Found problems: 606
2016 Chile TST IMO, 2
There are 2016 points near a line such that the distance from each point to the line is less than 1 cm, and the distance between any two points is always greater than 2 cm. Prove that there exist two points whose distance is at least 17 meters.
2017 Turkey EGMO TST, 1
Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation:
$3^m5^k=n^3+125$
2010 USA Team Selection Test, 9
Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and
\[\binom{3k}{k} \equiv 1 \pmod{p}.\]
2018 Bulgaria JBMO TST, 3
Find all positive integers $n$ such that the number
$$n^6 + 5n^3 + 4n + 116$$
is the product of two or more consecutive numbers.
2016 Azerbaijan JBMO TST, 3
Find all the pime numbers $(p,q)$ such that :
$p^{3}+p=q^{2}+q$
2024 Moldova Team Selection Test, 4
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2008 Junior Balkan Team Selection Tests - Moldova, 5
Find all natural pairs $ (x,y)$, such that $ x$ and $ y$ are relative prime and satisfy equality: $ 2x^2 \plus{} 5xy \plus{} 3y^2 \equal{} 41x \plus{} 62y \plus{} 21$.
2020 Azerbaijan IZHO TST, 3
Find all functions $u:R\rightarrow{R}$ for which there exists a strictly monotonic function $f:R\rightarrow{R}$ such that $f(x+y)=f(x)u(y)+f(y)$
for all $x,y\in{\mathbb{R}}$
2023 JBMO Shortlist, C2
There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other.
(Two rooks are not threatening each other if there is a block lying between them.)
2024 Israel TST, P1
Let $G$ be a connected (simple) graph with $n$ vertices and at least $n$ edges. Prove that it is possible to color the vertices of $G$ red and blue, so that the following conditions hold:
i. There is at least one vertex of each color,
ii. There is an even number of edges connecting a red vertex to a blue vertex, and
iii. If all such edges are deleted, one is left with two connected graphs.
2021 Romania Team Selection Test, 3
The external bisectors of the angles of the convex quadrilateral $ABCD$ intersect each other in $E,F,G$ and $H$ such that $A\in EH, \ B\in EF, \ C\in FG, \ D\in GH$. We know that the perpendiculars from $E$ to $AB$, from $F$ to $BC$ and from $G$ to $CD$ are concurrent. Prove that $ABCD$ is cyclic.
Russian TST 2018, P2
Let $\mathcal{F}$ be a finite family of subsets of some set $X{}$. It is known that for any two elements $x,y\in X$ there exists a permutation $\pi$ of the set $X$ such that $\pi(x)=y$, and for any $A\in\mathcal{F}$ \[\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.\]A bear and crocodile play a game. At a move, a player paints one or more elements of the set $X$ in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in $\mathcal{F}$ in his own color loses. If this does not happen and all the elements of $X$ have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.
2023 Azerbaijan JBMO TST, 2
Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that
$$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$
Proposed by [i]Petar Filipovski, Macedonia[/i]
2023 Azerbaijan IZhO TST, 1
In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$.
Proposed by Fatemeh Sajadi
2021 Romania Team Selection Test, 4
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following relationship for all real numbers $x$ and $y$\[f(xf(y)-f(x))=2f(x)+xy.\]
1998 China Team Selection Test, 1
Find $k \in \mathbb{N}$ such that
[b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots, \left( \begin{array}{c}
n\\
j + k - 1\end{array} \right)$ forms an arithmetic progression.
[b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots , \left( \begin{array}{c}
n\\
j + k - 2\end{array} \right)$ forms an arithmetic progression.
Find all $n$ which satisfies part [b]b.)[/b]
2024 Turkey Team Selection Test, 2
Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that
$$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$
for all real numbers $x,y$
2014 JBMO TST - Turkey, 4
Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m,$ $1 \le m \le 1000$ and after that directs $m$ undirected edges of $G$. The game ends when all edges are directed. If there is some directed cycle in $G$ Alice wins. Determine whether Alice has a winning strategy.
2023 Hong Kong Team Selection Test, Problem 2
Let $n$ be a positive integer. Show that if p is prime dividing $5^{4n}-5^{3n}+5^{2n}-5^{n}+1$, then $p\equiv 1 \;(\bmod\; 4)$.
2020 USA IMO Team Selection Test, 4
For a finite simple graph $G$, we define $G'$ to be the graph on the same vertex set as $G$, where for any two vertices $u \neq v$, the pair $\{u,v\}$ is an edge of $G'$ if and only if $u$ and $v$ have a common neighbor in $G$.
Prove that if $G$ is a finite simple graph which is isomorphic to $(G')'$, then $G$ is also isomorphic to $G'$.
[i]Mehtaab Sawhney and Zack Chroman[/i]
2025 Junior Balkan Team Selection Tests - Romania, P3
Let $n\geqslant 3$ be a positiv integer. Ana chooses the positive integers $a_1,a_2,\ldots,a_n$ and for any non-empty subset $A\subseteq\{1,2,\ldots,n\}$ she computes the sum \[s_A=\sum_{k
\in A}a_k.\]She orders these sums $s_1\leqslant s_2\leqslant\cdots\leqslant s_{2^n-1}.$ Prove that there exists a subset $B\subseteq\{1,2,\ldots,2^n-1\}$ with $2^{n-2}+1$ elements such that, regardless of the integers $a_1,a_2,\ldots,a_n$ chosen by Ana, these can be determined by only knowing the sums $s_i$ with $i\in B.$
2023 Israel TST, P3
Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which
\[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\]
holds for any $x,y\in \mathbb{Z}$.
2023 Chile TST Ibero., 4
Let \(ABC\) be a triangle with \(AB < AC\) and let \(\omega\) be its circumcircle. Let \(M\) denote the midpoint of side \(BC\) and \(N\) the midpoint of arc \(BC\) of \(\omega\) that contains \(A\). The circumcircle of triangle \(AMN\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Prove that \(BP = CQ\).
2015 Chile TST Ibero, 1
Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$:
\[
f(g(n)) = n + 2015,
\]
\[
g(f(n)) = n^2 + 2015.
\]
2018 Balkan MO Shortlist, A1
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:
$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$