Olympiad problems

1999 IMO Shortlist, 1

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.

1999 IMO Shortlist, 3

Tags: geometry , symmetry , 3D geometry , reflection , IMO , IMO 1999 , Convex hull

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

2018 Stars of Mathematics, 1

Tags: number theory , IMO Shortlist , Belarus , IMO 1999

Two natural numbers have the property that the product of their positive divisors are equal. Does this imply that they are equal? [i]Proposed by Belarus for the 1999th IMO[/i]

1999 IMO, 5

Tags: geometry , homothety , circumcircle , IMO , IMO 1999

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

1999 IMO, 6

Tags: algebra , functional equation , IMO , IMO 1999 , IMO Shortlist

Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

1999 IMO Shortlist, 5

Tags: algebra , functional equation , IMO , IMO 1999 , IMO Shortlist

Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

1999 IMO Shortlist, 5

Tags: combinatorics , Extremal combinatorics , matrix , IMO , IMO 1999 , IMO Shortlist

Let $n$ be an even positive integer. We say that two different cells of a $n \times n$ board are [b]neighboring[/b] if they have a common side. Find the minimal number of cells on the $n \times n$ board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.

1999 IMO Shortlist, 6

Tags: geometry , homothety , circumcircle , IMO , IMO 1999

Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.

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.

1999 IMO, 1

Tags: geometry , symmetry , 3D geometry , reflection , IMO , IMO 1999 , Convex hull

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

1999 IMO, 3

Tags: combinatorics , Extremal combinatorics , matrix , IMO , IMO 1999 , IMO Shortlist

Let $n$ be an even positive integer. We say that two different cells of a $n \times n$ board are [b]neighboring[/b] if they have a common side. Find the minimal number of cells on the $n \times n$ board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.

1999 IMO, 4

Tags: number theory , Divisibility , prime , IMO Shortlist , IMO , IMO 1999 , Hi

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

1999 IMO Shortlist, 1

Tags: number theory , Divisibility , prime , IMO Shortlist , IMO , IMO 1999 , Hi

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.