Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation

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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 $\subset$ Y $\subset$ S;

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

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.

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