3

Part of 2002 Poland - Second Round

Problems(2)

Source: ...

A positive integer

n

is given. In an association consisting of

n

6

commissions. Each commission contains at least

\large \frac{n}{4}

persons. Prove that there exist two commissions containing at least

\large \frac{n}{30}

persons in common.

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Find n for which inequality with x_i,y_i holds

Source:

Find all positive integers

n

such that for all real numbers

x_1,x_2,\ldots ,x_n,y_1,y_2,\ldots ,y_n

the following inequality holds:

x_1x_2\ldots x_n+y_1y_2\ldots y_n\le\sqrt{x_1^2+y_1^2}\cdot\sqrt{x_2^2+y_2^2}\cdot \cdots \sqrt{x_n^2+y_n^2}\cdot

inequalitiesinequalities proposed