Found problems: 112
2022 JBMO Shortlist, N6
Find all positive integers $n$ for which there exists an integer multiple of $2022$ such that the sum of the squares of its digits is equal to $n$.
2021 JBMO Shortlist, C4
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 [i]Nikola Velov, Macedonia[/i]
2022 JBMO Shortlist, G4
Given is an equilateral triangle $ABC$ and an arbitrary point, denoted by $E$, on the line segment $BC$. Let $l$ be the line through $A$ parallel to $BC$ and let $K$ be the point on $l$ such that $KE$ is perpendicular to $BC$. The circle with centre $K$ and radius $KE$ intersects the sides $AB$ and $AC$ at $M$ and $N$, respectively. The line perpendicular to $AB$ at $M$ intersects $l$ at $D$, and the line perpendicular to $AC$ at $N$ intersects $l$ at $F$. Show that the point of intersection of the angle bisectors of angles $MDA$ and $NFA$ belongs to the line $KE$.
2022 Romania Team Selection Test, 2
Let $n \geq 2$ be an integer and let \[M=\bigg\{\frac{a_1 + a_2 + ... + a_k}{k}: 1 \le k \le n\text{ and }1 \le a_1 < \ldots < a_k \le n\bigg\}\] be the set of the arithmetic means of the elements of all non-empty subsets of $\{1, 2, ..., n\}$. Find \[\min\{|a - b| : a, b \in M\text{ with } a \neq b\}.\]
2022 Romania Team Selection Test, 4
Can every positive rational number $q$ be written as
$$\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},$$
where $a, b, c, d$ are all positive integers?
[i]Proposed by Dominic Yeo, UK[/i]
2021 JBMO Shortlist, C1
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 [i]Savinien Kreczman and Martin Rakovsky, France[/i]
2023 Azerbaijan JBMO TST, 4
There are $200$ boxes on the table. In the beginning, each of the boxes contains a positive integer (the integers are not necessarily distinct). Every minute, Alice makes one move. A move consists of the following. First, she picks a box $X$ which contains a number $c$ such that $c = a + b$ for some numbers $a$ and $b$ which are contained in some other boxes. Then she picks a positive integer $k > 1$. Finally, she removes $c$ from $X$ and replaces it with $kc$. If she cannot make any mobes, she stops. Prove that no matter how Alice makes her moves, she won't be able to make infinitely many moves.
2022 Bulgaria EGMO TST, 4
Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively
defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there
exists some $i \in \mathbb{N}$ with $a_i = m^2$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
2021 Balkan MO Shortlist, A4
Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $x_0 = (0, 0)$, and for every $n \ge 1$, his jump is
$x_n - x_{n - 1} = (\pm f(n), \pm g(n))$ or $(\pm g(n), \pm f(n)),$
with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0, 0)$ infinitely many times when
(a) $f, g$ are polynomials with integer coefficients?
(b) $f, g$ are any pair of functions from the positive integers to the integers?
2021 JBMO Shortlist, N2
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).
2020 JBMO Shortlist, 7
Prove that there doesn’t exist any prime $p$ such that every power of $p$ is a palindrome (a palindrome is a number that is read the same from the left as it is from the right; in particular, a number that ends in one or more zeros cannot be a palindrome).
2021 JBMO Shortlist, N4
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 [i]Boris Stanković, Bosnia and Herzegovina[/i]
2019 JBMO Shortlist, N1
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number
$x^p + y^p + z^p - x - y - z$
is a product of exactly three distinct prime numbers.
2012 Romania Team Selection Test, 4
Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.
2022 JBMO Shortlist, C2
Let $n \ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ the difference of the numbers written in its extremities and by $p_s$ the number of all other drawn segments which intersect $s$ in its interior.
Find the greatest $n$ for which Alex can write the numbers on the circle such that $p_s \le |d_s|$, for each drawn segment $s$.
2022 Azerbaijan BMO TST, C3
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]
2016 Balkan MO Shortlist, G2
Let $ABCD$ be a cyclic quadrilateral with $AB<CD$. The diagonals intersect at the point $F$ and lines $AD$ and $BC$ intersect at the point $E$. Let $K$ and $L$ be the orthogonal projections of $F$ onto lines $AD$ and $BC$ respectively, and let $M$, $S$ and $T$ be the midpoints of $EF$, $CF$ and $DF$ respectively. Prove that the second intersection point of the circumcircles of triangles $MKT$ and $MLS$ lies on the segment $CD$.
[i](Greece - Silouanos Brazitikos)[/i]
2022 Greece Team Selection Test, 4
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]
2020 JBMO Shortlist, 3
Find all triples of positive real numbers $(a, b, c)$ so that the expression
$M = \frac{(a + b)(b + c)(a + b + c)}{abc}$
gets its least value.
2021 JBMO Shortlist, N7
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 [i]Borche Joshevski, Macedonia[/i]
2010 Balkan MO, 1
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]
2020 Junior Balkаn MO, 2
Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic.
Proposed by [i]Theoklitos Parayiou, Cyprus[/i]
2018 Balkan MO Shortlist, A4
Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that:
$$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$
2010 Balkan MO Shortlist, A1
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]
2021 JBMO Shortlist, C2
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 [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]