Olympiad problems

2023 Chile TST Ibero., 3

Determine the smallest positive integer $ n $ with the following property: for every triple of positive integers $ x, y, z $, with $ x $ dividing $ y^3 $, $ y $ dividing $ z^3 $, and $ z $ dividing $ x^3 $, it also holds that $ (xyz) $ divides $ (x + y + z)^n $.

2023 Hong Kong Team Selection Test, Problem 1

Tags: TST , hong Kong TST

Mandy needs to wake up early for attending a mathematics contest. She has set an alarm in her smartphone every 15 minutes since 5:30 am. If an alarm is not pressed off by her or her mother (or anything else), it will ring for a while, stop for a while, then will ring again 9 minutes later as the first ring, and so on (e.g. if the first alarm is not pressed off, it will ring again at 5:39 am). Also each alarm will work independently. Now suppose each ring-tone lasts for $x$ minutes, and the smartphone has eventually rung for 50 minutes before Mandy wakes up at 6:30 am (assume no one has pressed off any alarm before that). Find the value of $x$.

2019 India IMO Training Camp, P2

Tags: geometry , TST

Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$. Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$. [i]Proposed by Anant Mudgal[/i]

2020 Baltic Way, 18

Tags: number theory , AZE CMO TST , AZE EGMO TST , TST

Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that \begin{align*} x^2+y^2+z^2 &\equiv 0 \pmod n;\\ xyz &\equiv k \pmod n. \end{align*} Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.

2014 Turkey Team Selection Test, 3

Tags: geometry , combinatorics , TST , combinatorial geometry , Turkey

At the bottom-left corner of a $2014\times 2014$ chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.

2021 China Team Selection Test, 4

Tags: Inequality , algebra , TST

Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

2017 India IMO Training Camp, 1

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

2023 ISL, N3

Tags: IMO Shortlist , number theory , AZE BMO TST , AZE EGMO TST , TST , BRA Cono Sur TST

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 $E_{10}(m) < E_9(m)$.

2017 Turkey EGMO TST, 4

Tags: Turkey , EGMO , TST , geometry , contest problem

2025 China Team Selection Test, 2

Tags: geometry , TST

Suppose $\triangle ABC$ has $D$ as the midpoint of $BC$ and orthocenter $H$. Let $P$ be an arbitrary point on the nine point circle of $ABC$. The line through $P$ perpendicular to $AP$ intersects $BC$ at $Q$. The line through $A$ perpendicular to $AQ$ intersects $PQ$ at $X$. If $M$ is the midpoint of $AQ$, show that $HX \perp DM$.

2021 Romania Team Selection Test, 1

Tags: geometry , constant , romania , Romanian TST , TST

Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.

2024 Chile TST IMO, 2

Tags: number theory , Chile , TST

Find all natural numbers that have a multiple consisting only of the digit 9.

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]

2025 China Team Selection Test, 1

Tags: algebra , polynomial , TST

Show that the polynomial over variables $x,y,z$ \[ x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y) \] can't be written as a finite sum of squares of real polynomials over $x,y,z$.

2024 Indonesia TST, 2

Tags: algebra , Functional inequality , IMO Shortlist , AZE IMO TST , TST

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

2020 Junior Macedonian National Olympiad, 3

Tags: TST , number theory

Solve the following equation in the set of integers $x^5 + 2 = 3 \cdot 101^y$.

2023 India IMO Training Camp, 2

Tags: india , geometry , TST , anant mudgal geo

In triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to line $BC$. Point $K$ lies inside triangle $ABC$ such that $\angle KAB = \angle KCA$ and $\angle KAC = \angle KBA$. The line through $K$ perpendicular to like $DK$ meets the circle with diameter $BC$ at points $X,Y$. Prove that $AX \cdot DY = DX \cdot AY$

2015 Turkey Team Selection Test, 1

Tags: Turkey , TST , 2015 , number theory

Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.

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]

2011 Argentina Team Selection Test, 3

Tags: geometry , trapezoid , circumcircle , Argentina , TST

Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$. Show that $SO \perp XY$.

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: $$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$

2016 Ukraine Team Selection Test, 10

Tags: algebra , polynomial , calculus , TST

Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$ Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.

2012 USA TSTST, 2

Tags: geometry , TST

Let $ABCD$ be a quadrilateral with $AC = BD$. Diagonals $AC$ and $BD$ meet at $P$. Let $\omega_1$ and $O_1$ denote the circumcircle and the circumcenter of triangle $ABP$. Let $\omega_2$ and $O_2$ denote the circumcircle and circumcenter of triangle $CDP$. Segment $BC$ meets $\omega_1$ and $\omega_2$ again at $S$ and $T$ (other than $B$ and $C$), respectively. Let $M$ and $N$ be the midpoints of minor arcs $\widehat {SP}$ (not including $B$) and $\widehat {TP}$ (not including $C$). Prove that $MN \parallel O_1O_2$.

2015 USA Team Selection Test, 3

Tags: AMC , USA(J)MO , USAMO , graph theory , combinatorics , TST

A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible? [i]Proposed by Linus Hamilton[/i]

2025 Azerbaijan IZhO TST, 3

Tags: number theory , TST , AZE IzHO TST

Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]