Olympiad problems

2017 Saudi Arabia IMO TST, 3

Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.

2011 China Western Mathematical Olympiad, 4

Tags: geometric transformation , angle bisector , geometry , homothety

In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$

2004 Switzerland Team Selection Test, 8

Tags: sequence , divisibility , number theory , modular arithmetic

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]

2021 South East Mathematical Olympiad, 2

Tags: geometry

In $\triangle ABC$,$AB=AC>BC$, point $O,H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively,$G $ is the midpoint of segment $AH$ , $BE$ is the altitude on $AC$ . Prove that if $OE\parallel BC$, then $H$ is the incenter of $\triangle GBC$.

2020 Kosovo National Mathematical Olympiad, 4

Tags: geometry , circumcircle

Let $B'$ and $C'$ be points in the circumcircle of triangle $\triangle ABC$ such that $AB=AB'$ and $AC=AC'$. Let $E$ and $F$ be the foot of altitudes from $B$ and $C$ to $AC$ and $AB$, respectively. Show that $B'E$ and $C'F$ intersect on the circumcircle of triangle $\triangle ABC$.

2022 Durer Math Competition Finals, 4

Tags: perfect square , divisor , number theory

Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

2009 Moldova National Olympiad, 9.3

Tags: equilateral , parallel , geometry

Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.

2005 AMC 8, 20

Tags: least common multiple , number theory , modular arithmetic

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24 $

2018 Federal Competition For Advanced Students, P1, 3

Tags: decimal representation , digit , combinatorics , game , number theory

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

2016 Hong Kong TST, 1

Tags: circumcircle , geometry

Let $O$ be the circumcenter of a triangle $ABC$, and let $l$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$. Determine the value of $\angle BAC$ if the line $l$ goes through the midpoint of the line segment $AO$.

2008 F = Ma, 23

Tags:

Consider two uniform spherical planets of equal density but unequal radius. Which of the following quantities is the same for both planets? (a) The escape velocity from the planet’s surface. (b) The acceleration due to gravity at the planet’s surface. (c) The orbital period of a satellite in a circular orbit just above the planet’s surface. (d) The orbital period of a satellite in a circular orbit at a given distance from the planet’s center. (e) None of the above.

2005 QEDMO 1st, 10 (C3)

Tags: combinatorics

Let $n\geq 3$ be an integer. Let also $P_1,P_2,...,P_n$ be different two-element-subsets of $M=\{1,2,...,n\}$, such that when for $i,j \in M , i\neq j$ the sets $P_i,P_j$ are not totally disjoint, then there is a $k \in M$ with $P_k = \{ i,j\}$. Prove that every element of $M$ occurse in exactly $2$ of these subsets.

2003 ITAMO, 3

Tags: reflection , geometric transformation , circumcircle , homothety , geometry

Let a semicircle is given with diameter $AB$ and centre $O$ and let $C$ be a arbitrary point on the segment $OB$. Point $D$ on the semicircle is such that $CD$ is perpendicular to $AB$. A circle with centre $P$ is tangent to the arc $BD$ at $F$ and to the segment $CD$ and $AB$ at $E$ and $G$ respectively. Prove that the triangle $ADG$ is isosceles.

2015 MMATHS, 2

Tags: number theory

Determine, with proof, whether $22!6! + 1$ is prime.

2017 IFYM, Sozopol, 2

Tags: geometry

The lengths of the sides of a triangle are 19, 20, 21 cm. We can cut the triangle in a straight line into two parts. These two parts are put in a circle with radius $R$ cm without overlapping each other. Find the least possible value of $R$.

2022 ISI Entrance Examination, 8

Tags: inequalities , algebra , trigonometry

Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.

2022 CHMMC Winter (2022-23), 2

Tags: number theory

Select a number $X$ from the set of all $3$-digit natural numbers uniformly at random. Let $A \in [0,1]$ be the probability that $X$ is divisible by $11$, given that it is palindromic. Let $B \in [0,1]$ be the probability that X is palindromic, given that it is divisible by $11$. Compute $B-A$. Recall that a $3$-digit number is a palindrome if it reads the same left to right as right to left. For instance, $484$ is a palindrome, but $603$ is not a palindrome.

2014 Middle European Mathematical Olympiad, 2

Tags: geometry , combinatorics

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

2024 Princeton University Math Competition, A7

Tags: combinatorics

Let $\omega=e^{2\pi i/20}$ and let $S$ be the set $\{1, \omega, \ldots, \omega^{19}\}.$ How many subsets of $S$ sum to $0$? Include both $S$ and the empty set in your count.

2018 IMO Shortlist, C6

Tags: combinatorics

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

2023 Bulgaria National Olympiad, 1

Tags: graph theory , number theory , greatest common divisor , combinatorics

Let $G$ be a graph on $n\geq 6$ vertices and every vertex is of degree at least 3. If $C_{1}, C_{2}, \dots, C_{k}$ are all the cycles in $G$, determine all possible values of $\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|)$ where $|C|$ denotes the number of vertices in the cycle $C$.

2003 AMC 12-AHSME, 9

Tags: analytic geometry

A set $ S$ of points in the $ xy$-plane is symmetric about the origin, both coordinate axes, and the line $ y \equal{} x$. If $ (2, 3)$ is in $ S$, what is the smallest number of points in $ S$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 16$

I Soros Olympiad 1994-95 (Rus + Ukr), 9.3

Tags: geometry

Is there a quadrilateral in which the position of any vertex can be changed, leaving the other three in place, so that the resulting four points serve as the vertices of a quadrilateral equal to the original one?

2014 AMC 10, 10

Tags:

Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$? ${ \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7$

2015 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$