CWMO 2015

Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n$ be real numbers such that $\sum_{k=1}^nx_k$ be integer . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right|$ , $1\leq k\leq n$ .Find the maximum value of $\sum_{k=1}^nd_k$ .

Two circles $\left(\Omega_1\right),\left(\Omega_2\right)$ touch internally on the point $T$ . Let $M,N$ be two points on the circle $\left(\Omega_1\right)$ which are different from $T$ and $A,B,C,D$ be four points on $\left(\Omega_2\right)$ such that the chords $AB, CD$ pass through $M,N$ , respectively. Prove that if $AC,BD,MN$ have a common point $K$ , then $TK$ is the angle bisector of $\angle MTN$ .

• $\left(\Omega_2\right)$ is bigger than $\left(\Omega_1\right)$

Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$ .Prove that $$\left(\sum_{i=1}^n\frac{1}{1-x_i}\right)\left(\sum_{1\le i<j\le n} x_ix_j\right)\le \frac{n}{2}.$$

For $100$ straight lines on a plane, let $T$ be the set of all right-angled triangles bounded by some $3$ lines. Determine, with proof, the maximum value of $|T|$ .

Let $a,b,c,d$ are lengths of the sides of a convex quadrangle with the area equal to $S$ , set $S =\{x_1, x_2,x_3,x_4\}$ consists of permutations $x_i$ of $(a, b, c, d)$ . Prove that $$S \leq \frac{1}{2}(x_1x_2+x_3x_4).$$

For a sequence $a_1,a_2,...,a_m$ of real numbers, define the following sets $$A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}$$ Let $n$ be a given integer, and $n>2$ . For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set $A\triangle B$ , where $A\triangle B$ $= \left(A\cup B\right) \setminus \left(A\cap B\right).$

Let $a\in (0,1)$ , $f(z)=z^2-z+a, z\in \mathbb{C}$ . Prove the following statement holds: For any complex number z with $|z| \geq 1$ , there exists a complex number $z_0$ with $|z_0|=1$ , such that $|f(z_0)| \leq |f(z)|$ .

Let $k$ be a positive integer, and $n=\left(2^k\right)!$ .Prove that $\sigma(n)$ has at least a prime divisor larger than $2^k$ , where $\sigma(n)$ is the sum of all positive divisors of $n$ .