The Pidgeonhole Principle
Adeel Khan (chess64)
April 9, 2006
Sources: Mathematical Circles by Fomin, Genkin and Itenberg,
Mathematical Excalibur, Volume 1, Issue 1 byHong Kong University of
Science and Technology, Problem Solving Strategies by Engel, and
Mathematical Databaseat http://www.eng.mathdb.org/e main.html
Problems.
1. There are 100 pidgeonholes. How many pidgeons must you have
to guarantee that if all the pidgeons are putinto a pidgeonhole,
there is one pidgeonhole with at least 2 pidgeons?
2. There are 100 pidgeonholes. How many pidgeons must you have
to guarantee that if all the pidgeons are putinto a pidgeonhole,
there is one pidgeonhole with at least 10 pidgeons?
3. (Mathematical Circles) One million pine trees grow in a
forest. It is known that no pine tree has more than600000 pine
needles on it. Show that two pine trees in the forest must have the
same number of pine needles.
4. (Johns Hopkins Math Tournament 2006) Find the smallest n so
that no matter how n points are placed in aunit square, there
exists a pair of points separated by a distance no greater than
2
3.
5. (Mathematical Excalibur) Suppose 51 numbers are chosen from
1, 2, 3, . . . , 99, 100. Show that there are twowhich do not have
any common prime divisor.
6. (Problem Solving Strategies) A chessmaster has 77 days to
prepare for a tournament. He wants to play at leastone game per
day, but not more than 132 games. Prove that there is a sequence of
successive days in which heplays exactly 21 games.
7. (Mathematical Excalibur) Show that among any nine distinct
real numbers there are two, say a and b, suchthat
0
