CWMO 2018

Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy $x_i + x_j \geq (-1)^{i+j}$ for all $1 \leq i < j \leq 2018$ . Find the minimum possible value of $\sum_{i=1}^{2018} ix_i$ .

Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$ . Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$ Where $\{x\}$ denotes the fractional part of $x$ .

Let $M = \{1,2,\cdots , 10\}$ , and let $T$ be a set of 2-element subsets of $M$ . For any two different elements $\{a,b\}, \{x,y\}$ in $T$ , the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$ .

In acute angled $\triangle ABC$ , $AB > AC$ , points $E, F$ lie on $AC, AB$ respectively, satisfying $BF+CE = BC$ . Let $I_B, I_C$ be the excenters of $\triangle ABC$ opposite $B, C$ respectively, $EI_C, FI_B$ intersect at $T$ , and let $K$ be the midpoint of arc $BAC$ . Let $KT$ intersect the circumcircle of $\triangle ABC$ at $K,P$ . Show $T,F,P,E$ concyclic.

In acute triangle $ABC,$ $AB<AC,$ $O$ is the circumcenter of the triangle. $M$ is the midpoint of segment $BC,$ $(AOM)$ intersects the line $AB$ again at $D$ and intersects the segment $AC$ at $E.$ Prove that $DM=EC.$

Let $n \geq 2$ be an integer. Positive reals satisfy $a_1\geq a_2\geq \cdots\geq a_n.$ Prove that $$\left(\sum_{i=1}^n\frac{a_i}{a_{i+1}}\right)-n \leq \frac{1}{2a_1a_n}\sum_{i=1}^n(a_i-a_{i+1})^2,$$ where $a_{n+1}=a_1.$

Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that $$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$

Let $n,k$ be positive integers, satisfying $n$ is even, $k\geq 2$ and $n>4k.$ There are $n$ points on the circumference of a circle. If the endpoints of $\frac{n}{2}$ chords in a circle that do not intersect with each other are exactly the $n$ points, we call these chords a matching.Determine the maximum of integer $m,$ such that for any matching, there exists $k$ consecutive points, satisfying all the endpoints of at least $m$ chords are in the $k$ points.