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

2018 Azerbaijan JBMO TST, 1
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: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2016 Azerbaijan JBMO TST, 2
Tags: geometry , TST , AZE JBMO TST
Let the angle bisectors of $\angle BAC,$ $\angle CBA,$ and $\angle ACB$ meets the circumcircle of $\triangle ABC$ at the points $M,N,$ and $K,$ respectively. Let the segments $AB$ and $MK$ intersects at the point $P$ and the segments $AC$ and $MN$ intersects at the point $Q.$ Prove that $PQ\parallel BC$

2015 Azerbaijan JBMO TST, 4
Tags: number theory , prime numbers , AZE JBMO TST
Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2018 Azerbaijan JBMO TST, 4
Tags: combinatorics , TST , AZE JBMO TST
An $n\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the obtained grid (i.e. the side of any unit cell) is colored black so that any unit cell of the given square has exactly one black side. Find [b]a)[/b] the smallest [b]b)[/b] the greatest possible number of black unit segments.

2015 Azerbaijan JBMO TST, 2
Tags: number theory , AZE JBMO TST , TST
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$

2018 Azerbaijan JBMO TST, 2
Tags: geometry , Balkan MO Shortlist , TST , AZE JBMO TST
Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

2022 Kosovo Team Selection Test, 2
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]

2022 JBMO Shortlist, A2
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]

2015 District Olympiad, 1
Tags: inequalities , algebra , TST , AZE JBMO TST
If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality: $$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$

2024 Azerbaijan JBMO TST, 4
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 $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

2014 JBMO Shortlist, 3
Tags: number theory , AZE JBMO TST
Find all integer solutions to the equation $x^2=y^2(x+y^4+2y^2)$ .

2023 Bulgaria 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]

2024 Junior Balkan Team Selection Tests - Moldova, 4
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.

2018 Balkan MO Shortlist, G1
Tags: geometry , Balkan MO Shortlist , TST , AZE JBMO TST
Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

2024 Azerbaijan JBMO TST, 3
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.)

2017 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , TST , AZE JBMO TST
Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

2018 Azerbaijan JBMO TST, 1
Tags: geometry , TST , AZE JBMO TST
Let $\triangle ABC$ be an acute triangle. Let us denote the foot of the altitudes from the vertices $A, B$ and $C$ to the opposite sides by $D, E$ and $F,$ respectively, and the intersection point of the altitudes of the triangle $ABC$ by $H.$ Let $P$ be the intersection of the line $BE$ and the segment $DF.$ A straight line passing through $P$ and perpendicular to $BC$ intersects $AB$ at $Q.$ Let $N$ be the intersection of the segment $EQ$ with the perpendicular drawn from $A.$ Prove that $N$ is the midpoint of segment $AH.$

2024 Junior Balkan Team Selection Tests - Romania, P2
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.

2023 Azerbaijan JBMO TST, 3
Tags: geometry , AZE JBMO TST , TST
Let $ABC$ be a triangle and let $\Omega$ denote the circumcircle of $ABC$. The foot of altitude from $A$ to $BC$ is $D$. The foot of altitudes from $D$ to $AB$ and $AC$ are $K;L$ , respectively. Let $KL$ intersect $\Omega$ at $X;Y$, and let $AD$ intersect $\Omega$ at $Z$. Prove that $D$ is the incenter of triangle $XYZ$

2014 JBMO Shortlist, 5
Tags: number theory , AZE JBMO TST , TST
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$

2014 JBMO Shortlist, 4
Tags: number theory , prime numbers , AZE JBMO TST
Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2018 Azerbaijan JBMO TST, 2
Tags: algebra , Inequality , TST , AZE JBMO TST
a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

2018 Azerbaijan JBMO TST, 3
Tags: number theory , TST , AZE JBMO TST
Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

2016 Azerbaijan JBMO TST, 1
Tags: inequalities , algebra , TST , AZE JBMO TST
If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality: $$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$

2018 Azerbaijan JBMO TST, 4
Tags: combinatorics , TST , AZE JBMO TST
In the beginning, there are $100$ cards on the table, and each card has a positive integer written on it. An odd number is written on exactly $43$ cards. Every minute, the following operation is performed: for all possible sets of $3$ cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by $2^{2018}.$ Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by $2^{2018}.$