Olympiad problems

2013 SEEMOUS, Problem 2

Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that $$M^2=N^2=0_2\text{ and }MN+NM=I_2$$where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that $$M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.$$

2022 Caucasus Mathematical Olympiad, 7

Tags: geometry

Point $P$ is chosen on the leg $CB$ of right triangle $ABC$ ($\angle ACB = 90^\circ$). The line $AP$ intersects the circumcircle of $ABC$ at point $Q$. Let $L$ be the midpoint of $PB$. Prove that $QL$ is tangent to a fixed circle independent of the choice of point $P$.

2009 Harvard-MIT Mathematics Tournament, 4

Tags: number theory , least common multiple , greatest common divisor , algebra , polynomial , quadratics

Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.

2021 2nd Memorial "Aleksandar Blazhevski-Cane", 6

Tags: algebra , functional equation

Let $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all the functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x, y \in \mathbb{R}^{+}$, \[ f(x)f(y) = f(y)f(xf(y)) + \frac{1}{xy}. \]

2009 Stanford Mathematics Tournament, 9

Tags: geometry

Two circles with centers $A$ and $B$ intersect at points $X$ and $Y$. The minor arc $\angle{XY}=120$ degrees with respect to circle $A$, and $\angle{XY}=60$ degrees with respect to circle $B$. If $XY=2$, find the area shared by the two circles.

2018 Iran MO (2nd Round), 1

Tags: Iran , geometry

Let $P $ be the intersection of $AC $ and $BD $ in isosceles trapezoid $ABCD $ ($AB\parallel CD$ , $BC=AD $) . The circumcircle of triangle $ABP $ inersects $BC $ for the second time at $X $. Point $Y $ lies on $AX $ such that $DY\parallel BC $. Prove that $\hat {YDA} =2.\hat {YCA} $.

2011 Turkey Team Selection Test, 2

Tags: geometry , incenter , circumcircle , geometric transformation , reflection , rhombus , symmetry

Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of a triangle $ABC.$ If the point $E$ on the ray $BA$ and the point $F$ on the ray $CA$ satisfy the condition \[BE=CF=\frac{AB+BC+CA}{2}\] show that the lines $EF$ and $DI$ are perpendicular.

2025 Taiwan Mathematics Olympiad, 5

Tags: geometry , fixed circle , moving points

Two fixed circles $\omega$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $C$ and $D$ be two fixed points on the circle $\omega$. Let $P$ be a moving point on $\omega$. Line $PA$ meets circle $\Omega$ again at $Q$. Prove that the second intersection $R$ of two circumcircles of triangles $QPC$ and $QBD$ always lies on a fixed circle. [i]Proposed by buratinogigle[/i]

2020 BMT Fall, 7

Tags: number theory

Compute the number of ordered triples of positive integers $(a,b,c)$ such that $a + b + c + ab + bc + ac = abc + 1$.

2001 Vietnam National Olympiad, 2

Tags: number theory , relatively prime , number theory proposed

Let $N = 6^{n}$, where $n$ is a positive integer, and let $M = a^{N}+b^{N}$, where $a$ and $b$ are relatively prime integers greater than $1. M$ has at least two odd divisors greater than $1$ are $p,q$. Find the residue of $p^{N}+q^{N}\mod 6\cdot 12^{n}$.

2022 Bangladesh Mathematical Olympiad, 6

Tags: number theory

About $5$ years ago, Joydip was researching on the number $2017$. He understood that $2017$ is a prime number. Then he took two integers $a,b$ such that $0<a,b <2017$ and $a+b\neq 2017.$ He created two sequences $A_1,A_2,\dots ,A_{2016}$ and $B_1,B_2,\dots, B_{2016}$ where $A_k$ is the remainder upon dividing $ak$ by $2017$, and $B_k$ is the remainder upon dividing $bk$ by $2017.$ Among the numbers $A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}$ count of those that are greater than $2017$ is $N$. Prove that $N=1008.$

2002 Tournament Of Towns, 6

Tags: combinatorics proposed , combinatorics

The $52$ cards of a standard deck are placed in a $13\times 4$ array. If every two adjacent cards, vertically or horizontally, have the same suit or have the same value, prove that all $13$ cards of the same suit are in the same row.

2016 Brazil National Olympiad, 3

Tags: combinatorics , Brazilian Math Olympiad 2016 , Brazilian Math Olympiad , games , Combinatorial games

Let it $k$ be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number $N_0$ and starting with Alberto, they alternately do the following operation: change the number $n$ for a number $m$ such that $m < n$ and $m$ and $n$ differ, in its base-2 representation, in exactly $l$ consecutive digits for some $l$ such that $1 \leq l \leq k$. If someone can't play, he loses. We say a non-negative integer $t$ is a [i]winner[/i] number when the gamer who receives the number $t$ has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it [i]loser[/i]. Prove that, for every positive integer $N$, the total of non-negative loser integers lesser than $2^N$ is $2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}$

1991 Arnold's Trivium, 98

Tags: probability , function

In the game of "Fingers", $N$ players stand in a circle and simultaneously thrust out their right hands, each with a certain number of fingers showing. The total number of fingers shown is counted out round the circle from the leader, and the player on whom the count stops is the winner. How large must $N$ be for a suitably chosen group of $N/10$ players to contain a winner with probability at least $0.9$? How does the probability that the leader wins behave as $N\to\infty$?

2024 China Team Selection Test, 15

Tags: algebra , polynomial , 2024 CTST

$n>1$ is an integer. Let real number $x>1$ satisfy $$x^{101}-nx^{100}+nx-1=0.$$ Prove that for any real $0<a<b<1$, there exists a positive integer $m$ so that $a<\{x^m\}<b.$ [i]Proposed by Chenjie Yu[/i]

2018 CMIMC Number Theory, 10

Tags: algebra , polynomial

Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.

2017 Morocco TST-, 3

Tags: IMO Shortlist , geometry , mixtilinear incircle , projective geometry , median , geometry solved

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

2016 Junior Balkan MO, 3

Tags: number theory

Find all triplets of integers $(a,b,c)$ such that the number $$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$ is a power of $2016$. (A power of $2016$ is an integer of form $2016^n$,where n is a non-negative integer.)

2013 Bangladesh Mathematical Olympiad, 4

Tags: algebra , contests

Higher Secondary P4 If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.

2020 Princeton University Math Competition, A1/B3

Tags: geometry

Let $\gamma_1$ and $\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at point $Q$. Draw point $D$ on $\gamma_1$ and point $E$ on $\gamma_2$ such that line $DE$ is tangent to both circles. If the length $OQ = 1$ and the area of the quadrilateral $ODEP$ is $520$, then what is the value of length $PQ$?

2007 Putnam, 4

Tags: Putnam , algebra , polynomial , logarithms , quadratics , college contests

A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$

1996 China National Olympiad, 1

Tags: geometry , circumcircle , trigonometry , conics , similar triangles , China

Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.

1997 All-Russian Olympiad, 2

Tags: combinatorics proposed , combinatorics

An $n\times n$ square grid ($n\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells. [i]E. Poroshenko[/i]

PEN S Problems, 1

Tags: number theory , Diophantine equation , Miscellaneous Problems

a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` Now I can tell what they are.' What are the two numbers? b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?

2001 Portugal MO, 6

Tags: number theory , Digits

Let $n$ be a natural number. Prove that there is a multiple of $n$ that can be written only with the digits $0$ and $1$.