Olympiad problems

2003 AIME Problems, 6

In triangle $ABC,$ $AB=13,$ $BC=14,$ $AC=15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$

2024 Kyiv City MO Round 2, Problem 2

Tags: number theory , divisibility

Mykhailo wants to arrange all positive integers from $1$ to $2024$ in a circle so that each number is used exactly once and for any three consecutive numbers $a, b, c$ the number $a + c$ is divisible by $b + 1$. Can he do it? [i]Proposed by Fedir Yudin[/i]

2015 India National Olympiad, 3

Tags: function , trigonometry , functional equation , algebra

Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.

LMT Team Rounds 2010-20, 2020.S20

Tags:

Let $c_1<c_2<c_3$ be the three smallest positive integer values of $c$ such that the distance between the parabola $y=x^2+2020$ and the line $y=cx$ is a rational multiple of $\sqrt{2}$. Compute $c_1+c_2+c_3$.

2011 Belarus Team Selection Test, 2

Tags: equal angles , geometry

Two different points $X,Y$ are marked on the side $AB$ of a triangle $ABC$ so that $\frac{AX \cdot BX}{CX^2}=\frac{AY \cdot BY}{CY^2}$ . Prove that $\angle ACX=\angle BCY$. I.Zhuk

2007 ISI B.Math Entrance Exam, 7

Tags: trigonometry , function , inequalities

Let $ 0\leq \theta\leq \frac{\pi}{2}$ . Prove that $\sin \theta \geq \frac{2\theta}{\pi}$.

2011 Romania Team Selection Test, 3

Tags: combinatorics

Given a positive integer number $n$, determine the maximum number of edges a simple graph on $n$ vertices may have such that it contain no cycles of even length.

2021 Alibaba Global Math Competition, 8

Tags: complex analysis , inequalities , function

Let $f(z)$ be a holomorphic function in $\{\vert z\vert \le R\}$ ($0<R<\infty$). Define \[M(r,f)=\max_{\vert z\vert=r} \vert f(z)\vert, \quad A(r,f)=\max_{\vert z\vert=r} \text{Re}\{f(z)\}.\] Show that \[M(r,f) \le \frac{2r}{R-r}A(R,f)+\frac{R+r}{R-r} \vert f(0)\vert, \quad \forall 0 \le r<R.\]

2014 Mexico National Olympiad, 4

Tags: rectangle , geometric transformation , reflection , number theory , trigonometry , geometry

Problem 4 Let $ABCD$ be a rectangle with diagonals $AC$ and $BD$. Let $E$ be the intersection of the bisector of $\angle CAD$ with segment $CD$, $F$ on $CD$ such that $E$ is midpoint of $DF$, and $G$ on $BC$ such that $BG = AC$ (with $C$ between $B$ and $G$). Prove that the circumference through $D$, $F$ and $G$ is tangent to $BG$.

2002 Moldova National Olympiad, 1

Tags: function , triangle inequality , inequalities

Consider the real numbers $ a\ne 0,b,c$ such that the function $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ satisfies $ |f(x)|\le 1$ for all $ x\in [0,1]$. Find the greatest possible value of $ |a| \plus{} |b| \plus{} |c|$.

2013 National Olympiad First Round, 22

Tags:

For how many integers $0\leq n < 2013$, is $n^4+2n^3-20n^2+2n-21$ divisible by $2013$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ \text{None of above} $

2014 AMC 12/AHSME, 3

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Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? $ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $

2022 Purple Comet Problems, 8

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Find the number of divisors of $20^{22}$ that are perfect squares.

2015 District Olympiad, 4

Tags: modular arithmetic , number theory , arithmetic

[b]a)[/b] Show that the three last digits of $ 1038^2 $ are equal with $ 4. $ [b]b)[/b] Show that there are infinitely many perfect squares whose last three digits are equal with $ 4. $ [b]c)[/b] Prove that there is no perfect square whose last four digits are equal to $ 4. $

1992 Poland - Second Round, 6

Tags: algebra , sequence , recurrence relation

The sequences $(x_n)$ and $(y_n)$ are defined as follows: $$ x_{n+1} = \frac{x_n+2}{x_n+1},\quad y_{n+1}=\frac{y_n^2+2}{2y_n} \quad \text{ for } n= 0,1,2,\ldots.$$ Prove that for every integer $ n\geq 0 $ the equality $ y_n = x_{2^n-1} $ holds.

2006 Taiwan National Olympiad, 1

Tags: number theory

Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

1986 Balkan MO, 1

Tags: incenter , circumcircle , geometry

A line passing through the incenter $I$ of the triangle $ABC$ intersect its incircle at $D$ and $E$ and its circumcircle at $F$ and $G$, in such a way that the point $D$ lies between $I$ and $F$. Prove that: $DF \cdot EG \geq r^{2}$.

2005 IMO Shortlist, 4

Tags: algebra , functional equation

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$. [i]Proposed by B.J. Venkatachala, India[/i]

2022 Poland - Second Round, 6

Tags: combinatorics , graph theory

$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.

2010 Iran MO (3rd Round), 3

Tags: geometry , geometric transformation , reflection , circumcircle , homothety , ratio , power of a point

in a quadrilateral $ABCD$ digonals are perpendicular to each other. let $S$ be the intersection of digonals. $K$,$L$,$M$ and $N$ are reflections of $S$ to $AB$,$BC$,$CD$ and $DA$. $BN$ cuts the circumcircle of $SKN$ in $E$ and $BM$ cuts the circumcircle of $SLM$ in $F$. prove that $EFLK$ is concyclic.(20 points)

2009 Jozsef Wildt International Math Competition, W. 7

Tags: integration , inequalities , calculus , logarithm

If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$

2016 Junior Regional Olympiad - FBH, 2

Tags: root , algebra

If $$w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}$$ prove that $w=\sqrt{3}-1$

2002 Germany Team Selection Test, 1

Tags: function , quadratic , algebra

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

2000 Harvard-MIT Mathematics Tournament, 36

Tags: geometry , inradius , trigonometry , trig identities , law of cosines

If, in a triangle of sides $a, b, c$, the incircle has radius $\frac{b+c-a}{2}$, what is the magnitude of $\angle A$?

2017 QEDMO 15th, 6

Tags: number theory , diophantine , diophantine equation

Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.