Understand the problem

We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$. Prove that all the sets have a common element.

Source of the problem

Austrian-polish mathematical Olympiad 1978

Combinatorics
Easy
Suggested Book
An Excursion in Mathematics

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 $S$ as one of the sets. There are 1977 other sets and only 40 elements in $S$, so some element $x\in S$ is contained in at least 50 other sets. Call these sets $S_1,S_2,\cdots S_{50}$.

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

Let $T$ be a set such that $x\notin T$. Consider the sets $\{T\cap S_i\}_{1\le i\le 50}$. They are all singleton sets and they are pairwise distinct (prove it!). This means that $T$ has more than 50 elements, which is a contradiction.

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