Olympiad problems

2021 Middle European Mathematical Olympiad, 4

Let $n \ge 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n-1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots,n-1 \}$. Prove that if after $\lceil \tfrac{n}{2} \rceil$ jumps Zagi has visited $\lceil \tfrac{n}{2} \rceil+1$ distinct vertices, then after $n-1$ jumps Zagi will have visited all of the vertices. ([i]Remark.[/i] For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)

2008 District Olympiad, 4

Tags: geometry , parallelogram , trigonometry , Gauss , geometric transformation , reflection , ratio

Let $ ABCD$ be a cyclic quadrilater. Denote $ P\equal{}AD\cap BC$ and $ Q\equal{}AB \cap CD$. Let $ E$ be the fourth vertex of the parallelogram $ ABCE$ and $ F\equal{}CE\cap PQ$. Prove that $ D,E,F$ and $ Q$ lie on the same circle.

2015 Geolympiad Spring, 1

Tags: Geolympiad Spring 2015

Let $ABC$ be a triangle. Suppose $P,Q$ are on lines $AB, AC$ (on the same side of A) with $AP=AC$ and $AB=AQ$. Now suppose points $X,Y$ move along the sides $AB, AC$ of $ABC$ so that $XY || PQ$. Determine the locus of the circumcenters of the variable triangle $AXY$.

1962 All-Soviet Union Olympiad, 3

Tags: Russia , algebra , Sequences

Given integers $a_0,a_1, ... , a_{100}$, satisfying $a_1>a_0$, $a_1>0$, and $a_{r+2}=3 a_{r+1}-2a_r$ for $r=0, 1, ... , 98$. Prove $a_{100}>299$

2006 Sharygin Geometry Olympiad, 3

Tags: geometry , construction , Circumcenter

The map shows sections of three straight roads connecting the three villages, but the villages themselves are located outside the map. In addition, the fire station located at an equal distance from the three villages is not indicated on the map, although its location is within the map. Is it possible to find this place with the help of a compass and a ruler, if the construction is carried out only within the map?

2016 Mathematical Talent Reward Programme, MCQ: P 8

Tags: prime numbers , number theory

Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is [list=1] [*] 283 [*] 307 [*] 593 [*] 691 [/list]

2016 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry

Let $ABC$ be a acute triangle where $\angle BAC =60$. Prove that if the Euler's line of $ABC$ intersects $AB,AC$ in $D,E$, then $ADE$ is equilateral.

1989 Poland - Second Round, 6

Tags: geometry , tangential

In the triangle $ ABC $, the lines $ CP $, $ AP $, $ BP $ are drawn through the internal point $ P $ and intersect the sides $ AB $, $ BC $, $ CA $ at points $ K $, $ L $, $ M$, respectively. Prove that if circles can be inscribed in the quadrilaterals $ AKPM $ and $ KBLP $, then a circle can also be inscribed in the quadrilateral $ LCMP $.

2020 CMIMC Team, 2

Tags: team , 2020

Find all sets of five positive integers whose mode, mean, median, and range are all equal to $5$.

2008 Vietnam National Olympiad, 2

Tags: trigonometry , geometry unsolved , geometry

Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \angle BME \equal{} \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac{MC}{AB}$ in terms of $ \theta.$

2012 Moldova Team Selection Test, 12

Tags: combinatorics

Let $k \in \mathbb{N}$. Prove that \[ \binom{k}{0} \cdot (x+k)^k - \binom{k}{1} \cdot (x+k-1)^k+...+(-1)^k \cdot \binom{k}{k} \cdot x^k=k! ,\forall k \in \mathbb{R}\]

2016 Latvia Baltic Way TST, 12

Tags: geometry , points

For what positive numbers $m$ and $n$ do there exist points $A_1, ..., Am$ and $B_1 ..., B_n$ in the plane such that, for any point $P$, the equation $$|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2 $$ holds true?

2017 Israel National Olympiad, 5

Tags: national olympiad , geometry , pentagon , circumcircle

A regular pentagon $ABCDE$ is given. The point $X$ is on his circumcircle, on the arc $\overarc{AE}$. Prove that $|AX|+|CX|+|EX|=|BX|+|DX|$. [u][b]Remark:[/b][/u] Here's a more general version of the problem: Prove that for any point $X$ in the plane, $|AX|+|CX|+|EX|\ge|BX|+|DX|$, with equality only on the arc $\overarc{AE}$.

2014 Tuymaada Olympiad, 4

Tags: geometry , parallelogram , induction , combinatorics unsolved , combinatorics , Tuymaada

A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

OIFMAT III 2013, 5

Tags: concurrency , concurrent , geometry

In an acute triangle $ ABC $ with circumcircle $ \Omega $ and circumcenter $ O $, the circle $ \Gamma $ is drawn, passing through the points $ A $, $ O $ and $ C $ together with its diameter $ OQ $, then the points $ M $ and $ N $ are chosen on the lines $ AQ $ and $ AC $, respectively, in such a way that the quadrilateral $ AMBN $ is a parallelogram. Prove that the point of intersection of the lines $ MN $ and $ BQ $ lies on the circle $ \Gamma $.

1980 AMC 12/AHSME, 23

Tags: trigonometry , Pythagorean Theorem , geometry , similar triangles

Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}2$. The length of the hypotenuse is $\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}$

2021 CMIMC, 1.8

Tags: algebra , number theory

There are integers $v,w,x,y,z$ and real numbers $0\le \theta < \theta' \le \pi$ such that $$\cos 3\theta = \cos 3\theta' = v^{-1}, \qquad w+x\cos \theta + y\cos 2\theta = z\cos \theta'.$$ Given that $z\ne 0$ and $v$ is positive, find the sum of the $4$ smallest possible values of $v$. [i]Proposed by Vijay Srinivasan[/i]

2020 JBMO TST of France, 4

Tags: algebra , Inequality , TST

$a, b, c$ are real positive numbers for which $a+b+c=3$. Prove that $a^{12}+b^{12}+c^{12}+8(ab+bc+ca) \geq 27$

2010 IFYM, Sozopol, 5

Tags: geometry , excircle

We are given $\Delta ABC$, for which the excircle to side $BC$ is tangent to the continuations of $AB$ and $AC$ in points $E$ and $F$ respectively. Let $D$ be the reflection of $A$ in line $EF$. If it is known that $\angle BAC=2\angle BDC$, then determine $\angle BAC$.

2015 VTRMC, Problem 4

Tags: series

Consider the harmonic series $\sum_{n\ge1}\frac1n=1+\frac12+\frac13+\ldots$. Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms $\frac1k$.)

2023 AIME, 12

Tags: AMC , AIME , AIME II

In $\triangle ABC$ with side lengths $AB=13$, $BC=14$, and $CA=15$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}$. There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ$. Then $AQ$ can be written as $\frac{m}{\sqrt{n}}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2020 BMT Fall, 17

Tags: combinatorics

Shrek throws $5$ balls into $5$ empty bins, where each ball’s target is chosen uniformly at random. After Shrek throws the balls, the probability that there is exactly one empty bin can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

2020 Harvard-MIT Mathematics Tournament, 7

Tags:

Find the sum of all positive integers $n$ for which \[\frac{15\cdot n!^2+1}{2n-3}\] is an integer. [i]Proposed by Andrew Gu.[/i]

2010 Switzerland - Final Round, 2

Tags: geometry , geometry proposed

Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$. Show that $ PMID$ ist cyclic.

1991 AMC 12/AHSME, 20

Tags: geometry , algebra , polynomial , Vieta , logarithms , AMC

The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is $ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $