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# 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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#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

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