CN-SMO 2014

Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$ ,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$ ,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$

Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$ . Prove that$$x_1+x_2+\cdots+x_n\le \frac{5}{3}.$$

Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions: $(1)$ Square bar $AB$ is bisected by square bar $CD$ $(2)$ At least one square of $AB$ lay on both sides of $CD$ $(3)$ At least one square of $CD$ lay on both sides of $AB$ There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet