# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]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.

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.26.6" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.26.6"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.26.6" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.26.6"]Fix one set and use PHP to find out how many times its elements are repeated. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.26.6"]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 $S_1,S_2,\cdots S_{50}$.

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.26.6"]

Prove that the element $x$ found in hint 2 is common to all the sets.  [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.26.6"]

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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