# Understand the problem

We are given sets of size each. The size of the intersection of any two sets is exactly . Prove that all the sets have a common element.

##### Source of the problem

Austrian-polish mathematical Olympiad 1978

##### Topic

Combinatorics

##### Difficulty Level

Easy

##### Suggested Book

An Excursion in Mathematics

# Start with hints

Do you really need a hint? Try it first!

Fix one set and use PHP to find out how many times its elements are repeated.

Let us fix as one of the sets. There are 1977 other sets and only 40 elements in , so some element is contained in at least 50 other sets. Call these sets .

Prove that the element found in hint 2 is common to all the sets.

Let be a set such that . Consider the sets . They are all singleton sets and they are pairwise distinct (prove it!). This means that has more than 50 elements, which is a contradiction.

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#### Math Olympiad Program

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