$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$. (1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets. (2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.

Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).

The numbers $ 0$, $ 1$, $ \dots$, $ n$ ($ n \ge 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $ g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $ g(n)$ for every $ n$.

Let $G$ be a graph whose vertices are the integers. Assume that any two integers are connected by a finite path in $G$. For two integers $x$ and $y$, we denote by $d(x, y)$ the length of the shortest path from $x$ to $y$, where the length of a path is the number of edges in it. Assume that $d(x, y) \mid x-y$ for all integers $x, y$ and define $S(G)=\{d(x, y) | x, y \in \mathbb{Z}\}$. Find all possible sets $S(G)$.

Let $P = \{(x, y) | x, y \in \{0, 1, 2,... , 2015\}\}$ be a set of points on the plane. Straight wires of unit length are placed to connect points in $P$ so that each piece of wire connects exactly two points in $P$, and each point in $P$ is an endpoint of exactly one wire. Prove that no matter how the wires are placed, it is always possible to draw a straight line parallel to either the horizontal or vertical axis passing through midpoints of at least $506$ pieces of wire.

Determine all non-negative real numbers $a$, for which $f(a)=0$ for all functions $f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0} $, who satisfy the equation $f(f(x) + f(y)) = yf(1 + yf(x))$ for all non-negative real numbers $x$ and $y$.

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.

Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is [asy]draw((2.7,3.99)--(0,3)--(0,0)); draw((3.7,3.99)--(1,3)--(1,0)); draw((4.7,3.99)--(2,3)--(2,0)); draw((5.7,3.99)--(3,3)--(3,0)); draw((0,0)--(3,0)--(5.7,0.99)); draw((0,1)--(3,1)--(5.7,1.99)); draw((0,2)--(3,2)--(5.7,2.99)); draw((0,3)--(3,3)--(5.7,3.99)); draw((0,3)--(3,3)--(3,0)); draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33)); draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66)); draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99)); [/asy] $\textbf{(A)}\ 19\text{ sq.cm} \qquad \textbf{(B)}\ 24\text{ sq.cm} \qquad \textbf{(C)}\ 30\text{ sq.cm} \qquad \textbf{(D)}\ 54\text{ sq.cm} \qquad \textbf{(E)}\ 72\text{ sq.cm}$

A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$?

A merchant placed on display some dresses, each with a marked price. He then posted a sign “$ \frac{1}{3}$ off on these dresses.” The cost of the dresses was $ \frac{3}{4}$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was: $ \textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{3}{4}$

Two cubes are inscribed in a sphere of radius $R$. Calculate the sum of squares of all segments connecting the vertices of one cube with the vertices of the other cube

For a vertex $X$ of a quadrilateral, let $h(X)$ be the sum of the distances from $X$ to the two sides not containing $X$. Show that if a convex quadrilateral $ABCD$ satisfies $h(A) = h(B) = h(C) = h(D)$, then it must be a parallelogram.

\[\int_0^\infty 1+\cos\left(\tfrac 1{\sqrt x}\right)-2\cos\left(\tfrac 1{\sqrt {2x}}\right)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

For every integers $ a,b,c$ whose greatest common divisor is $n$, if \[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a} \\ {2x \plus{} y \minus{} 2z \equal{} b} \\ {3x \plus{} y \plus{} 5z \equal{} c} \end{array} \] has a solution in integers, what is the smallest possible value of positive number $ n$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ \text{None}$

Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.