Olympiad problems

2014 USAMO, 5

Tags: geometry , circumcircle , vector , geometric transformation , parallelogram , trigonometry , complex bash

Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

2014 NIMO Problems, 3

Tags: calculus , integration

In land of Nyemo, the unit of currency is called a [i]quack[/i]. The citizens use coins that are worth $1$, $5$, $25$, and $125$ quacks. How many ways can someone pay oﬀ $125$ quacks using these coins? [i]Proposed by Aaron Lin[/i]

2025 Harvard-MIT Mathematics Tournament, 25

Tags: guts

Let $ABCD$ be a trapezoid such that $AB \parallel CD, AD=13, BC=15, AB=20,$ and $CD=34.$ Point $X$ lies inside the trapezoid such that $\angle{XAB}=2\angle{XBA}$ and $\angle{XDC}=2\angle{XCD}.$ Compute $XD-XA.$

1966 IMO Longlists, 2

Tags: inequalities

Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove \[ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.\]

2011 Paraguay Mathematical Olympiad, 2

Tags: ratio , vector , similar triangles , geometry

In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AC$ and $BC$ respectively. The distance from the midpoint of $BD$ to the midpoint of $AE$ is $4.5$. What is the length of side $AB$?

2017 Brazil National Olympiad, 6.

Tags: number theory , divisibility , prime , group theory

[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.

PEN G Problems, 3

Tags: irrational number

Prove that there exist positive integers $ m$ and $ n$ such that \[ \left\vert\frac{m^{2}}{n^{3}}\minus{}\sqrt{2001}\right\vert <\frac{1}{10^{8}}.\]

2011 Laurențiu Duican, 2

Tags: limit , integration , real analysis , squeeze theorem , definite integral , calculus

$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $ [i]Gabriela Boeriu[/i]

2011 Romania National Olympiad, 4

Tags: function , integration , real analysis

Let $ f,F:\mathbb{R}\longrightarrow\mathbb{R} $ be two functions such that $ f $ is nondecreasing, $ F $ admits finite lateral derivates in every point of its domain, $$ \lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} , $$ for all real numbers $ y, $ and $ F(0)=0. $ Prove that $ F(x)=\int_0^x f(t)dt, $ for all real numbers $ x. $

2017 CentroAmerican, 1

Tags: geometry , geometric transformation , reflection , angle bisector

$ABC$ is a right-angled triangle, with $\angle ABC = 90^{\circ}$. $B'$ is the reflection of $B$ over $AC$. $M$ is the midpoint of $AC$. We choose $D$ on $\overrightarrow{BM}$, such that $BD = AC$. Prove that $B'C$ is the angle bisector of $\angle MB'D$. NOTE: An important condition not mentioned in the original problem is $AB<BC$. Otherwise, $\angle MB'D$ is not defined or $B'C$ is the external bisector.

2022 Austrian MO Beginners' Competition, 2

Tags: combinatorics , tiles , tiling

You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible? (Walther Janous)

2016 Purple Comet Problems, 18

Tags:

Find the least positive integer $N$ that is 50 times the number of positive integer divisors that $N$ has.

2011 Croatia Team Selection Test, 2

Tags: induction , group theory , abstract algebra , graph theory , combinatorics

There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).

2023 Simon Marais Mathematical Competition, B4

Tags: number theory

[i](The following problem is open in the sense that the answer to part (b) is not currently known.)[/i] [list=a] [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a positive real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [/list]

2006 Baltic Way, 7

Tags: combinatorics

A photographer took some pictures at a party with $10$ people. Each of the $45$ possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?

1993 Dutch Mathematical Olympiad, 2

Tags: ratio , geometry

In a triangle $ ABC$ with $ \angle A\equal{}90^{\circ}$, $ D$ is the midpoint of $ BC$, $ F$ that of $ AB$, $ E$ that of $ AF$ and $ G$ that of $ FB$. Segment $ AD$ intersects $ CE,CF$ and $ CG$ in $ P,Q$ and $ R$, respectively. Determine the ratio: $ \frac{PQ}{QR}$.

2008 AMC 12/AHSME, 14

Tags: logarithm

A circle has a radius of $ \log_{10}(a^2)$ and a circumference of $ \log_{10}(b^4)$. What is $ \log_ab$? $ \textbf{(A)}\ \frac {1}{4\pi} \qquad \textbf{(B)}\ \frac {1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}$

1981 Tournament Of Towns, (009) 3

Tags: diagonal , geometry , cyclic , perpendicular , area , bisects area

$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area. (V Varvarkin)

2024 ELMO Shortlist, N8

Tags: number theory

Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that: [list] [*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and [*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$. [/list] [i]Allen Wang[/i]

2014 Belarus Team Selection Test, 3

Tags: combinatorial geometry , combinatorics , coloring

$n$ points are marked on a plane. Each pair of these points is connected with a segment. Each segment is painted one of four different colors. Find the largest possible value of $n$ such that one can paint the segments so that for any four points there are four segments (connecting these four points) of four different colors. (E. Barabanov)

2012-2013 SDML (Middle School), 3

Tags:

Jeffrey jogs $3$ miles at $4$ miles per hour. Then, he walks $4$ miles at $3$ miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth.

JOM 2014, 5.

Tags: geometry

Given $\triangle ABC$ with circumcircle $\Gamma$ and circumcentre $O$, let $X$ be a point on $\Gamma$. Let $XC_1$, $XB_1$ to be feet of perpendiculars from $X$ to lines $AB$ and $AC$. Define $\omega_C$ as the circle with centre the midpoint of $AB$ and passing through $C_1$ . Define $\omega_B$ similarly. Prove that $\omega_B$ and $\omega_C$ has a common point on $XO$.

2015 CentroAmerican, Problem 3

Tags: geometry , cyclic quadrilateral , circumcircle

Let $ABCD$ be a cyclic quadrilateral with $AB<CD$, and let $P$ be the point of intersection of the lines $AD$ and $BC$.The circumcircle of the triangle $PCD$ intersects the line $AB$ at the points $Q$ and $R$. Let $S$ and $T$ be the points where the tangents from $P$ to the circumcircle of $ABCD$ touch that circle. (a) Prove that $PQ=PR$. (b) Prove that $QRST$ is a cyclic quadrilateral.

2008 All-Russian Olympiad, 4

Tags: geometry , 3d geometry , tetrahedron , sphere , circumcircle , function , analytic geometry

Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

1991 Arnold's Trivium, 78

Tags: trigonometry

Solve the Cauchy problem \[\frac{\partial ^2A}{\partial t^2}=9\frac{\partial^2 A}{\partial x^2}-2B,\;\frac{\partial^2 B}{\partial t^2}=6\frac{\partial^2 B}{\partial x^2}-2A\] \[A|_{t=0}=\cos x,\; B|_{t=0}=0,\; \left.\frac{\partial A}{\partial t}\right|_{t=0}=\left.\frac{\partial B}{\partial t}\right|_{t=0}=0\]