Olympiad problems

2025 Azerbaijan Junior NMO, 3

Alice and Bob take turns taking balloons from a box containing infinitely many balloons. In the first turn, Alice takes $k_1$ amount of balloons, where $\gcd(30;k_1)\neq1$. Then, on his first turn, Bob takes $k_2$ amount of ballons where $k_1<k_2<2k_1$. After first turn, Alice and Bob alternately takes as many balloons as his/her partner has. Is it possible for Bob to take $k_2$ amount of balloons at first, such that after a finite amount of turns, one of them have a number of balloons that is a multiple of $2025^{2025}$?

2023 Azerbaijan National Mathematical Olympiad, 3

Tags: algebra , AZE JUNIOR NMO

Find all the real roots of the system of equations: $$ \begin{cases} x^3+y^3=19 \\ x^2+y^2+5x+5y+xy=12 \end{cases} $$

2021 Azerbaijan Junior NMO, 2

Tags: number theory , JBMO Shortlist , AZE JUNIOR NMO

Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.

2017 Azerbaijan Junior National Olympiad, P2

Tags: number theory , Sequence , AZE JUNIOR NMO

For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order:$4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your answer)

2020 Azerbaijan National Olympiad, 1

Tags: combinatorics , AZE JUNIOR NMO

$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.[b]Example:[/b]$\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers?

2015 Azerbaijan National Olympiad, 3

Tags: algebra , polynomial , AZE JUNIOR NMO

Find all polynomials $P(x)$ with real coefficents such that \[P(P(x))=(x^2+x+1)\cdot P(x)\] where $x \in \mathbb{R}$

2025 Azerbaijan Junior NMO, 6

Tags: geometry , AZE JUNIOR NMO

Let $T$ be a point outside circle $\omega$ centered at $O$. Tangents from $T$ to $\omega$ touch $\omega$ at $A;B$. Line $TO$ intersects bigger $AB$ arc at $C$.The line drawn from $T$ parallel to $AC$ intersects $CB$ at $E$. Ray $TE$ intersects small $BC$ arc at $F$. Prove that the circumcircle of $OEF$ is tangent to $\omega$.