Olympiad problems

2022 Stanford Mathematics Tournament, 1

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Compute \[\frac{5+\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{7+\sqrt{12}}{\sqrt{3}+\sqrt{4}}+\dots+\frac{63+\sqrt{992}}{\sqrt{31}+\sqrt{32}}.\]

2016 Dutch IMO TST, 3

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

2008 iTest Tournament of Champions, 2

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Find the value of $|xy|$ given that $x$ and $y$ are integers and \[6x^2y^2+5x^2-18y^2=17253.\]

2002 JBMO ShortLists, 13

Tags: inequalities , rectangle , geometry

Let $ A_1,A_2,...,A_{2002}$ be arbitrary points in the plane. Prove that for every circle of radius $ 1$ and for every rectangle inscribed in this circle, there exist $3$ vertices $ M,N,P$ of the rectangle such that $ MA_1 + MA_2 + \cdots + MA_{2002} + $ $NA_1 + NA_2 + \cdots + NA_{2002} + $ $PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006$.

2004 Estonia National Olympiad, 3

Tags: combinatorics , combinatorial geometry

From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.

2005 VTRMC, Problem 3

Tags: recurrence relation , counting , combinatorics

We wish to tile a strip of $n$ $1$-inch by $1$-inch squares. We can use dominos which are made up of two tiles that cover two adjacent squares, or $1$-inch square tiles which cover one square. We may cover each square with one or two tiles and a tile can be above or below a domino on a square, but no part of a domino can be placed on any part of a different domino. We do not distinguish whether a domino is above or below a tile on a given square. Let $t(n)$ denote the number of ways the strip can be tiled according to the above rules. Thus for example, $t(1)=2$ and $t(2)=8$. Find a recurrence relation for $t(n)$, and use it to compute $t(6)$.

2016 Dutch BxMO TST, 3

Tags: geometry , incircle , collinear , right triangle

Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$. (a) Prove that $\vartriangle ABC \sim \vartriangle DEF$. (b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.

2010 Contests, 3

Tags: set , set theory , interval , union , intersection

Let $I_1, I_2, I_3$ be three open intervals of $\mathbb{R}$ such that none is contained in another. If $I_1\cap I_2 \cap I_3$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.

2025 Belarusian National Olympiad, 9.6

Tags: algebra , inequality

Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$ [i]M. Karpuk[/i]

2020 Peru EGMO TST, 5

Tags: angle bisector , geometry

Let $AD$ be the diameter of a circle $\omega$ and $BC$ is a chord of $\omega$ which is perpendicular to $AD$. Let $M,N,P$ be points on the segments $AB,AC,BC$ respectively, such that $MP\parallel AC$ and $PN\parallel AB$. The line $MN$ cuts the line $PD$ in the point $Q$ and the angle bisector of $\angle MPN$ in the point $R$. Prove that the points $B,R,Q,C$ are concyclic.

2023 USA TSTST, 5

Tags: algebra , complex number

Suppose $a,\,b,$ and $c$ are three complex numbers with product $1$. Assume that none of $a,\,b,$ and $c$ are real or have absolute value $1$. Define \begin{tabular}{c c c} $p=(a+b+c)+\left(\dfrac 1a+\dfrac 1b+\dfrac 1c\right)$ & \text{and} & $q=\dfrac ab+\dfrac bc+\dfrac ca$. \end{tabular} Given that both $p$ and $q$ are real numbers, find all possible values of the ordered pair $(p,q)$. [i]David Altizio[/i]

2008 Bulgaria National Olympiad, 1

Tags: quadratic , number theory

Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number.

2012 Romania National Olympiad, 1

Tags: trigonometry , algebra

[color=darkred]Let $M=\{x\in\mathbb{C}\, |\, |z|=1,\ \text{Re}\, z\in\mathbb{Q}\}\, .$ Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set $M$ .[/color]

2018 Moldova Team Selection Test, 3

Tags: geometry , angle chasing , projective geometry , similar triangles

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2011 NIMO Problems, 15

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Let \[ N = \sum_{a_1 = 0}^2 \sum_{a_2 = 0}^{a_1} \sum_{a_3 = 0}^{a_2} \dots \sum_{a_{2011} = 0}^{a_{2010}} \left [ \prod_{n=1}^{2011} a_n \right ]. \] Find the remainder when $N$ is divided by 1000. [i]Proposed by Lewis Chen [/i]

2011 Sharygin Geometry Olympiad, 3

Tags: geometry , perpendicular , circumcircle , concurrency

The line passing through vertex $A$ of triangle $ABC$ and parallel to $BC$ meets the circumcircle of $ABC$ for the second time at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the perpendiculars from $A_1, B_1, C_1$ to $BC, CA, AB$ respectively concur.

2019 PUMaC Geometry B, 4

Tags: geometry , probability , number theory , relatively prime

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

2014 NIMO Summer Contest, 10

Tags: probability , expected value

Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$. [i]Proposed by Evan Chen[/i]

2018 CCA Math Bonanza, L4.3

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$ABC$ is an isosceles triangle with $AB=AC$. Point $D$ is constructed on AB such that $\angle{BCD}=15^\circ$. Given that $BC=\sqrt{6}$ and $AD=1$, find the length of $CD$. [i]2018 CCA Math Bonanza Lightning Round #4.3[/i]

2022 HMNT, 20

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Let $\triangle{ABC}$ be an isosceles right triangle with $AB=AC=10.$ Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $BM.$ Let $AN$ hit the circumcircle of $\triangle{ABC}$ again at $T.$ Compute the area of $\triangle{TBC}.$

2007 Hungary-Israel Binational, 3

Tags: trigonometry , function , calculus , derivative , inequalities , geometry

Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.

2013 Saudi Arabia GMO TST, 4

Tags: altitude , geometry , perpendicular , circles

In acute triangle $ABC$, points $D$ and $E$ are the feet of the perpendiculars from $A$ to $BC$ and $B$ to $CA$, respectively. Segment $AD$ is a diameter of circle $\omega$. Circle $\omega$ intersects sides $AC$ and $AB$ at $F$ and $G$ (other than $A$), respectively. Segment $BE$ intersects segments $GD$ and $GF$ at $X$ and $Y$ respectively. Ray $DY$ intersects side $AB$ at $Z$. Prove that lines $XZ$ and $BC$ are perpendicular

V Soros Olympiad 1998 - 99 (Russia), 8.5

Tags: geometry , collinear

Points $A$, $B$ and $C$ lie on one side of the angle with the vertex at point $O$, and points $A'$, $B'$ and $C'$ lie on the other. It is known that$ B$ is the midpoint of the segment $AC$, $B'$ is the midpoint of the segment $A'C'$, and lines $AA'$, $BB'$ and $CC'$ are parallel (fig.). Prove that the centers of the circles circumscribed around the triangles $OAC$, $OA'C$ and $OBB'$ lie on the same straight line. [img]https://cdn.artofproblemsolving.com/attachments/d/6/92831077781bc45f25e9f71077034f84753a59.png[/img]

2021 Malaysia IMONST 2, 3

Tags: recurrence

Given a sequence of positive integers $$a_1, a_2, a_3, a_4, a_5, \dots$$ such that $a_2 > a_1$ and $a_{n+2} = 3a_{n+1} - 2a_n$ for all $n \geq 1$. Prove that $a_{2021} > 2^{2019}$.

2022 MOAA, 13

Tags: floor function , algebra

Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$ . Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.