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 $n$.