Olympiad problems

1940 Putnam, B7

Tags: inequalities

Which is greater $$\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}$$ where $n>8?$

2015 Math Prize for Girls Problems, 10

Tags:

Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.

2010 Sharygin Geometry Olympiad, 5

Tags: geometry , equal segments , right triangle , altitude , circumcircle

Let $BH$ be an altitude of a right-angled triangle $ABC$ ($\angle B = 90^o$). The incircle of triangle $ABH$ touches $AB,AH$ in points $H_1, B_1$, the incircle of triangle $CBH$ touches $CB,CH$ in points $H_2, B_2$, point $O$ is the circumcenter of triangle $H_1BH_2$. Prove that $OB_1 = OB_2$.

1971 IMO Longlists, 29

Tags: rhombus , geometry

A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.

2021 Purple Comet Problems, 19

Tags: number theory

For some integers $u$,$ v$, and $w$, the equation $$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$ holds for all real numbers a, b, and c that satisfy $$au + bv + cw = 0$$ Find the minimum possible value of $u^2 + v^2 + w^2$.

2024 Bulgarian Winter Tournament, 12.1

Tags: combinatorics

Maria and Bilyana play the following game. Maria has $2024$ fair coins and Bilyana has $2023$ fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?

2011 VTRMC, Problem 6

Tags: set

Let $S$ be a set with an asymmetric relation $<$; this means that if $a,b\in S$ and $a<b$, then we do not have $b<a$. Prove that there exists a set $T$ containing $S$ with an asymmetric relation $\prec$ with the property that if $a,b\in S$, then $a<b$ if and only if $a\prec b$, and if $x,y\in T$ with $x\prec y$, then there exists $t\in T$ such that $x\prec t\prec y$.

2014 Cezar Ivănescu, 3

Tags: vieta , linear algebra

Let $ A,B,C,D $ be four $ 2\times 2 $ complex matrices such that $ A-D $ is invertible and such that $$ A^2+BA+C=0=D^2+BD+C. $$ Prove that $ \text{tr} (A+D) =-\text{tr} B $ and $ \det (AD) =\det C. $

2014 Bundeswettbewerb Mathematik, 2

Tags: geometry , 3d geometry , prism , algorithm , geometric transformation , reflection , induction

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

2023 ELMO Shortlist, N2

Tags: number theory

Determine the greatest positive integer $n$ for which there exists a sequence of distinct positive integers $s_1$, $s_2$, $\ldots$, $s_n$ satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\] [i]Proposed by Holden Mui[/i]

2016 Fall CHMMC, 2

Tags: analytic geometry , probability

Alice and Bob find themselves on a coordinate plane at time $t=0$ at $A(1,0)$ and $B(-1,0)$ respectively. They have no sense of direction, but they want to find each other. They each pick a direction independently and with uniform random probability. Both Alice and Bob travel at a constant speed of $1 \frac{unit}{min}$ in their chosen directions. They continue on their straight line paths forever, each hoping to catch sight of the other. They both have a $1$ unit radius of view; they can see something if and only if its distance from them is at most $1$ unit. What is the probability they never see each other?

2019 Auckland Mathematical Olympiad, 1

Tags: geometry , angle

Given a convex quadrilateral $ABCD$ in which $\angle BAC = 20^o$, $\angle CAD = 60^o$, $\angle ADB = 50^o$ , and $\angle BDC = 10^o$. Find $\angle ACB$.

2022 HMNT, 1

Tags: probability

Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.

2012 Romania National Olympiad, 1

Tags: geometry , square , distance

Let $P$ be a point inside the square $ABCD$ and $PA = 1$, $PB = \sqrt2$ and $PC =\sqrt3$. a) Determine the length of segment $[PD]$. b) Determine the angle $\angle APB$.

2005 Alexandru Myller, 2

Tags: superior algebra , linear algebra , matrix , algebra , polynomial

Let $A\in M_4(\mathbb R)$ be an invertible matrix s.t. $\det(A+^tA)=5\det A$ and $\det (A-^tA)=\det A$. Prove that for every complex root $\omega$ of order 5 of unitity (i.e. $\omega^5=1,\omega\not\in\mathbb R$) the following relation holds $\det(\omega A+^tA)=0$. [i]Dan Popescu[/i]

2022 Stanford Mathematics Tournament, 8

Tags:

Let $\Gamma$ and $\Omega$ be circles that are internally tangent at a point $P$ such that $\Gamma$ is contained entirely in $\Omega$. Let $A,B$ be points on $\Omega$ such that the lines $PB$ and $PA$ intersect the circle $\Gamma$ at $Y$ and $X$ respectively, where $X,Y\neq P$. Let $O_1$ be the circle with diameter $AB$ and $O_2$ be the circle with diameter $XY$. Let $F$ be the foot of $Y$ on $XP$. Let $T$ and $M$ be points on $O_1$ and $O_2$ respectively such that $TM$ is a common tangent to $O_1$ and $O_2$. Let $H$ be the orthocenter of $\triangle ABP$. Given that $PF=12$, $FX=15$, $TM=18$, $PB=50$, find the length of $AH$.

2024 Kazakhstan National Olympiad, 3

Tags: geometry

An acute triangle $ABC$ ($AB\ne AC$) is inscribed in the circle $\omega$ with center at $O$. The point $M$ is the midpoint of the side $BC$. The tangent line to $\omega$ at point $A$ intersects the line $BC$ at point $D$. A circle with center at point $M$ with radius $MA$ intersects the extensions of sides $AB$ and $AC$ at points $K$ and $L$, respectively. Let $X$ be such a point that $BX\parallel KM$ and $CX\parallel LM$. Prove that the points $X$, $D$, $O$ are collinear.

2022 European Mathematical Cup, 1

Tags: number theory , modular arithmetic , divisibility , divisor

Determine all positive integers $n$ for which there exist positive divisors $a$, $b$, $c$ of $n$ such that $a>b>c$ and $a^2 - b^2$, $b^2 - c^2$, $a^2 - c^2$ are also divisors of $n$.

2020-2021 OMMC, 8

Tags: geometry

Let triangle $MAD$ be inscribed in circle $O$ with diameter $85$ such that $MA = 68$ and $DA = 40$. The altitudes from $M, D$ to sides $AD$ and $MA$, respectively, intersect the tangent to circle $O$ at $A$ at $X$ and $Y$ respectively. $XA \times YA$ can be expressed as $\frac{a}{b}$, where $a$ and $ b$ are relatively prime positive integers. Find $a + b$.

2014 Dutch IMO TST, 3

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

Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.

2023 JBMO TST - Turkey, 2

Tags: chessboard , combinatorics

A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?

2014 Olympic Revenge, 1

Tags: circumcircle , projective geometry , geometry

Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.

2013 National Olympiad First Round, 18

Tags: modular arithmetic

What is remainder when the sum \[\binom{2013}{1}+2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013}\] is divided by $41$? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None} $

2021 Purple Comet Problems, 2

Tags:

A furniture store set the sticker price of a table $40$ percent higher than the wholesale price that the store paid for the table. During a special sale, the table sold for $35$ percent less than this sticker price. Find the percent the final sale price was of the original wholesale price of the table.

2008 Balkan MO Shortlist, G3

Tags: geometry , perimeter , area of a triangle , bisector , orthocenter , angle

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.