CGMO 2017

(1) Find all positive integer $n$ such that for any odd integer $a$ , we have $4\mid a^n-1$ (2) Find all positive integer $n$ such that for any odd integer $a$ , we have $2^{2017}\mid a^n-1$

Given quadrilateral $ABCD$ such that $\angle BAD+2 \angle BCD=180 ^ \circ .$ Let $E$ be the intersection of $BD$ and the internal bisector of $\angle BAD$ . The perpendicular bisector of $AE$ intersects $CB,CD$ at $X,Y,$ respectively. Prove that $A,C,X,Y$ are concyclic.

Given $a_i\ge 0,x_i\in\mathbb{R},(i=1,2,\ldots,n)$ . Prove that $$((1-\sum_{i=1}^n a_i\cos x_i)^2+(1-\sum_{i=1}^n a_i\sin x_i)^2)^2\ge 4(1-\sum_{i=1}^n a_i)^3$$

Partition $\frac1{2002},\frac1{2003},\frac1{2004},\ldots,\frac{1}{2017}$ into two groups. Define $A$ the sum of the numbers in the first group, and $B$ the sum of the numbers in the second group. Find the partition such that $|A-B|$ attains it minimum and explains the reason.

Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number $C$ such that for any positive integer $n$ , we have $$\sum_{k=1}^n x^2_k (x_k - x_{k-1})>C$$

Given a finite set $X$ , two positive integers $n,k$ , and a map $f:X\to X$ . Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$ , $i=1,2,3,\ldots$ . It is known that for any $x\in X$ , $f^{(n)}(x)=x$ . Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$ . Prove that: (1) $\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$ (2) $\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.

This is a very classical problem. Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$ .Lines $AC$ and $BD$ intersect at point $E$ ,and lines $AD$ , $BC$ intersect at point $F$ .Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$ .Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.

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