CWMO 2016

Let $a,b,c,d$ be real numbers such that $abcd>0$ . Prove that:There exists a permutation $x,y,z,w$ of $a,b,c,d$ such that $$2(xy+zw)^2>(x^2+y^2)(z^2+w^2)$$ .

Let $n$ and $k$ be integers with $k\leq n-2$ . The absolute value of the sum of elements of any $k$ -element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$ , then for any $2\leq i \leq n$ , we have: $$|a_1|+|a_i|\leq2$$

For an $n$ -tuple of integers, define a transformation to be: $$(a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1)$$ Find all ordered pairs of integers $(n,k)$ with $n,k\geq 2$ , such that for any $n$ -tuple of integers $(a_1,a_2,\cdots,a_{n-1},a_n)$ , after a finite number of transformations, every element in the of the $n$ -tuple is a multiple of $k$ .

Prove that there exist infinitely many positive integer triples $(a,b,c)$ such that $a ,b,c$ are pairwise relatively prime ,and $ab+c ,bc+a ,ca+b$ are pairwise relatively prime .

Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers , $S_k= \sum\limits_{i=1}^{k}a_i$ $(1\le k\le n)$ .Prove that $$\sum\limits_{i=1}^{n}\left(a_iS_i\sum\limits_{j=i}^{n}a^2_j\right)\le \sum\limits_{i=1}^{n}\left(a_iS_i\right)^2$$

For any given integers $m,n$ such that $2\leq m<n$ and $(m,n)=1$ . Determine the smallest positive integer $k$ satisfying the following condition: for any $m$ -element subset $I$ of $\{1,2,\cdots,n\}$ if $\sum_{i\in I}i> k$ , then there exists a sequence of $n$ real numbers $a_1\leq a_2 \leq \cdots \leq a_n$ such that $$\frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i$$