Olympiad problems

1996 Abels Math Contest (Norwegian MO), 3

Per and Kari each have $n$ pieces of paper. They both write down the numbers from $1$ to $2n$ in an arbitrary order, one number on each side. Afterwards, they place the pieces of paper on a table showing one side. Prove that they can always place them so that all the numbers from $1$ to $2n$ are visible at once.

2000 Tuymaada Olympiad, 1

Tags: analytic geometry , LaTeX , topology , set theory , combinatorics solved , combinatorics

Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?

2001 AIME Problems, 14

Tags: pigeonhole principle , AIME , AIME I , recursion , combinatorics , combinatorics solved

A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

1987 Canada National Olympiad, 4

Tags: combinatorics proposed , combinatorics , combinatorics solved , graph theory , induction

On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?

2004 Romania Team Selection Test, 15

Tags: Euler , combinatorics solved , combinatorics

Some of the $n$ faces of a polyhedron are colored in black such that any two black-colored faces have no common vertex. The rest of the faces of the polyhedron are colored in white. Prove that the number of common sides of two white-colored faces of the polyhedron is at least $n-2$.

2005 Taiwan TST Round 1, 1

Tags: geometry , rectangle , combinatorics solved , combinatorics

More than three quarters of the circumference of a circle is colored black. Prove that there exists a rectangle such that all of its vertices are black. Actually the result holds if "three quarters" is replaced by "one half"...

1980 Bundeswettbewerb Mathematik, 3

Tags: combinatorial geometry , combinatorics solved , combinatorics

Given 2n+3 points in the plane, no three on a line and no four on a circle, prove that it is always possible to find a circle C that goes through three of the given points and splits the other 2n in half, that is, has n on the inside and n on the outside.

2004 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , combinatorics solved , combinatorics

Given is a convex polygon with $n\geq 5$ sides. Prove that there exist at most $\displaystyle \frac{n(2n-5)}3$ triangles of area 1 with the vertices among the vertices of the polygon.

2010 Benelux, 1

Tags: combinatorics proposed , combinatorics , combinatorics solved , symmetry , Subsets , Bounding

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

2016 Israel Team Selection Test, 4

Tags: combinatorics , combinatorics solved

A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?