Olympiad problems

2019 IMO Shortlist, N1

Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \] [i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]

1993 IMO, 5

Tags: function , algebra , functional equation , IMO , IMO 1993

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

2004 IMO Shortlist, 1

Tags: inequalities , IMO 2004 , triangle inequality , IMO Shortlist , n-variable inequality , IMO

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

1981 IMO Shortlist, 7

Tags: function , algebra , functional equation , IMO , IMO 1981

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

2012 IMO Shortlist, N7

Tags: IMO , IMO 2012 , number theory , equation , IMO Shortlist , Dusan Djukic

Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1. \] [i]Proposed by Dusan Djukic, Serbia[/i]

1992 IMO, 2

Tags: function , algebra , functional equation , IMO , IMO 1992

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

1963 IMO Shortlist, 1

Tags: algebra , equation , parametric equation , IMO , IMO 1963

Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

1976 IMO Longlists, 41

Tags: number theory , Product , maximization , IMO , imo 1976

Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

2020 IMO, 2

Tags: inequalities , IMO 2020 , IMO , algebra , IMO Shortlist 2020 , Mount Inquality , Hi

The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

1975 IMO Shortlist, 6

Tags: modular arithmetic , Divisibility , Digits , decimal representation , IMO , IMO 1975 , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

1998 IMO Shortlist, 5

Tags: matrix , combinatorics , IMO , Inequality , IMO 1998 , IMO Shortlist , Set systems

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

1984 IMO, 1

Tags: geometry , convex quadrilateral , circle , parallel , diameter , IMO , IMO 1984

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

1999 IMO, 2

Tags: inequalities , algebra , IMO , IMO 1999 , maximization , IMO Shortlist , Marcin Kuczma

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

1968 IMO, 1

Tags: trigonometry , algebra , Triangle , triangle inequality , IMO , IMO 1968

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

1960 IMO Shortlist, 2

Tags: inequalities , algebra , IMO , IMO 1960

For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]

1969 IMO Shortlist, 67

Tags: Inequality , IMO , IMO 1969 , algebra

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

1985 IMO, 5

Tags: geometry , circumcircle , IMO , IMO 1985 , angle , Triangle

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

1988 IMO, 3

Tags: algebra , functional equation , binary representation , function , IMO , IMO 1988 , david monk

A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

2005 IMO Shortlist, 4

Tags: geometry , IMO Shortlist , IMO 2005 , IMO , Waldemar Pompe , Spiral Similarity , Harry Potter

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

1973 IMO, 1

Tags: geometry , triangle inequality , minimization , Triangle , optimization , IMO , IMO 1973

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

2007 IMO, 3

Tags: algorithm , graph theory , IMO , combinatorics , IMO 2007 , Clique number , IMO Shortlist

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a [i]clique[/i] if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its [i]size[/i]. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room. [i]Author: Vasily Astakhov, Russia[/i]

1984 IMO, 3

Tags: geometry , combinatorial geometry , Coloring , Ramsey Theory , IMO , IMO 1984

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.