Found problems: 606
2023 Junior Balkan Team Selection Tests - Moldova, 11
Find all prime $x,y$ and $z,$ such that $x^5 +y^3 -(x+y)^2=3z^3$
2021 Junior Balkan Team Selection Tests - Romania, P4
Let $n\geq 2$ be a positive integer. On an $n\times n$ board, $n$ rooks are placed in such a manner that no two attack each other. All rooks move at the same time and are only allowed to move in a square adjacent to the one in which they are located. Determine all the values ​​of $n$ for which there is a placement of the rooks so that, after a move, the rooks still do not attack each other.
[i]Note: Two squares are adjacent if they share a common side.[/i]
2017 Turkey EGMO TST, 5
In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?
[b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.
2023 Hong Kong Team Selection Test, Problem 3
A point $P$ lies inside an equilateral triangle $ABC$ such that $AP=15$ and $BP=8$. Find the maximum possible value of the sum of areas of triangles $ABP$ and $BCP$.
2021 Bolivian Cono Sur TST, 2
Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules:
$\cdot$ The frog can jump only in points of $M$
$\cdot$ The frog can't jump more than $1$ time over the same point.
$\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$
In how many ways the Frog can make his target?
2023 Thailand October Camp, 2
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
2023 USA EGMO Team Selection Test, 4
Let $ABC$ be a triangle with $AB+AC=3BC$. The $B$-excircle touches side $AC$ and line $BC$ at $E$ and $D$, respectively. The $C$-excircle touches side $AB$ at $F$. Let lines $CF$ and $DE$ meet at $P$. Prove that $\angle PBC = 90^{\circ}$.
[i]Ray Li[/i]
2021 Serbia JBMO TSTs, 2
Solve the following equation in natural numbers:
\begin{align*}
x^2=2^y+2021^z
\end{align*}
2019 JBMO Shortlist, A2
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 China Team Selection Test, 2
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
2010 Junior Balkan Team Selection Tests - Romania, 3
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that:
$$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.
2014 JBMO TST - Turkey, 1
Find all real values of $a$ for which the equation $x(x+1)^3=(2x+a)(x+a+1)$ has four distinct real roots.
2014 Serbia JBMO TST, 2
Let $a,b,c,d$ be the natural numbers such that
$2^a+4^b+5^c=2014^d.$
Find all $(a,b,c,d).$
2017 Turkey Team Selection Test, 8
In a triangle $ABC$ the bisectors through vertices $B$ and $C$ meet the sides $\left [ AC \right ]$ and $\left [ AB \right ]$ at $D$ and $E$ respectively. Let $I_{c}$ be the center of the excircle which is tangent to the side $\left [ AB \right ]$ and $F$ the midpoint of $\left [ BI_{c} \right ]$. If $\left | CF \right |^2=\left | CE \right |^2+\left | DF \right |^2$, show that $ABC$ is an equilateral triangle.
2024 Thailand TST, 2
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2023 ISL, G3
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD, PM$, and $BC$ are concurrent.
2022 Bulgaria JBMO TST, 1
Are there positive integers $a$, $b$, $c$ and $d$ such that:
a) $a^{2021} + b^{2023} = 11(c^{2022} + d^{2024})$;
b) $a^{2022} + b^{2022} = 11(c^{2022} + d^{2022})$?
2022 Serbia JBMO TST, 2
Let $I$ be the incenter, $A_1$ and $B_1$ midpoints of sides $BC$ and $AC$ of a triangle $\Delta ABC$. Denote by $M$ and $N$ the midpoints of the arcs $AC$ and $BC$ of circumcircle of $\Delta ABC$ which do contain the other vertex of the triangle. If points $M$, $I$ and $N$ are collinear prove that:
\begin{align*}
\angle AIB_1=\angle BIA_1=90^{\circ}
\end{align*}
2018 USA Team Selection Test, 1
Let $n \ge 2$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Prove that the $n^{\text{th}}$ smallest positive integer relatively prime to $n$ is at least $\sigma(n)$, and determine for which $n$ equality holds.
[i]Proposed by Ashwin Sah[/i]
2015 Chile TST Ibero, 3
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
2005 Germany Team Selection Test, 2
Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations
\[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\]
Prove the inequality
\[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]
2024 Thailand TST, 1
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
2018 Junior Balkan Team Selection Tests - Moldova, 2
Let $x$,$y$ be positive real numbers such that $\frac{1}{1+x+x^2}+\frac{1}{1+y+y^2}+\frac{1}{1+x+y}=1$.Prove that $xy=1.$
2018 Polish MO Finals, 3
Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that
$$f(f(x)+f(y))+cxy=f(x+y).$$
2018 Turkey Team Selection Test, 2
Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that
$$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$ for all real numbers $x,y$.