Mathematical and Physical Journal
for High Schools
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Problem A. 451. (March 2008)

A. 451. Let \mathcal{U}\subset P(S) be an upset and let \mathcal{D}\subset P(S) be a downset on a set S of n elements, i.e.

(a) for every X\in\mathcal{U} we have Y\in\mathcal{U} for all X\subsetY\subsetS;

(b) for every X\in\mathcal{D} and Y\subsetX, Y\in\mathcal{D}.

Prove that |\mathcal{U}| \cdot |\mathcal{D}|
\ge 2^n \cdot |\mathcal{U}\cap\mathcal{D}|.

(5 pont)

Deadline expired on April 15, 2008.


4 students sent a solution.
5 points:Lovász László Miklós, Nagy 235 János, Tomon István, Wolosz János.

Problems in Mathematics of KöMaL, March 2008