JBMO-SL 2021

Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for which the following property holds: If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$ ), then $x_1 = . . . = x_n = 0$ .

Proposed by Vincent Jugé and Théo Lenoir, France

Let $n$ be a positive integer. A finite set of integers is called $n$ -divided if there are exactly $n$ ways to partition this set into two subsets with equal sums. For example, the set $\{1, 3, 4, 5, 6, 7\}$ is $2$ -divided because the only ways to partition it into two subsets with equal sums is by dividing it into $\{1, 3, 4, 5\}$ and $\{6, 7\}$ , or $\{1, 5, 7\}$ and $\{3, 4, 6\}$ . Find all the integers $n > 0$ for which there exists a $n$ -divided set.

Proposed by Martin Rakovsky, France

In Mathcity, there are infinitely many buses and infinitely many stations. The stations are indexed by the powers of $2: 1, 2, 4, 8, 16, ...$ Each bus goes by finitely many stations, and the bus number is the sum of all the stations it goes by. For simplifications, the mayor of Mathcity wishes that the bus numbers form an arithmetic progression with common difference $r$ and whose first term is the favourite number of the mayor. For which positive integers $r$ is it always possible that, no matter the favourite number of the mayor, given any $m$ stations, there is a bus going by all of them?

Proposed by Savinien Kreczman and Martin Rakovsky, France

Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.

Proposed by Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina

Attachments:

We have a set of $343$ closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least $1$ and at most $7$ . No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or vice versa. It is allowed to choose a triple of positive integers $(b; y; r) \in \{1; 2; ...; 7\}^3$ and flip all the jars whose number of blue, yellow and red marbles differ by not more than $1$ from $b, y, r$ , respectively. After $n$ moves all the jars turned out to be with the caps down. Find the number of all possible values of $n$ , if $n \le 2021$ .

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?

Proposed by Nikola Velov, Macedonia

Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. Determine the winner for all cases of $m$ and $n$ where $m, n \ge 3$ .

Proposed by Toghrul Abbasov, Azerbaijan

Attachments:

Let $P$ be an interior point of the isosceles triangle $ABC$ with $\hat{A} = 90^{\circ}$ . If $$\widehat{PAB} + \widehat{PBC} + \widehat{PCA} = 90^{\circ},$$ prove that $AP \perp BC$ .

Proposed by Mehmet Akif Yıldız, Turkey

Let $ABC$ be an acute triangle with circumcircle $\omega$ and circumcenter $O$ . The perpendicular from $A$ to $BC$ intersects $BC$ and $\omega$ at $D$ and $E$ , respectively. Let $F$ be a point on the segment $AE$ , such that $2 \cdot FD = AE$ . Let $l$ be the perpendicular to $OF$ through $F$ . Prove that $l$ , the tangent to $\omega$ at $E$ , and the line $BC$ are concurrent.

Proposed by Stefan Lozanovski, Macedonia

Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$ . Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$ . On the extension of $CD$ , beyond the point $D$ , we pick a point $Z$ such that $MZ = \frac{AE}{2}$ . Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$ . The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$ . If $I$ is the point of intersection of $BZ$ with $AE$ , prove that the lines $BL$ and $CI$ intersect on the line $AZ$ .

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ . Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$ , respectively, such that $PQ$ is parallel to $BC$ . Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$ . The segments $BQ$ and $CP$ intersect at $S$ . The line $LS$ intersects $\omega$ at $K$ . Prove that $\angle BKP = \angle CKQ$ .

Proposed by Ervin Macić, Bosnia and Herzegovina

Find all positive integers $a, b, c$ such that $ab + 1$ , $bc + 1$ , and $ca + 1$ are all equal to factorials of some positive integers.

Proposed by Nikola Velov, Macedonia

The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$ . a) Determine the smallest and the largest possible values of $xy + yz - xz$ . b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).

Dragos, the early ruler of Moldavia, and Maria the Oracle play the following game. Firstly, Maria chooses a set $S$ of prime numbers. Then Dragos gives an infinite sequence $x_1, x_2, ...$ of distinct positive integers. Then Maria picks a positive integer $M$ and a prime number $p$ from her set $S$ . Finally, Dragos picks a positive integer $N$ and the game ends. Dragos wins if and only if for all integers $n \ge N$ the number $x_n$ is divisible by $p^M$ ; otherwise, Maria wins. Who has a winning strategy if the set S must be: $\hspace{5px}$ a) finite; $\hspace{5px}$ b) infinite?

Proposed by Boris Stanković, Bosnia and Herzegovina

Find all pairs of integers $(x, y)$ such that $x^2 + 5y^2 = 2021y$ .

Given a positive integer $n \ge 2$ , we define $f(n)$ to be the sum of all remainders obtained by dividing $n$ by all positive integers less than $n$ . For example dividing $5$ with $1, 2, 3$ and $4$ we have remainders equal to $0, 1, 2$ and $1$ respectively. Therefore $f(5) = 0 + 1 + 2 + 1 = 4$ . Find all positive integers $n \ge 3$ such that $f(n) = f(n - 1) + (n - 2)$ .

Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$ . Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$ ; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$ ; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$ . Who has a winning strategy?

Proposed by Borche Joshevski, Macedonia