Olympiad problems

1985 All Soviet Union Mathematical Olympiad, 412

One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle circumscribed around $AMD$ triangle has radius $R$.

1964 AMC 12/AHSME, 31

Tags: algebra , polynomial , function

Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: $\textbf{(A)}\ \dfrac{1}{2}f(n) \qquad \textbf{(B)}\ f(n)\qquad \textbf{(C)}\ 2f(n)+1 \qquad \textbf{(D)}\ f^2(n) \qquad \textbf{(E)}\ \dfrac{1}{2}(f^2(n)-1)$

2001 AMC 10, 20

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A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length $ 2000$. What is the length of each side of the octagon? $ \textbf{(A)}\ \frac{1}{3}(2000) \qquad \textbf{(B)}\ 2000(\sqrt2\minus{}1) \qquad \textbf{(C)}\ 2000(2\minus{}\sqrt2)$ $ \textbf{(D)}\ 1000 \qquad \textbf{(E)}\ 1000\sqrt2$

2014 ASDAN Math Tournament, 1

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Points $A$, $B$, $C$, and $D$ lie in the plane with $AB=AD=7$, $CB=CD=4$, and $BD=6$. Compute the sum of all possible values of $AC$.

2012 India National Olympiad, 5

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

Let $ABC$ be an acute angled triangle. Let $D,E,F$ be points on $BC, CA, AB$ such that $AD$ is the median, $BE$ is the internal bisector and $CF$ is the altitude. Suppose that $\angle FDE=\angle C, \angle DEF=\angle A$ and $\angle EFD=\angle B.$ Show that $ABC$ is equilateral.

2005 China Western Mathematical Olympiad, 8

Tags: graph theory , ramsey theory , combinatorics

For $n$ people, if it is known that (a) there exist two people knowing each other among any three people, and (b) there exist two people not knowing each other among any four people. Find the maximum of $n$. Here, we assume that if $A$ knows $B$, then $B$ knows $A$.

2025 CMIMC Team, 2

Tags: team

We are searching for the number $7$ in the following binary tree: [center] [img] https://cdn.artofproblemsolving.com/attachments/8/c/70ad159d239e9fd8dd9775e6391965e1016f03.png [/img] [/center] We use the following algorithm (which terminates with probability $1$): [list=1] [*] Write down the number currently at the root node. [*] If we wrote down $7,$ terminate. [*] Else, pick a random edge, and swap the two numbers at the endpoints of that edge [*] Go back to step $1.$ [/list] Let $p(a)$ be the probability that we ever write down the number $a$ after running the algorithm once. Find $$p(1)+p(2)+p(3)+p(5)+p(6).$$

2025 Harvard-MIT Mathematics Tournament, 2

Tags: geometry

In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.

2019 China Team Selection Test, 5

Tags: geometry

Let $M$ be the midpoint of $BC$ of triangle $ABC$. The circle with diameter $BC$, $\omega$, meets $AB,AC$ at $D,E$ respectively. $P$ lies inside $\triangle ABC$ such that $\angle PBA=\angle PAC, \angle PCA=\angle PAB$, and $2PM\cdot DE=BC^2$. Point $X$ lies outside $\omega$ such that $XM\parallel AP$, and $\frac{XB}{XC}=\frac{AB}{AC}$. Prove that $\angle BXC +\angle BAC=90^{\circ}$.

2012 ELMO Shortlist, 1

Tags: inequalities , trigonometry

Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that \[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\] [i]Ray Li, Max Schindler.[/i]

2016 South East Mathematical Olympiad, 3

Tags: combinatorics

Given any integer $n\geq 3$. A finite series is called $n$-series if it satisfies the following two conditions $1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$ $2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$ How many $n$-series are there $?$

1950 Putnam, A3

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The sequence $x_0, x_1, x_2, \ldots$ is defined by the conditions \[ x_0 = a, x_1 = b, x_{n+1} = \frac{x_{n - 1} + (2n - 1) ~x_n}{2n}\] for $n \ge 1,$ where $a$ and $b$ are given numbers. Express $\lim_{n \to \infty} x_n$ concisely in terms of $a$ and $b.$

2024 HMNT, 15

Tags: guts

Compute the sum of the three smallest positive integers $n$ for which $$\frac{1+2+3+\cdots+(2024n-1)+2024n}{1+2+3+\cdots+(4n-1)+4n}$$ is an integer.

2018 Peru Cono Sur TST, 2

Tags: algebra

Let $ x $ be a positive real number such that the numbers $ x^{-1} $, $ x $, and $ x^{2018} $ have the same fractional part: $$ \{x^{-1}\} = \{x\} = \{x^{2018}\}. $$ Prove that $ x = 1 $. [b]Note:[/b] If $ x $ is a real number, its fractional part is $ \{x\} = x - \lfloor x \rfloor $, where $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $ x $.

2018 Thailand TSTST, 1

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients satisfying: $P(2017) = 2016$ and $$(P(x)+1)^2=P(x^2+1).$$

1996 IberoAmerican, 2

Tags: circumcircle , geometry

Let $\triangle{ABC}$ be a triangle, $D$ the midpoint of $BC$, and $M$ be the midpoint of $AD$. The line $BM$ intersects the side $AC$ on the point $N$. Show that $AB$ is tangent to the circuncircle to the triangle $\triangle{NBC}$ if and only if the following equality is true: \[\frac{{BM}}{{MN}} =\frac{({BC})^2}{({BN})^2}.\]

1994 IMO Shortlist, 3

Tags: combinatorics , prime number , combinatorial number theory , number theory , modular arithmetic

Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

2014 AMC 10, 9

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For real numbers $w$ and $z$, \[ \frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014. \] What is $\tfrac{w+z}{w-z}$ ? ${ \textbf{(A)}\ \ -2014\qquad\textbf{(B)}\ \frac{-1}{2014}\qquad\textbf{(C)}\ \frac{1}{2014}\qquad\textbf{(D)}}\ 1\qquad\textbf{(E)}\ 2014$

2012 Cuba MO, 2

Tags: geometry , parallel

Given the triangle $ABC$, let $L$, $M$ and $N $be the midpoints of $BC$, $CA$ and $AB$ respectively. The lines $LM$ and $LN$ cut the tangent to the circumcircle at $A$ at $P$ and $Q$ respectively . Prove that $CP \parallel BQ$.

2024 JHMT HS, 1

Tags: number theory

Compute the smallest positive integer $N$ for which $N \cdot 2^{2024}$ is a multiple of $2024$.

KoMaL A Problems 2020/2021, A. 787

Tags: divisibility , komal prob , binomial coefficient , number theory

Let $p_n$ denote the $n^{\text{th}}$ prime number and define $a_n=\lfloor p_n\nu\rfloor$ for all positive integers $n$ where $\nu$ is a positive irrational number. Is it possible that there exist only finitely many $k$ such that $\binom{2a_k}{a_k}$ is divisible by $p_i^{10}$ for all $i=1,2,\ldots,2020?$ [i]Proposed by Superguy and ayan.nmath[/i]

2022 Korea National Olympiad, 5

Tags: incenter , geometry

For a scalene triangle $ABC$ with an incenter $I$, let its incircle meets the sides $BC, CA, AB$ at $D, E, F$, respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$, and $Q$ the intersection of the lines $BI$ and $EF$. Prove that $\overline{PQ}=\overline{CD}$.

Dumbest FE I ever created, 5.

Tags: algebra , algebruh , rutthee ahh function , function

Find all non decreasing function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ and $m,n \in \mathbb{N}_0$ such that $m+n \neq 0$ there exist $m',n' \in \mathbb{N}_0$ such that $m'+n'=m+n+1$ and $$f(f^m(x)+f^n(y))=f^{m'}(x)+f^{n'}(y)$$ . Note : $f^0(x)=x$ and $f^{n}(x)=f(f^{n-1}(x))$ for all $n \in \mathbb{N}$ . [hide=original]Find all non decreasing functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(x))+y$$ .[/hide]

2022 Thailand Mathematical Olympiad, 3

Tags: combinatorics

Let $\Omega$ be a circle in a plane. $2022$ pink points and $2565$ blue points are placed inside $\Omega$ such that no point has two colors and no two points are collinear with the center of $\Omega$. Prove that there exists a sector of $\Omega$ such that the angle at the center is acute and the number of blue points inside the sector is greater than the number of pink points by exactly $100$. (Note: such sector may contain no pink points.)

LMT Team Rounds 2021+, 10

Tags: combinatorics

There are $15$ people attending math team: $12$ students and $3$ captains. One of the captains brings $33$ identical snacks. A nonnegative number of names (students and/or captains) are written on the NO SNACK LIST. At the end of math team, all students each get n snacks, and all captains get $n +1$ snacks, unless the person’s name is written on the board. After everyone’s snacks are distributed, there are none left. Find the number of possible integer values of $n$.