Found problems: 606
2021 China Team Selection Test, 5
Determine all $ f:R\rightarrow R $ such that
$$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$
2016 USA Team Selection Test, 3
Let $p$ be a prime number. Let $\mathbb F_p$ denote the integers modulo $p$, and let $\mathbb F_p[x]$ be the set of polynomials with coefficients in $\mathbb F_p$. Define $\Psi : \mathbb F_p[x] \to \mathbb F_p[x]$ by \[ \Psi\left( \sum_{i=0}^n a_i x^i \right) = \sum_{i=0}^n a_i x^{p^i}. \] Prove that for nonzero polynomials $F,G \in \mathbb F_p[x]$, \[ \Psi(\gcd(F,G)) = \gcd(\Psi(F), \Psi(G)). \] Here, a polynomial $Q$ divides $P$ if there exists $R \in \mathbb F_p[x]$ such that $P(x) - Q(x) R(x)$ is the polynomial with all coefficients $0$ (with all addition and multiplication in the coefficients taken modulo $p$), and the gcd of two polynomials is the highest degree polynomial with leading coefficient $1$ which divides both of them. A non-zero polynomial is a polynomial with not all coefficients $0$. As an example of multiplication, $(x+1)(x+2)(x+3) = x^3+x^2+x+1$ in $\mathbb F_5[x]$.
[i]Proposed by Mark Sellke[/i]
2013 Junior Balkan Team Selection Tests - Romania, 3
Let $ABCD$ be a cyclic quadrilateral and $\omega_1, \omega_2$ the incircles of triangles $ABC$ and $BCD$. Show that the common external tangent line of $\omega_1$ and $\omega_2$, the other one than $BC$, is parallel with $AD$
2019 Moldova Team Selection Test, 3
On the table there are written numbers $673, 674, \cdots, 2018, 2019.$ Nibab chooses arbitrarily three numbers $a,b$ and $c$, erases them and writes the number $\frac{\min(a,b,c)}{3}$, then he continues in an analogous way. After Nibab performed this operation $673$ times, on the table remained a single number $k$. Prove that $k\in(0,1).$
2023 Indonesia TST, 1
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.
2024 Azerbaijan IZhO TST, 2
Find all positive integers $n$ such that one can place checkers on a $n\times n$ checkerboard such that any square chosen from the checkerboard has exactly $2$ adjacent squares with checkers on it. Two squares are considered adjacent if they both share a common side
2013 Chile TST Ibero, 1
Prove that the equation
\[
x^z + y^z = z^z
\]
has no solutions in postive integers.
2014 Chile TST Ibero, 1
Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$:
\[
f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.
\]
Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.
2023 Junior Balkan Team Selection Tests - Romania, P2
Given is a triangle $ABC$. Let the points $P$ and $Q$ be on the sides $AB, AC$, respectively, so that $AP=AQ$, and $PQ$ passes through the incenter $I$. Let $(BPI)$ meet $(CQI)$ at $M$, $PM$ meets $BI$ at $D$ and $QM$ meets $CI$ at $E$. Prove that the line $MI$ passes through the midpoint of $DE$.
2024 Junior Macedonian Mathematical Olympiad, 5
The shapes in the image consist of six unit cubes. Which of the following 3D objects can be filled up with the aforementioned shapes:
a) a cube with side length $3$, from which one edge has been removed (i.e. three layers of the shape [img]https://i.imgur.com/vUqgHS2.png[/img] )?
b) a rectangular prism of size $5 \times 4 \times 3$, from which two edges of length $3$ have been removed from one of the $5 \times 3$ sides (i.e. three layers of the shape [img]https://imgur.com/W4pfEfz.png[/img] )?
We can use each of shapes at most once, no two shapes can overlap, nor protrude from the 3D object and every unit cube of the 3D object must be covered by a unit cube of one of the constituent shapes.
[center][img]https://imgur.com/evAmuep.png[/img][/center]
[i]Proposed by Ilija Jovčeski[/i]
2020 Bulgaria Team Selection Test, 2
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
2024 Chile TST Ibero., 5
Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.
2023 Israel TST, P1
Toph wants to tile a rectangular $m\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?
2024 Thailand TST, 2
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2020 Vietnam Team Selection Test, 5
Find all positive integers $k$, so that there are only finitely many positive odd numbers $n$ satisfying $n~|~k^n+1$.
2023 Hong Kong Team Selection Test, Problem 3
Let $n\ge 4$ be a positive integer. Consider any set $A$ formed by $n$ distinct real numbers such that the following condition holds: for every $a\in A$, there exist distinct elements $x, y, z \in A$ such that $\left| x-a \right|, \left| y-a \right|, \left| z-a \right| \ge 1$. For each $n$, find the greatest real number $M$ such that $\sum_{a\in A}^{}\left| a \right|\ge M$.
2010 Romania Team Selection Test, 2
Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$.
[i]Cezar Lupu & Vlad Matei[/i]
2023 Romania Team Selection Test, P1
Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.
2023 Junior Balkan Team Selection Tests - Moldova, 9
Let $ AD $, $ BE $ and $ CF $ be the altitudes of $ \Delta ABC $. The points $ P, \, \, Q, \, \, R $ and $ S $ are the feet of the perpendiculars drawn from the point $ D $ on the segments $ BA $, $ BE $, $ CF $ and $ CA $, respectively. Prove that the points $ P, \, \, Q, \, \, R $ and $ S $ are collinear.
2024 Azerbaijan BMO TST, 5
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 $\max(a_1,a_{2023})\ge 507$.
2024 Junior Balkan Team Selection Tests - Romania, P2
Let $ABC$ be a scalene triangle, with circumcircle $\omega$ and incentre $I.{}$ The tangent line at $C$ to $\omega$ intersects the line $AB$ at $D.{}$ The angle bisector of $BDC$ meets $BI$ at $P{}$ and $AI{}$ at $Q{}.$ Let $M{}$ be the midpoint of the segment $PQ.$ Prove that the line $IM$ passes through the middle of the arc $ACB$ of $\omega.$
[i]Dana Heuberger[/i]
2018 Azerbaijan JBMO TST, 1
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:
$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$
2018 Macedonia JBMO TST, 4
Determine all pairs $(p, q)$, $p, q \in \mathbb {N}$, such that
$(p + 1)^{p - 1} + (p - 1)^{p + 1} = q^q$.
2016 Azerbaijan JBMO TST, 2
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 $PQ\parallel BC$
2017 China Team Selection Test, 5
Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.