Found problems: 606
2021 Greece Junior Math Olympiad, 3
Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.
2018 Iran Team Selection Test, 4
Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$.
Prove that there exist infinitely many positive integers which they are not "useful but not optimized".
(e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number)
[i]Proposed by Mohsen Jamali[/i]
2016 Chile TST IMO, 3
A set \( A \) of integers is said to be \textit{admissible} if it satisfies the property:
\[
\text{If } x, y \in A, \text{ then } x^2 + kxy + y^2 \in A \text{ for all } k \in \mathbb{Z}.
\]
Determine all pairs \( (m, n) \) of nonzero integers such that the only admissible set containing both \( m \) and \( n \) is the set of all integers.
2024 Azerbaijan IMO TST, 1
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.
[i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]
2021 Romania Team Selection Test, 2
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$
2017 China Team Selection Test, 2
In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.