Olympiad problems

2007 Iran MO (3rd Round), 2

Tags: circumcircle , geometry

a) Let $ ABC$ be a triangle, and $ O$ be its circumcenter. $ BO$ and $ CO$ intersect with $ AC,AB$ at $ B',C'$. $ B'C'$ intersects the circumcircle at two points $ P,Q$. Prove that $ AP\equal{}AQ$ if and only if $ ABC$ is isosceles. b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.

2011 LMT, 17

Tags: geometry

Let $ABC$ be a triangle with $AB = 15$, $AC = 20$, and right angle at $A$. Let $D$ be the point on $\overline{BC}$ such that $\overline{AD}$ is perpendicular to $\overline{BC}$, and let $E$ be the midpoint of $\overline{AC}$. If $F$ is the point on $\overline{BC}$ such that $\overline{AD} \parallel \overline{EF}$, what is the area of quadrilateral $ADFE$?

2020 Tuymaada Olympiad, 8

Tags: combinatorics

In a horizontal strip $1 \times n$ made of $n$ unit squares the vertices of all squares are marked. The strip is partitioned into parts by segments connecting marked points and not lying on the sides of the strip. The segments can not have common inner points; the upper end of each segment must be either above the lower end or further to the right. Prove that the number of all partitions is divisible by $2^n$. (The partition where no segments are drawn, is counted too.) [i](E. Robeva, M. Sun)[/i]

2007 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities

Let $a, b, c$ three positive reals such that \[\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\geq 1. \] Show that \[a+b+c\geq ab+bc+ca. \]

2002 Stanford Mathematics Tournament, 1

Tags: algebra , polynomial

Completely factor the polynomial $x^4-x^3-5x^2+3x+6$

2022 Austrian Junior Regional Competition, 3

Tags: geometry , equal segments , semicircle

A semicircle is erected over the segment $AB$ with center $M$. Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$. The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$. Prove that $PM = PS$ holds. [i](Karl Czakler)[/i]

1987 Polish MO Finals, 3

Tags: polynomial , sequence , sum of digits , number theory , algebra

$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .

2019 India PRMO, 14

Tags: prime , perfect square , relation

Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.

2010 China National Olympiad, 2

Tags: combinatorics

There is a deck of cards placed at every points $A_1, A_2, \ldots , A_n$ and $O$, where $n \geq 3$. We can do one of the following two operations at each step: $1)$ If there are more than 2 cards at some points $A_i$, we can withdraw three cards from that deck and place one each at $A_{i-1}, A_{i+1}$ and $O$. (Here $A_0=A_n$ and $A_{n+1}=A_1$); $2)$ If there are more than or equal to $n$ cards at point $O$, we can withdraw $n$ cards from that deck and place one each at $A_1, A_2, \ldots , A_n$. Show that if the total number of cards is more than or equal to $n^2+3n+1$, we can make the number of cards at every points more than or equal to $n+1$ after finitely many steps.

2003 AMC 12-AHSME, 12

Tags: algebra , polynomial

What is the largest integer that is a divisor of \[ (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) \]for all positive even integers $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 165$

2005 IMO Shortlist, 2

Tags: geometry , ratio , triangle , concurrency

Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent. [i]Bogdan Enescu, Romania[/i]

2015 AoPS Mathematical Olympiad, 5

Tags: circumcircle , geometry

Let $ABC$ be a triangle with orthocenter $h$. Let $AH$, $BH$, and $CH$ intersect the circumcircle of $\triangle ABC$ at points $D$, $E$, and $F$. Find the maximum value of $\frac{[DEF]}{[ABC]}$. (Here $[X]$ denotes the area of $X$.) [i]Proposed by tkhalid.[/i]

2011 China Team Selection Test, 2

Tags: induction , number theory

Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and \[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\] Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

2011 Mathcenter Contest + Longlist, 1 sl1

Tags: inequalities , algebra

Let $a,b,c \in \mathbb{R}$. Prove that $$\sum_{cyc} (a^3-b^3)^2+3\sum_{cyc}(a^2-b^2)^2+6(a-b)(b-c)(c-a)(ab+ bc+ca) \ge 0.$$ [i](LightLucifer)[/i]

2023 Grand Duchy of Lithuania, 1

Tags: functional equation , algebra

Given a non-zero real number $a$. Find all functions $f : R \to R$, such that $$f(f(x + y)) = f(x + y) + f(x)f(y) + axy$$ for all $x, y \in R$.

2010 Tournament Of Towns, 4

Tags: geometry , induction , rectangle

A square board is dissected into $n^2$ rectangular cells by $n-1$ horizontal and $n-1$ vertical lines. The cells are painted alternately black and white in a chessboard pattern. One diagonal consists of $n$ black cells which are squares. Prove that the total area of all black cells is not less than the total area of all white cells.

2020 IMEO, Problem 5

Tags: number theory , function

For a positive integer $n$ with prime factorization $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ let's define $\lambda(n) = (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_k}$. Define $L(n)$ as sum of $\lambda(x)$ over all integers from $1$ to $n$. Define $K(n)$ as sum of $\lambda(x)$ over all [b]composite[/b] integers from $1$ to $n$. For some $N>1$, we know, that for every $2\le n \le N$, $L(n)\le 0$. Prove that for this $N$, for every $2\le n \le N$, $K(n)\ge 0$. [i]Mykhailo Shtandenko[/i]

1924 Eotvos Mathematical Competition, 2

Tags: geometry , locus , fixed

If $O$ is a given point, $\ell$ a given line, and $a$ a given positive number, find the locus of points $P$ for which the sum of the distances from $P$ to $O$ and from $P$ to $\ell$ is $a$.

2011 AMC 12/AHSME, 20

Tags: algebra , polynomial

Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5 $

Durer Math Competition CD 1st Round - geometry, 2018.C3

Tags: geometry , midpoint , isosceles

In the isosceles triangle $ABC$, $AB = AC$. Let $E$ be on side $AB$ such that $\angle ACE = \angle ECB = 18^o$, and let $D$ be the midpoint of side $CB$. If we know the length of $AD$ is $3$ units, what is the length of $CE$?

2013 IMC, 3

Tags:

There are $\displaystyle{2n}$ students in a school $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$. Each week $\displaystyle{n}$ students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen? [i]Proposed by Oleksandr Rybak, Kiev, Ukraine.[/i]

2008 Baltic Way, 11

Tags: number theory

Consider a subset $A$ of $84$ elements of the set $\{1,\,2,\,\dots,\,169\}$ such that no two elements in the set add up to $169$. Show that $A$ contains a perfect square.

2014 Contests, 3

Tags: greatest common divisor , modular arithmetic , number theory

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.

2020 Canada National Olympiad, 5

Tags: combinatorics , graph theory

Simple graph $G$ has $19998$ vertices. For any subgraph $\bar G$ of $G$ with $9999$ vertices, $\bar G$ has at least $9999$ edges. Find the minimum number of edges in $G$

2023 Yasinsky Geometry Olympiad, 2

Tags: geometry , incenter , construction

Let $I$ be the incenter of triangle $ABC$. $K_1$ and $K_2$ are the points on $BC$ and $AC$ respectively, at which the inscribed circle is tangent. Using a ruler and a compass, find the center of the inscribed circle for triangle $CK_1K_2$ in the minimal possible number of steps (each step is to draw a circle or a line). (Hryhorii Filippovskyi)