Find all positive integer $n$ and nonnegative integer $a_1,a_2,\dots,a_n$ satisfying: $i$ divides exactly $a_i$ numbers among $a_1,a_2,\dots,a_n$, for each $i=1,2,\dots,n$. ($0$ is divisible by all integers.)

Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.

Find all $n>1$ and integers $a_1,a_2,\dots,a_n$ satisfying the following three conditions: (i) $2<a_1\le a_2\le \cdots\le a_n$ (ii) $a_1,a_2,\dots,a_n$ are divisors of $15^{25}+1$. (iii) $2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)$

4. Let $a \geq b \geq c \geq d>0$. Show that \[ \frac{b^3}{a} + \frac{c^3}{b} + \frac{d^3}{c} + \frac{a^3}{d} + 3 \left( ab+bc+cd+da \right) \geq 4 {\left( a^2 + b^2 + c^2 +d^2 \right)}. \]Other problems (in Korean) are also available at https://www.facebook.com/KoreanMathOlympiad

Given an integer $n\ge 2$, show that there exist two integers $a,b$ which satisfy the following. For all integer $m$, $m^3+am+b$ is not a multiple of $n$.

Let triangle $ABC$ be an acute scalene triangle, and denote $D,E,F$ by the midpoints of $BC,CA,AB$, respectively. Let the circumcircle of $DEF$ be $O_1$, and its center be $N$. Let the circumcircle of $BCN$ be $O_2$. $O_1$ and $O_2$ meet at two points $P, Q$. $O_2$ meets $AB$ at point $K(\neq B)$ and meets $AC$ at point $L(\neq C)$. Show that the three lines $EF,PQ,KL$ are concurrent.

Prove that there is no function $f:\mathbb{R}_{\ge0}\rightarrow\mathbb{R}$ satisfying: $f(x+y^2)\ge f(x)+y$ for all two nonnegative real numbers $x,y$.

For a positive integer $n$, there is a school with $n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ well-formed. If the maximum number of students in a well-formed set is $k$, show that the maximum number of well-formed sets is not greater than $3^{(n+k)/3}$. Here, an empty set and a set with one student is regarded as well-formed as well.