ققنوس

1	Let $ABC$ be a triangle, let $AB > AC$ . Its incircle touches side $BC$ at point $E$ . Point $D$ is the second intersection of the incircle with segment $AE$ (different from $E$ ). Point $F$ (different from $E$ ) is taken on segment $AE$ such that $CE = CF$ . The ray $CF$ meets $BD$ at point $G$ . Show that $CF = FG$ .	S
2	The sequence $\{x_{n}\}$ is defined by $x_{1} = 2,x_{2} = 12$ , and $x_{n + 2} = 6x_{n + 1} - x_{n}$ , $(n = 1,2,\ldots)$ . Let $p$ be an odd prime number, let $q$ be a prime divisor of $x_{p}$ . Prove that if $q\neq2,3,$ then $q\geq 2p - 1$ .	S
3	Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $a_{1},\frac {a_{1} + a_{2}}{2},a_{2},\frac {a_{2} + a_{3}}{2},a_{3},\frac {a_{3} + a_{4}}{2},\cdots$ has the same color.	S

Day 2

4	Prove that for arbitary positive integer $n\geq 4$ , there exists a permutation of the subsets that contain at least two elements of the set $G_{n} = \{1,2,3,\cdots,n\}$ : $P_{1},P_{2},\cdots,P_{2^n - n - 1}$ such that $\|P_{i}\cap P_{i + 1}\| = 2,i = 1,2,\cdots,2^n - n - 2.$	S
5	For two given positive integers $m,n > 1$ , let $a_{ij} (i = 1,2,\cdots,n,j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $f$ , where $f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i = 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{i = j}^{m}a_{ij}^2}$ .	S
6	Find the maximal constant $M$ , such that for arbitrary integer $n\geq 3,$ there exist two sequences of positive real number $a_{1},a_{2},\cdots,a_{n},$ and $b_{1},b_{2},\cdots,b_{n},$ satisfying (1): $\sum_{k = 1}^{n}b_{k} = 1,2b_{k}\geq b_{k - 1} + b_{k + 1},k = 2,3,\cdots,n - 1;$ (2): $a_{k}^2\leq 1 + \sum_{i = 1}^{k}a_{i}b_{i},k = 1,2,3,\cdots,n, a_{n}\equiv M$ .

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