roots. Is it possible for the equation $\lfloor x^2\rfloor+px+q=0$ with $p\neq 0$ to have more than 100 roots? Alexey Tolpygo

Let $O{}$ be the circumcenter of an acute triangle $ABC$. Let $M{}$ be the midpoint of $AC$. The straight line $BO$ intersects the altitudes $AA_1{}$ and $CC_1{}$ at the points $H_a$ and $H_c$ respectively. The circumcircles of the triangles $BH_aA$ and $BH_cC$ have a second point of intersection $K{}$. Prove that $K{}$ lies on the straight line $BM$. Mikhail Evdokimov

Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers? Boris Frenkin

Let $AX$ and $BZ$ be altitudes of the triangle $ABC$. Let $AY$ and $BT$ be its angle bisectors. It is given that angles $XAY$ and $ZBT$ are equal. Does this necessarily imply that $ABC$ is isosceles? The Jury

Is it possible to split a square into 4 isosceles triangles such that no two are congruent? Is it possible to split an equilateral triangle into 4 isosceles triangles such that no two are congruent? Vladimir Rastorguev

There are several dominoes on a board such that each domino occupies two adjacent cells and none of the dominoes are adjacent by side or vertex. The bottom left and top right cells of the board are free. A token starts at the bottom left cell and can move to a cell adjacent by side: one step to the right or upwards at each turn. Is it always possible to move from the bottom left to the top right cell without passing through dominoes if the size of the board is a) $100 \times 101$ cells and b) $100 \times 100$ cells? Nikolay Chernyatiev