This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 52

2023 JBMO Shortlist, G2
Tags: JBMO Shortlist , geometry , AZE JBMO TST
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.

2017 Azerbaijan JBMO TST, 1
Tags: inequalities , algebra , TST , AZE JBMO TST
Let $x,y,z,t$ be positive numbers.Prove that $\frac{xyzt}{(x+y)(z+t)}\leq\frac{(x+z)^2(y+t)^2}{4(x+y+z+t)^2}.$

2018 Azerbaijan JBMO TST, 3
Tags: number theory , TST , AZE JBMO TST
Find all nonnegative integers $(x,y,z,u)$ with satisfy the following equation: $2^x + 3^y + 5^z = 7^u.$

2024 Junior Balkan Team Selection Tests - Moldova, 10
Tags: JBMO Shortlist , algebra , AZE JBMO TST , Inequality , TST
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

2022 Azerbaijan JBMO TST, A2
Tags: algebra , TST , Inequality , AZE JBMO TST
For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that: $$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$

2022 Greece JBMO TST, 1
Tags: number theory , AZE JBMO TST , JBMO Shortlist , TST
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

2024 Azerbaijan JBMO TST, 1
Tags: JBMO Shortlist , number theory , AZE JBMO TST
Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$. Find the minimum possible value of $|A|$.

2015 Azerbaijan JBMO TST, 3
Tags: geometry , circumcircle , AZE JBMO TST
Let $ABC$ be a triangle such that $AB$ is not equal to $AC$. Let $M$ be the midpoint of $BC$ and $H$ be the orthocenter of triangle $ABC$. Let $D$ be the midpoint of $AH$ and $O$ the circumcentre of triangle $BCH$. Prove that $DAMO$ is a parallelogram.

2023 Azerbaijan JBMO TST, 1
Tags: number theory , LCM , Junior , Balkan , shortlist , AZE JBMO TST , Inequality
Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

2014 JBMO Shortlist, 1
Tags: number theory , AZE JBMO TST
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]

2016 Azerbaijan JBMO TST, 3
Tags: combinatorics , TST , AZE JBMO TST
All cells of the $m\times n$ table are colored either white or black such that all corner cells of any rectangle containing the cells of this table with sides greater than one cell are not the same color. For values $m = 2, 3, 4,$ find all $n$ such that the mentioned coloring is possible.

2016 Azerbaijan JBMO TST, 3
Tags: number theory , TST , AZE JBMO TST
Find all the pime numbers $(p,q)$ such that : $p^{3}+p=q^{2}+q$

2022 JBMO Shortlist, N2
Tags: number theory , LCM , Junior , Balkan , shortlist , AZE JBMO TST , Inequality
Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

2023 JBMO Shortlist, C2
Tags: JBMO Shortlist , combinatorics , AZE JBMO TST , TST
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.)

2023 Azerbaijan JBMO TST, 2
Tags: algebra , JBMO Shortlist , TST , AZE JBMO TST
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]

2018 Balkan MO Shortlist, A1
Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2019 JBMO Shortlist, A2
Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2024 Azerbaijan JBMO TST, 2
Tags: JBMO Shortlist , geometry , AZE JBMO TST
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.

2015 Azerbaijan JBMO TST, 2
Tags: number theory , AZE JBMO TST
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]

2023 JBMO Shortlist, N3
Tags: JBMO Shortlist , number theory , AZE JBMO TST
Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$. Find the minimum possible value of $|A|$.

2016 Azerbaijan JBMO TST, 4
Tags: combinatorics , TST , AZE JBMO TST
There are three stacks of tokens on the table: the first contains $a,$ the second contains $b,$ and the third contains $c$ where $a \ge b \ge c.$ Players $A$ and $B$ take turns playing a game of swapping tokens. $A$ starts first. On each turn, the player who gets his turn chooses two stacks, then takes at least one token from the stack with the lowest number of tokens and places them on the stack with the highest number of tokens. If the number of tokens in the two piles he/she chooses is equal, then he/she takes at least one token from any of them and puts it in the other. If only one pile remains after a player's move, that player is considered a winner. At what values of $a, b, c$ who has the winning strategy ($A$ or $B$)?

2022 Azerbaijan JBMO TST, N1
Tags: number theory , AZE JBMO TST , JBMO Shortlist , TST
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

2023 JBMO Shortlist, A5
Tags: JBMO Shortlist , algebra , AZE JBMO TST , Inequality , TST
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

2021 JBMO Shortlist, N1
Tags: number theory , AZE JBMO TST , JBMO Shortlist , TST
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

2015 Azerbaijan JBMO TST, 4
Tags: number theory , AZE JBMO TST
Find all integer solutions to the equation $x^2=y^2(x+y^4+2y^2)$ .