Found problems: 32
2025 Azerbaijan Junior NMO, 1
A teacher creates a fraction using numbers from $1$ to $12$ (including $12$). He writes some of the numbers on the numerator, and writes $\times$ (multiplication) between each number. Then he writes the rest of the numbers in the denominator and also writes $\times$ between each number. There is at least one number both in numerator and denominator. The teacher ensures that the fraction is equal to the smallest possible integer possible.
What is this positive integer, which is also the value of the fraction?
2021 Azerbaijan Junior NMO, 3
$a,b,c $ are positive real numbers . Prove that
$\sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}} +\sqrt[7]{\frac{b}{c+a}+\frac{c}{b+a}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}}\geq 3$
2023 Azerbaijan National Mathematical Olympiad, 1
For any natural number, let's call the numbers formed from its digits and have the same "digit" arrangement with the initial number as the "partial numbers". For example, the partial numbers of $149$ are ${1, 4, 9, 14,19, 49, 149},$ and the partial numbers of $313$ are ${3, 1, 31,33, 13, 313}.$ Find all natural numbers whose partial numbers are all prime. Justify your opinion.
2025 Azerbaijan Junior NMO, 4
A $3\times3$ square is filled with numbers $1;2;3...;9$.The numbers inside four $2\times2$ squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
$\text{a)}$ $24,24,25,25$
$\text{b)}$ $20,23,26,29$
2021 Azerbaijan Junior NMO, 4
Initially, the numbers $1,1,-1$ written on the board.At every step,Mikail chooses the two numbers $a,b$ and substitutes them with $2a+c$ and $\frac{b-c}{2}$ where $c$ is the unchosen number on the board. Prove that at least $1$ negative number must be remained on the board at any step.
2024 Azerbaijan National Mathematical Olympiad, 2
Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$:
$$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$
$$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$
$$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.
2025 Azerbaijan Junior NMO, 2
Find all $4$ consecutive even numbers, such that the sum of their squares divides the square of their product.
2020 Azerbaijan National Olympiad, 4
There is a non-equilateral triangle $ABC$.Let $ABC$'s Incentri $I$.Point $D$ is on the $BC$ side.The circle drawn outside the triangle $IBD$ and $ICD$ intersects the sides $AB$ and $AC$ at points $E$ and $F.$The circle drawn outside the triangle $DEF$ intersects the sides $AB$ and $AC$ at $N$ and $M$.Prove that $EM\parallel FN $.
2024 Azerbaijan National Mathematical Olympiad, 4
A $9 \times 10$ board is divided into $90$ unit cells. There are certain rules for moving a non-standard chess queen from one square to another:
[list]
[*]The queen can only move along the column or row it is in each step.
[*]For any natural number $n$, if $x$ cells move made in $(2n-1)$th step, then $9-x$ cells move will be done in $(2n)$th step. The last cell it stops at during these steps is considered the visited cell.
[/list]
Is it possible for the queen to move from any square on the board and return to the square where it started after visiting all the squares of the board exactly once?
Note: At each step, the queen chooses the right, left, up, and down direction within the above condition can choose.
2021 Azerbaijan Junior NMO, 1
At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 22 \times 23$ in order to make it a perfect square?
2021 Azerbaijan Senior NMO, 3
In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$.
$\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$
2020 Azerbaijan National Olympiad, 5
$a,b,c$ are non-negative integers.
Solve: $a!+5^b=7^c$
[i]Proposed by Serbia[/i]
2018 Azerbaijan Junior NMO, 3
$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$
2023 Azerbaijan National Mathematical Olympiad, 4
Solve the following diophantine equation in the set of nonnegative integers:
$11^{a}5^{b}-3^{c}2^{d}=1$.
2020 JBMO Shortlist, 1
Determine whether there is a natural number $n$ for which $8^n + 47$ is prime.
2019 JBMO Shortlist, N6
$a,b,c$ are non-negative integers.
Solve: $a!+5^b=7^c$
[i]Proposed by Serbia[/i]
2023 Azerbaijan National Mathematical Olympiad, 2
Let $I$ be the incenter in the acute triangle $ABC.$ Rays $BI$ and $CI$ intersect the circumcircle of triangle $ABC$ at points $S$ and $T,$ respectively. The segment $ST$ intersects the sides $AB$ and $AC$ at points $K$ and $L,$ respectively. Prove that $AKIL$ is a rhombus.
2024 Azerbaijan National Mathematical Olympiad, 3
Find all the natural numbers $a, b, c$ satisfying the following equation:
$$a^{12} + 3^b = 1788^c$$.
2025 Azerbaijan Junior NMO, 5
For positive real numbers $x;y;z$ satisfying $0<x,y,z<2$, find the biggest value the following equation could acquire:
$$(2x-yz)(2y-zx)(2z-xy)$$
2024 Azerbaijan National Mathematical Olympiad, 1
Alice thinks about a natural number in her mind. Bob tries to find that number by asking him the following 10 questions:
[list]
[*]Is it divisible by 1?
[*]Is it divisible by 2?
[*]Is it divisible by 3?
[*]...
[*]Is it divisible by 9?
[*]Is it divisible by 10?
[/list]
Alice's answer to all questions except one was "yes". When she answers "no", she adds that "the greatest common factor of the number I have in mind and the divisor in the question you asked is 1”. According to this information, to which question did Alice answer "no"?
2024 Azerbaijan National Mathematical Olympiad, 5
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
2024 Azerbaijan Senior NMO, 3
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
2023 Azerbaijan National Mathematical Olympiad, 5
Baklavas with nuts are laid out on the table in a row at the Nowruz celebration. Kosa and Kechel saw this and decided to play a game. Kosa eats one baklava from either the beginning or the end of the row in each move. Kechel either doesn't touch anything in each move or chooses the baklava he wants and just eats the nut on it. They agree that the first Kosa will start the game and make $20$ moves in each step, and the Kechel will only make $1$ move in each step. If the last baklava eaten by the Kosa is a nut, he wins the game. It is given that the number of baklavas is a multiple of $20.$
$A)$ If the number of baklavas is $400,$ prove that Kosa will win the game regardless of which strategy Kechel chooses.
$B)$ Is it always true that no matter how many baklavas there are and what strategy Kechel chooses, Kosa will always win the game?
2022 Azerbaijan Junior National Olympiad, A3
Let $x,y,z \in \mathbb{R}^{+}$ and $x^2+y^2+z^2=x+y+z$. Prove that
$$x+y+z+3 \ge 6 \sqrt[3]{\frac{xy+yz+zx}{3}}$$
2021 Azerbaijan Junior NMO, 5
In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$.
$\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$