UA-UMO 2023

Oleksiy placed positive integers in the cells of the $8\times 8$ chessboard. For each pair of adjacent-by-side cells, Fedir wrote down the product of the numbers in them and added all the products. Oleksiy wrote down the sum of the numbers in each pair of adjacent-by-side cells and multiplied all the sums. It turned out that the last digits of both numbers are equal to $1$ . Prove that at least one of the boys made a mistake in the calculation.

For example, for a square $3\times 3$ and the arrangement of numbers shown below, Fedir would write the following numbers: $2, 6, 8, 24, 15, 35, 2, 6, 8, 20, 18, 42$ , and their sum ends with a digit $6$ ; Oleksiy would write the following numbers: $3, 5, 6, 10, 8, 12, 3, 5, 6, 9, 9, 13$ , and their product ends with a digit $0$ .

\begin{tabular}{| c| c | c |} \hline 1 & 2 & 3 \hline 2 & 4 & 6 \hline 3 & 5 & 7 \hline \end{tabular}

Proposed by Oleksiy Masalitin and Fedir Yudin

In one country, a one-round tennis tournament was held (everyone played with everyone exactly once). Participants received $1$ point for winning a match, and $0$ points for losing. There are no draws in tennis. At the end of the tournament, Oleksiy saw the number of points scored by each participant, as well as the schedule of all the matches in the tournament, which showed the pairs of players, but not the winners. He chooses matches one by one in any order he wants and tries to guess the winner, after which he is told if he is correct. Prove that Oleksiy can act in such a way that he is guaranteed to guess the winners of more than half of the matches.

Proposed by Oleksiy Masalitin

Positive integers $x, y$ satisfy the following conditions: $$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$ Show that $x = y$ .

Here $\{x\}$ denotes the fractional part of $x$ . For example, $\{3.14\} = 0.14$ .

Proposed by Anton Trygub

Point $T$ is chosen in the plane of a rhombus $ABCD$ so that $\angle ATC + \angle BTD = 180^\circ$ , and circumcircles of triangles $ATC$ and $BTD$ are tangent to each other. Show that $T$ is equidistant from diagonals of $ABCD$ .

Proposed by Fedir Yudin

Do there exist $10$ real numbers, not all of which are equal, each of which is equal to the square of the sum of the remaining $9$ numbers?

Proposed by Bogdan Rublov

In a convex pentagon $ABCDE$ the following conditions hold : $AB \parallel CD$ , $BC \parallel DE$ , and $\angle BAE = \angle AED$ . Prove that $AB + BC = CD + DE$ Proposed by Anton Trygub

The country has $n \ge 3$ airports, some pairs of which are connected by bidirectional flights. Every day, the government closes the airport with the strictly highest number of flights going out of it. What is the maximum number of days this can continue?

Proposed by Fedir Yudin

You are given a set of $m$ integers, all of which give distinct remainders modulo some integer $n$ . Show that for any integer $k \le m$ you can split this set into $k$ nonempty groups so that the sums of elements in these groups are distinct modulo $n$ .

Proposed by Anton Trygub

$n \ge 4$ real numbers are arranged in a circle. It turned out that for any four consecutive numbers $a, b, c, d$ , that lie on the circle in this order, holds $a+d = b+c$ . For which $n$ does it follow that all numbers on the circle are equal?

Proposed by Oleksiy Masalitin

Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$ , where $a_{102} = a_1$ . What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$ ?

Proposed by Oleksiy Masalitin

You are given an acute triangle $ABC$ with circumcircle $\omega$ . Points $F$ on $AC$ , $E$ on $AB$ and $P, Q$ on $\omega$ are chosen so that $\angle AFB = \angle AEC = \angle APE = \angle AQF = 90^\circ$ . Show that lines $BC, EF, PQ$ are concurrent or parallel.

Proposed by Fedir Yudin

Find the smallest real number $C$ , such that for any positive integers $x \neq y$ holds the following: $$\min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C$$ Here $\{x\}$ denotes the fractional part of $x$ . For example, $\{3.14\} = 0.14$ .

Proposed by Anton Trygub

A point $O$ lies inside $\triangle ABC$ so that $\angle BOC=90-\angle BAC$ . Let $BO, CO$ meet $AC, AB$ at $K, L$ . Points $K_1, L_1$ lie on the segments $CL, BK$ so that $K_1B=K_1K$ and $L_1C=L_1L$ . If $M$ is the midpoint of $BC$ , then prove that $\angle K_1ML_1=90^{o}$ .

Proposed by Anton Trygub

You are given $n \ge 2$ distinct positive integers. Let's call a pair of these integers elegant if their sum is an integer power of $2$ . For every $n$ find the largest possible number of elegant pairs.

Proposed by Oleksiy Masalitin

What is the largest possible number of edges in a graph on $2n$ nodes, if there exists exactly one way to split its nodes into $n$ pairs so that the nodes from each pair are connected by an edge?

Proposed by Anton Trygub