# [Solutions] Caucasus Mathematical Olympiad 2021

## Juniors

1. Let $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$.
2. In a triangle $ABC$ let $K$ be a point on the median $BM$ such that $CK=CM$. It appears that $\angle CBM = 2 \angle ABM$. Prove that $BC=MK$.
3. We have $n>2$ non-zero integers such that each one of them is divisible by the sum of the other $n-1$ numbers. Prove that the sum of all the given numbers is zero.
4. A square grid $2n \times 2n$ is constructed of matches (each match is a segment of length 1). By one move Peter can choose a vertex which (at this moment) is the endpoint of 3 or 4 matches and delete two matches whose union is a segment of length 2. Find the least possible number of matches that could remain after a number of Peter's moves.
5. Let $a, b, c$ be positive integers such that the product$$\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a)$$is a perfect square. Prove that the product$$\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a)$$is also a perfect square.
6. A row of $2021$ balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?
7. An acute triangle $ABC$ is given. Let $AD$ be its altitude, let $H$ and $O$ be its orthocenter and its circumcenter, respectively. Let $K$ be the point on the segment $AH$ with $AK=HD$; let $L$ be the point on the segment $CD$ with $CL=DB$. Prove that line $KL$ passes through $O$.
8. Let us call a set of positive integers nice, if its number of elements is equal to the average of all its elements. Call a number $n$ amazing, if one can partition the set $\{1,2,\ldots,n\}$ into nice subsets.
a) Prove that any perfect square is amazing.
b) Prove that there exist infinitely many positive integers which are not amazing.

## Seniors

1. Integers from 1 to 100 are placed in a row in some order. Let us call a number large-right, if it is greater than each number to the right of it; let us call a number large-left, is it is greater than each number to the left of it. It appears that in the row there are exactly $k$ large-right numbers and exactly $k$ large-left numbers. Find the maximal possible value of $k$.
2. We have $n>2$ non-zero integers such that each one of them is divisible by the sum of the other $n-1$ numbers. Prove that the sum of all the given numbers is zero.
3. Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked. (Recall that the vertices of a triangle are not collinear.)
4. In an acute triangle $ABC$ let $AH_a$ and $BH_b$ be altitudes. Let $H_aH_b$ intersect the circumcircle of $ABC$ at $P$ and $Q$. Let $A'$ be the reflection of $A$ in $BC$, and let $B'$ be the reflection of $B$ in $CA$. Prove that $A', B'$, $P$, $Q$ are concyclic.
5. A triangle $\Delta$ with sidelengths $a\leq b\leq c$ is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of $\Delta$. Prove that $b^2>ac$.
6. A row of $2021$ balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?
7. $4$ tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens. Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi$. Prove that $\Pi$ is a square.
8. An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots$ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots$ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots$, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to $1$.
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