Olympiad problems

1988 Tournament Of Towns, (176) 2

Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

2024 Assara - South Russian Girl's MO, 6

Tags: geometry

In the regular hexagon $ABCDEF$, a point $X$ was marked on the diagonal $AD$ such that $\angle AEX = 65^\circ$. What is the degree measure of the angle $\angle XCD$? [i]A.V.Smirnov, I.A.Efremov[/i]

2010 Tournament Of Towns, 1

Tags: ratio , number theory

Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?

2003 China Team Selection Test, 3

Tags: number theory

Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.

2022 Saudi Arabia BMO + EGMO TST, 2.3

Tags: number theory , combinatorics

Let $n$ be an even positive integer. On a board n real numbers are written. In a single move we can erase any two numbers from the board and replace each of them with their product. Prove that for every $n$ initial numbers one can in finite number of moves obtain $n$ equal numbers on the board.

2009 Indonesia TST, 3

Tags: circumcircle , geometry

Let $ C_1$ be a circle and $ P$ be a fixed point outside the circle $ C_1$. Quadrilateral $ ABCD$ lies on the circle $ C_1$ such that rays $ AB$ and $ CD$ intersect at $ P$. Let $ E$ be the intersection of $ AC$ and $ BD$. (a) Prove that the circumcircle of triangle $ ADE$ and the circumcircle of triangle $ BEC$ pass through a fixed point. (b) Find the the locus of point $ E$.

2019 Greece National Olympiad, 2

Tags: geometry

Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center of minor arc $\overarc{AB}$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at $Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$.

2023 MMATHS, 7

Tags:

A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x,y).$ For every distinct pair of lights $(x_1, y_1), (x_2, y_2),$ with $x_1<x_2$ and $y_1>y_2,$ all lights strictly between them (i.e. $x_1<x<x_2$ and $y_2<y<y_1$) are toggled. After this procedure is done, how many lights are on?

1986 ITAMO, 1

Tags: parallel , circles , tangent , geometry

Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$. The lines connecting a point $R$ on $\beta$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. Prove that $ST$ is parallel to the line tangent to $\beta$ at $R$.

2012 239 Open Mathematical Olympiad, 3

Tags:

There are $n$ points in the space such that none $4$ of them lie on a plane. You can select two points $A$ and $B$ and move point $A$ to the midpoint of line segment $AB$. It turned out that, after several moves, the points took the same places (possibly in a different order). What is the smallest value of $n$ that this could happen for some $n$ points?

2021 Brazil Undergrad MO, Problem 1

Tags: matrix algebra , square matrix , linear algebra

Consider the matrices like $$M= \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)$$ such that $det(M) = 1$. Show that a) There are infinitely many matrices like above with $a,b,c \in \mathbb{Q}$ b) There are finitely many matrices like above with $a,b,c \in \mathbb{Z}$

2008 International Zhautykov Olympiad, 2

Tags: symmetry , circumcircle , trigonometry , cyclic quadrilateral , geometry

Let $ A_1A_2$ be the external tangent line to the nonintersecting cirlces $ \omega_1(O_1)$ and $ \omega_2(O_2)$,$ A_1\in\omega_1$,$ A_2\in\omega_2$.Points $ K$ is the midpoint of $ A_1A_2$.And $ KB_1$ and $ KB_2$ are tangent lines to $ \omega_1$ and $ \omega_2$,respectvely($ B_1\neq A_1$,$ B_2\neq A_2$).Lines $ A_1B_1$ and $ A_2B_2$ meet in point $ L$,and lines $ KL$ and $ O_1O_2$ meet in point $ P$. Prove that points $ B_1,B_2,P$ and $ L$ are concyclic.

2001 JBMO ShortLists, 7

Tags: modular arithmetic , number theory

Prove that there are are no positive integers $x$ and $y$ such that $x^5+y^5+1=(x+2)^5+(y-3)^5$. [hide="Note"] The restriction $x,y$ are positive isn't necessary.[/hide]

1983 IMO Longlists, 38

Tags: polynomial , algebra

Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula \[u_{n+2 }= u_n - u_{n+1}.\] [b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined. [b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.

2019 India IMO Training Camp, P1

Tags: number theory , polynomial

Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$ [i] Proposed by Anant Mudgal [/i]

1974 AMC 12/AHSME, 24

Tags: probability

A fair die is rolled six times. The probability of rolling at least a five at least five times is $ \textbf{(A)}\ \frac{13}{729} \qquad\textbf{(B)}\ \frac{12}{729} \qquad\textbf{(C)}\ \frac{2}{729} \qquad\textbf{(D)}\ \frac{3}{729} \qquad\textbf{(E)}\ \text{none of these} $

2022 Purple Comet Problems, 4

Tags:

A jar contains red, blue, and yellow candies. There are $14\%$ more yellow candies than blue candies, and $14\%$ fewer red candies than blue candies. Find the percent of candies in the jar that are yellow.

2013 HMNT, 8

Tags: algebra

Define the sequence $\{x_i\}_{i \ge 0}$ by $x_0 = x_1 = x_2 = 1$ and $x_k = \frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k > 2$. Find $x_{2013}$.

2012 India Regional Mathematical Olympiad, 6

Tags: subset , set theory , combinatorics , combinatorics of set

Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$. We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$. Define [i]WSUM [/i] for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if $A = \{2, 5, 7, 8\}$, [i]WSUM [/i] is $3(2 + 7) + 2(5 + 8)$.) Find the sum of [i]WSUMs[/i] over all the subsets of S. (Assume that WSUM for the null set is $0$.)

1993 Swedish Mathematical Competition, 6

Tags: function , algebra

For real numbers $a$ and $b$ define $f(x) = \frac{1}{ax+b}$. For which $a$ and $b$ are there three distinct real numbers $x_1,x_2,x_3$ such that $f(x_1) = x_2$, $f(x_2) = x_3$ and $f(x_3) = x_1$?

2016 India Regional Mathematical Olympiad, 3

Tags: number theory , algebra , polynomial

Find all integers $k$ such that all roots of the following polynomial are also integers: $$f(x)=x^3-(k-3)x^2-11x+(4k-8).$$

2021 LMT Spring, A23 B24

Tags: number theory

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. A group of haikus Some have one syllable less Sixteen in total. The group of haikus Some have one syllable more Eighteen in total. What is the largest Total count of syllables That the group can’t have? (For instance, a group Sixteen, seventeen, eighteen Fifty-one total.) (Also, you can have No sixteen, no eighteen Syllable haikus) [i]Proposed by Jeff Lin[/i]

1988 Tournament Of Towns, (198) 1

Tags: combinatorial geometry , chessboard , combinatorics

What is the smallest number of squares of a chess board that can be marked in such a manner that (a) no two marked squares may have a common side or a common vertex, and (b) any unmarked square has a common side or a common vertex with at least one marked square? Indicate a specific configuration of marked squares satisfying (a) and (b) and show that a lesser number of marked squares will not suffice. (A. Andjans, Riga)

2000 AIME Problems, 10

Tags: trigonometry , trapezoid , angle bisector , geometry

A circle is inscribed in quadrilateral $ABCD,$ tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q.$ Given that $AP=19, PB=26, CQ=37,$ and $QD=23,$ find the square of the radius of the circle.

2009 Belarus Team Selection Test, 2

Tags: algebra , sequence

a) Prove that there is not an infinte sequence $(x_n)$, $n=1,2,...$ of positive real numbers satisfying the relation $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_{n}}$, $\forall n \in N$ (*) b) Do there exist sequences satisfying (*) and containing arbitrary many terms? I.Voronovich