Olympiad problems

1997 Tuymaada Olympiad, 1

The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.

2023 Chile National Olympiad, 3

Tags: geometry , Equilateral , areas

Let $\vartriangle ABC$ be an equilateral triangle with side $1$. $1011$ points $P_1$, $P_2$, $P_3$, $...$, $P_{1011}$ on the side $AC$ and $1011$ points $Q_1$, $Q_2$, $Q_3$, $...$ ,$ Q_{1011}$ on side AB (see figure) in such a way as to generate $2023$ triangles of equal area. Find the length of the segment $AP_{1011}$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/fea495c16a0b626e0c3882df66d66011a1a3af.png[/img] PS. Harder version of [url=https://artofproblemsolving.com/community/c4h3323135p30741470]2023 Chile NMO L1 P3[/url]

1995 Italy TST, 4

Tags: geometry , circumcircle , geometric transformation , reflection , vector , geometry proposed

In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.

2018 IMO Shortlist, C6

Tags: IMO Shortlist , combinatorics , IMO shortlist 2018

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

2022 Dutch IMO TST, 2

Tags: geometry , concurrency , common tangents

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

2011 Morocco National Olympiad, 1

Tags: inequalities , calculus , derivative , inequalities unsolved

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

2017 AIME Problems, 2

Tags: AMC , AIME , 2017 AIME I

When each of 702, 787, and 855 is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of 412, 722, and 815 is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Fine $m+n+r+s$.

2012 JBMO ShortLists, 1

Tags: geometry , circumcircle

Let $ABC$ be an equilateral triangle , and $P$ be a point on the circumcircle of the triangle but distinct from $A$ ,$B$ and $C$. The lines through $P$ and parallel to $BC$ , $CA$ , $AB$ intersect the lines $CA$ , $AB$ , $BC$ at $M$ , $N$ and $Q$ respectively .Prove that $M$ , $N$ and $Q$ are collinear .

2018 Israel National Olympiad, 2

Tags: arithmetic sequence , geometric sequence , number theory , algebra , geometry

An [i]arithmetic sequence[/i] is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$. A [i]geometric sequence[/i] is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$. [list=a] [*] Does every arithmetic sequence of [b]integers[/b] have an infinite subsequence which is geometric? [*] Does every arithmetic sequence of [b]real numbers[/b] have an infinite subsequence which is geometric? [/list]

2010 AMC 12/AHSME, 25

Tags: function , AoPSwiki , AMC

For every integer $ n\ge 2$, let $ \text{pow}(n)$ be the largest power of the largest prime that divides $ n$. For example $ \text{pow}(144)\equal{}\text{pow}(2^4\cdot 3^2)\equal{}3^2$. What is the largest integer $ m$ such that $ 2010^m$ divides \[ \prod_{n\equal{}2}^{5300}\text{pow}(n)\text{?}\] $ \textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78$

2012 Thailand Mathematical Olympiad, 3

Tags: floor function , number theory , Even , Integer

Let $m, n > 1$ be coprime odd integers. Show that $$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$ is an even integer, where $\phi$ is Euler’s totient function.

2011 Mongolia Team Selection Test, 3

Tags: function , graph theory , combinatorics unsolved , combinatorics

Let $n$ and $d$ be positive integers satisfying $d<\dfrac{n}{2}$. There are $n$ boys and $n$ girls in a school. Each boy has at most $d$ girlfriends and each girl has at most $d$ boyfriends. Prove that one can introduce some of them to make each boy have exactly $2d$ girlfriends and each girl have exactly $2d$ boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa) (proposed by B. Batbaysgalan, folklore).

2016 Kurschak Competition, 2

Tags: combinatorics

Prove that for any finite set $A$ of positive integers, there exists a subset $B$ of $A$ satisfying the following conditions: [list][*]if $b_1,b_2\in B$ are distinct, then neither $b_1$ and $b_2$ nor $b_1+1$ and $b_2+1$ are multiples of each other, and [*] for any $a\in A$, we can find a $b\in B$ such that $a$ divides $b$ or $b+1$ divides $a+1$.[/list]

Maryland University HSMC part II, 2023.3

Tags: number theory , combination , Divisibility

Let $p$ be a prime, and $n > p$ be an integer. Prove that \[ \binom{n+p-1}{p} - \binom{n}{p} \] is divisible by $n$.

2013 District Olympiad, 1

Tags: limit , integration , calculus , calculus computations

Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$

2016 Saudi Arabia BMO TST, 1

Tags: SAU , Divisibility

Let $ a > b > c > d $ be positive integers such that \begin{align*} a^2 + ac - c^2 = b^2 + bd - d^2 \end{align*} Prove that $ ab + cd $ is a composite number.

1998 Belarus Team Selection Test, 1

Tags: geometry , Locus , circles , angles

Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

2011 India IMO Training Camp, 1

Tags: geometry , circumcircle , reflection , parallelogram , IMO Shortlist

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2011 AMC 12/AHSME, 2

Tags: inequalities , AMC

Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal? $ \textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95 $

2009 Junior Balkan Team Selection Tests - Romania, 3

Tags: concurrency , concurrent , Segment , geometry

Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.

2022 Iran MO (3rd Round), 1

Tags: combinatorics , graph theory , Tournament , vertex degree

For each natural number $k$ find the least number $n$ such that in every tournament with $n$ vertices, there exists a vertex with in-degree and out-degree at least $k$. (Tournament is directed complete graph.)

1996 All-Russian Olympiad Regional Round, 8.7

Tags: number theory , algebra

Dunno wrote several different natural numbers on the board and divided (in his head) the sum of these numbers by their product. After this, Dunno erased the smallest number and divided (again in his mind) the amount of the remaining numbers by their product. The second result was $3$ times greater than the first. What number did Dunno erase?

2019 Sharygin Geometry Olympiad, 20

Tags: geometry

Let $O$ be the circumcenter of triangle ABC, $H$ be its orthocenter, and $M$ be the midpoint of $AB$. The line $MH$ meets the line passing through $O$ and parallel to $AB$ at point $K$ lying on the circumcircle of $ABC$. Let $P$ be the projection of $K$ onto $AC$. Prove that $PH \parallel BC$.

2010 F = Ma, 8

Tags: 2010 , Problem 8

A car attempts to accelerate up a hill at an angle $\theta$ to the horizontal. The coefficient of static friction between the tires and the hill is $\mu > \tan \theta$. What is the maximum acceleration the car can achieve (in the direction upwards along the hill)? Neglect the rotational inertia of the wheels. (A) $g \tan \theta$ (B) $g(\mu \cos \theta - \sin \theta)$ (C) $g(\mu - \sin \theta)$ (D) $g \mu \cos \theta$ (E) $g(\mu \sin \theta - \cos \theta)$

2001 Cuba MO, 6

Tags: algebra , inequalities

The roots of the equation $ax^2 - 4bx + 4c = 0$ with $ a > 0$ belong to interval $[2, 3]$. Prove that: a) $a \le b \le c < a + b.$ b) $\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} .$