In a school, there are totally $n$ students, with $n \ge 2$. The students take part in $m$ clubs and in each club, there are at least $2$ members (a student may take part in more than $1$ club). Eventually, the Principal notices that: If $2$ clubs share at least $2$ common members then they have different numbers of members. Prove that $$m \le (n - 1)^2$$

[b]Problem 3[/b] Let $ P(x) = x^{2015} -2x^{2014}+1$ and $ Q(x) = x^{2015} -2x^{2014}-1$. Determine for each of the polynomials $P$ and $Q$ whether it is a divisor of some nonzero polynomial $c_0 + c_{1}x +\ldots + c_{n}x^n$ n whose coefficients $c_i$ are all in the set $ \{ -1, 1\}$.

Given a fixed integer $k>0,r=k+0.5$,define $f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$ where $[x]$ denotes the smallest integer not less than $x$. prove that there exists integer $m$ such that $f^m(r)$ is an integer.

Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that $Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) $ for some complex number $w$ with $||w||=\frac{\sqrt3}{2}$ , $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.

The number of significant digits in the measurement of the side of a square whose computed area is $ 1.1025$ square inches to the nearest ten-thousandth of a square inch is: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 1$

Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.

For real numbers $x \ge 0$ and $y \ge 0$, write $A= \frac{x+y}{2}$ for the arithmetic mean and $G=\sqrt{xy}$ for the geometric mean of $x$ and $y$. Furthermore, let $W= \frac{\sqrt{x}+\sqrt{y}}{2}$ be the arithmetic mean of $\sqrt{x}$ and $\sqrt{y}$. Prove that $$G\le W^2 \le A.$$ Determine all $x$ and $y$ such that $G= W^2 = A$

Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.

Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$

Prove the inequality $$\frac{a + b}{a^2 -ab + b^2} \le \frac{4}{ |a + b|}$$ for all real numbers $a$ and $b$ with $a + b\ne 0$. When does equality hold?

You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is \( S \). Find all possible values of \( S \) that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.

Consider an alphabetized list of all the arrangements of the letters in the word BETWEEN. Then BEEENTW would be in position $1$ in the list, BEEENWT would be in position $2$ in the list, and so forth. Find the position that BETWEEN would be in the list.

For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.

A square of area $40$ is inscribed in a semicircle as shown. What is the area of the semicircle? [asy] defaultpen(linewidth(0.8)); real r=sqrt(50), s=sqrt(10); draw(Arc(origin, r, 0, 180)); draw((r,0)--(-r,0), dashed); draw((s,0)--(s,2*s)--(-s,2*s)--(-s,0));[/asy] $ \textbf{(A) }20\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }30\pi\qquad\textbf{(D) }40\pi\qquad\textbf{(E) }50\pi $

Find the maximum possible volume of a tetrahedron which has three faces with area $1$.

Fix positive integers $m$ and $n$. Suppose that $a_1, a_2, \dots, a_m$ are reals, and that pairwise distinct vectors $v_1, \dots, v_m\in \mathbb{R}^n$ satisfy $$\sum_{j\neq i} a_j \frac{v_j-v_i}{||v_j-v_i||^3}=0$$ for $i=1,2,\dots,m$. Prove that $$\sum_{1\le i<j\le m} \frac{a_ia_j}{||v_j-v_i||}=0.$$

Let $S$ be the maximum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4} \] given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$. Given that $S$ can be written in the form $m/n$ where $m,n$ are coprime positive integers, find $100m+n$. [i]Proposed by Kaan Dokmeci[/i]

In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.

Let $ ABC$ be a triangle with sides $ BC\equal{}a, CA\equal{}b,AB\equal{}c$ and let $ D$ and $ E$ be the midpoints of $ AC$ and $ AB$, respectively. Prove that the medians $ BD$ and $ CE$ are perpendicular to each other if and only if $ b^2\plus{}c^2\equal{}5a^2$.

The following is a code and is meant to be broken. 2 707 156 377 38 2 328 17 185 2 713 73 566 1130 328 73 38 259 471 38 17 566 2 134 707 38 274 377 328 38 1130 40 377 566 73 820 566 566 134 11 2 328 38 185 2 713 566 134 328 2 918 134 11 713 134 274 707 713 73 38 1130 17 134 707 11 820 707 707 38 17 713 73 38 134 566 40 2 918 377 566 134 713 38 328 820 274 4 38 566 707 156 377 38 707 40 2 918 377 566 134 713 38 328 820 274 4 38 566 134 707 713 73 38 2 328 707 991 38 566 713 377 713 73 38 707 38 918 38 328 713 73 707 73 377 566 713 2 328 707 991 38 566 532 820 38 707 713 134 377 328 377 328 713 73 134 707 713 38 707 713 185 2 713 73 566 1130 328 707 40 2 918 377 566 134 713 38 328 820 274 4 38 566 134 707 713 73 38 2 328 707 991 38 566 713 377 713 73 38 707 38 11 377 328 17 259 377 328 79 2 328 707 991 38 566 532 820 38 707 713 134 377 328 377 328 713 73 134 707 713 38 707 713 991 73 2 713 134 707 713 73 38 707 820 274 377 40 713 73 38 134 566 40 2 918 377 566 134 713 38 328 820 274 4 38 566 707

$ABCD$ is a convex quadrilateral. Angles $A$ and $C$ are equal. Points $M$ and $N$ are on the sides $AB$ and $BC$ such that $MN||AD$ and $MN=2AD$. Let $K$ be the midpoint of $MN$ and $H$ be the orthocenter of $\triangle ABC$. Prove that $HK$ is perpendicular to $CD$.

Find the smallest positive integer whose cube ends in 888.

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.

The familiar $3$-dimensional cube has $6$ $2$-dimensional faces, $12$ $1$-dimensional edges, and $8$ $0$-dimensional vertices. Find the number of $9$-dimensional sub-subfaces in a $12$-dimensional cube.