In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.

On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$

Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number \[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\] is a perfect square.

Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$.

Let $a, b, c$ be positive reals such that $a+b+c=1$. Prove that the inequality \[a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1\] holds.

In quadrilateral $ABCD$ sides $AD$ and $BC$ aren't parallel. Diagonals $AC$ and $BD$ intersect in $E$. $F$ and $G$ are points on sides $AB$ and $DC$ such $\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}$ Prove that if $E, F, G$ are collinear then $ABCD$ is cyclic.