Olympiad problems

2023 Hong Kong Team Selection Test, Problem 1

Tags: combinatorics , TST

Given a $24 \times 24$ square grid, initially all its unit squares are coloured white. A move consists of choosing a row, or a column, and changing the colours of all its unit squares, from white to black, and from black to white. Is it possible that after finitely many moves, the square grid contains exactly $574$ black unit squares?

2022 Azerbaijan EGMO/CMO TST, N4

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 $f(2020)$.

2022 Israel TST, 1

Tags: combinatorics , TST , Israel

Bilbo, Gandalf, and Nitzan play the following game. First, Nitzan picks a whole number between $1$ and $2^{2022}$ inclusive and reveals it to Bilbo. Bilbo now compiles a string of length $4044$ built from the three letters $a,b,c$. Nitzan looks at the string, chooses one of the three letters $a,b,c$, and removes from the string all instances of the chosen letter. Only then is the string revealed to Gandalf. He must now guess the number Nitzan chose. Can Bilbo and Gandalf work together and come up with a strategy beforehand that will always allow Gandalf to guess Nitzan's number correctly, no matter how he acts?

2019 Kosovo Team Selection Test, 2

Tags: functional equation , algebra , function , bounded , Reals , TST , Kosovo

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ $$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$

2022 Junior Balkan Team Selection Tests - Moldova, 3

Tags: geometry , TST

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A straight line is drawn through point $B$, which again intersects circles $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Point $E$, located on circle $\omega_1$ , satisfies the relation $CE = CB$ , and point $F$, located on circle $\omega_2$, satisfies the relation $DB = DF$. The line $BF$ intersects again the circle $\omega_1$ at the point $P$, and the line $BE$ intersects again the circle $\omega_2$ at the point $Q$. Prove that the points $A, P$, and $Q$ are collinear.

2015 Turkey Team Selection Test, 3

Tags: Turkey , TST , combinatorics , algebra , Set partition

Let $m, n$ be positive integers. Let $S(n,m)$ be the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that \[S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)\]

2024 Junior Balkan Team Selection Tests - Romania, P1

Tags: Inequality , JBMO Shortlist , algebra , TST

For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that $$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$

2019 Slovenia Team Selection Test, 1

Tags: TST , geometry

Let $ABC$ be a non-right isosceles triangle such that $AC = BC$. Let $D$ be such a point on the perpendicular bisector of $AB$, that $AD$ is tangent on the $ABC$ circumcircle. Let $E$ be such a point on $AB$, that $CE$ and $AD$ are perpendicular and let $F$ be the second intersection of line $AC$ and the circle $CDE$. Prove that $DF$ and $AB$ are parallel.

2017 Iran Team Selection Test, 2

Tags: geometry , TST

Let $P$ be a point in the interior of quadrilateral $ABCD$ such that: $$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$ Prove that: $$\angle PBD= \left | \angle BCA - \angle PCA \right |$$ [i]Proposed by Ali Zamani[/i]

2023 Israel TST, P1

Tags: TST , algebra , functional equation , function

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds: \[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]

2023 Israel TST, P3

Tags: geometry , trapezoid , TST , circumcircle

Let $ABC$ be a fixed triangle. Three similar (by point order) isosceles trapezoids are built on its sides: $ABXY, BCZW, CAUV$, such that the sides of the triangle are bases of the respective trapezoids. The circumcircles of triangles $XZU, YWV$ meet at two points $P, Q$. Prove that the line $PQ$ passes through a fixed point independent of the choice of trapezoids.

2023 Israel TST, P3

Tags: TST , geometry , triangle centers , circumcircle , incenter

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and incenter $I$. The midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$ is denoted $S$. Points $E, F$ were chosen on line $OI$ for which $BE$ and $CF$ are both perpendicular to $OI$. Point $X$ was chosen so that $XE\perp AC$ and $XF\perp AB$. Point $Y$ was chosen so that $YE\perp SC$ and $YF\perp SB$. $D$ was chosen on $BC$ so that $DI\perp BC$. Prove that $X$, $Y$, and $D$ are collinear.

2017 Turkey EGMO TST, 6

Tags: Turkey , EGMO , TST , number theory , contest problem , order of an element

Find all pairs of prime numbers $(p,q)$, such that $\frac{(2p^2-1)^q+1}{p+q}$ and $\frac{(2q^2-1)^p+1}{p+q}$ are both integers.

2018 Azerbaijan IZhO TST, 4

Tags: combinatorics , TST , AZE IzHO TST

There are $10$ cities in each of the three countries. Each road connects two cities from two different countries (there is at most one road between any two cities.) There are more than $200$ roads between these three countries. Prove that three cities, one city from each country, can be chosen such that there is a road between any two of these cities.

2019 Iran Team Selection Test, 2

Tags: Iranian TST , TST , Iran

$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square. Prove $a$ is one of $a_1,\dots ,a_n$ [i]Proposed by Mohsen Jamali[/i]

2022 Bolivia Cono Sur TST, P2

Tags: geometry , TST , Bolivia , Inversion

On $\triangle ABC$ if there existed a point $D$ in $AC$ such that $\angle CBD=\angle ABD+60$ and $\angle BDC=30$ and $AB \cdot BC=BD^2$, then find the angles inside the triangle $\triangle ABC$

2020 Thailand TST, 1

Tags: geometry , IMO Shortlist , TST

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

2012 Serbia National Math Olympiad, 2

Tags: number theory , TST , AZE IzHO TST

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

2021 Azerbaijan IZhO TST, 1

Tags: algebra , Inequality , TST , AZE IzHO TST

Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

2020 USA EGMO Team Selection Test, 5

Tags: graph theory , graph cycles , spanning tree , combinatorics , TST

Let $G = (V, E)$ be a finite simple graph on $n$ vertices. An edge $e$ of $G$ is called a [i]bottleneck[/i] if one can partition $V$ into two disjoint sets $A$ and $B$ such that [list] [*] at most $100$ edges of $G$ have one endpoint in $A$ and one endpoint in $B$; and [*] the edge $e$ is one such edge (meaning the edge $e$ also has one endpoint in $A$ and one endpoint in $B$). [/list] Prove that at most $100n$ edges of $G$ are bottlenecks. [i]Proposed by Yang Liu[/i]

2021 Israel TST, 1

Tags: sum of digits , TST , number theory

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

2022 Germany Team Selection Test, 1

Tags: geometry , perpendicular bisector , Triangle Geometry , TST

Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$. The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$. The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$. The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$. (Yes, these definitions have the symmetries you would expect.) Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.

2018 China Team Selection Test, 3

Tags: combinatorics , TST , China TST

Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.

2022 Israel TST, 3

Tags: geometry , TST , Israel , circumcircle , incenter

Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.

Russian TST 2022, P2

Tags: functional equation , algebra , TST

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(xy+f(x))+f(y)=xf(y)+f(x+y),\]for all real numbers $x,y$.