There are 2009 committees, each with 45 members,
any two of which have exactly one person in common.
Show that there is one person who is a member of all the committees.
Consider an equilateral triangle having side-lengths one inch,
and any set of five points on the interior of that triangle.
Show that two of those points must be within one-half inch of each other.
Consider the set S={1,2,…,2n}.
Prove that any (n+1)-element subset of S
contains two members such that one is a multiple of the other.
St. Petersburg City Math Olympiad 2001 (P&B)
In each square of a 10×10 checkerboard,
a positive integer from the set {2,3,…,11} is written
in such a way that any two numbers that appear
in adjacent or diagonally adjacent squares of the board
don’t share a common factor greater than 1.
Prove that some number appears at least 17 times.
Let G be a graph with n vertices and k edges,
such that the edges are labelled 1,…,k in some fashion.
Prove that there must exist a path in G (with repeated vertices allowed)
of length at least 2k/n for which the labels of edges
in the path occur in increasing order.
Given a positive integer n, what is the largest k
such that the numbers 1,2,…,n can be put into k boxes
so that the sum of the numbers in each box is the same?
[When n=8, the example {1,2,3,6},{4,8},{5,7}
shows that the largest k is at least 3.]
Let aj,bj,cj be integers for 1≤j≤N.
Assume for each j, at least one of aj,bj,cj is odd.
Show that there exist integers r,s,t such that
raj+sbj+tcj is odd for at least 4N/7 values of j,1≤j≤N.
There are 2010 boxes labeled B1,B2,…,B2010.
For some positive integer n, a total of 2010n balls
have been distributed among the boxes in some way.
You may redistribute the balls by a sequence of moves of this form:
choose an i and move exactly i balls
from box Bi to another box of your choice.
For which values of n is it possible,
through a sequence of these moves,
to reach the distribution with exactly n balls in each box
regardless of the initial distribution of balls?