Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P$ and $Q$ is $d$. Find $d$.

Let $n\in\mathbb{N}$, the set $A=\{(x_1,x_2...,x_n)|x_i\in\mathbb{R}_{+}, i=1,2,...,n\}$ and the function $$f:A\rightarrow\mathbb{R}, f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{1}{nx_n}.$$ Prove that $f(\textstyle\binom{n}{1},\binom{n}{2},...,\binom{n}{n-1},\binom{n}{n})=f(2^{n-1},2^{n-2},...,2,1).$

Show that \[ (2a-1) \sin x + (1-a) \sin(1-a)x \geq 0 \] for $0 \leq a \leq 1$ and $0 \leq x \leq \pi$.

Find all positive integers $ (x,y)$ such that $ \frac{y^2x}{x\plus{}y}$ is a prime number

Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for a) $N = 2011$ b) $N = 2012$ ?

Evaluate the product \[(\sqrt 5+\sqrt6+\sqrt7)(-\sqrt 5+\sqrt6+\sqrt7)(\sqrt 5-\sqrt6+\sqrt7)(\sqrt 5+\sqrt6-\sqrt7).\]

The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$? $ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$

Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$. Calculate the area of the quadrilateral.

The equation $ax^5 + bx^4 + c = 0$ has three distinct roots. Show that so does the equation $cx^5 +bx+a = 0$.

Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal, \[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of \[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]

A sphere is inscribed inside a pyramid with a square as a base whose height is $\frac{\sqrt{15}}{2}$ times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube?

If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$, compute the value of the expression \[ \left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times \left( \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}} - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}} \right). \]

Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$. Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$. If $AD = 16$ and $BC = 20$, what is the length of $PQ$?

In a triangle $ABC$, $AB > AC$, the external bisector of angle $A$ meets the circumcircle of triangle $ABC$ at $E$, and $F$ is the foot of the perpendicular from $E$ onto $AB$. Prove that $2AF = AB - AC$

Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, $20$ children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, $25$ children raised their hands. Prove that some child that raised her hand had two green neighbors.

At one school, $85$ percent of the students are taking mathematics courses, $55$ percent of the students are taking history courses, and $7$ percent of the students are taking neither mathematics nor history courses. Find the percent of the students who are taking both mathematics and history courses.

Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor .$ [i]Petre Rău[/i]

[b]p1.[/b] Solve the equation $$\frac{\sqrt{x^2 - 2x + 1}}{x^2 - 1}+\frac{x^2 - 1}{\sqrt{x^2 - 2x + 1}}=\frac52.$$ [b]p2.[/b] Solve the inequality: $\frac{1 - 2\sqrt{1-x^2}}{x} \le 1$. [b]p3.[/b] Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{|AD| + |BC|}{2}$. Prove that $AD$ is parallel to $BC$. [b]p4.[/b] Solve the equation: $\frac{1}{\cos x}+\frac{1}{\sin x}= 2\sqrt2$. [b]p5.[/b] Long, long ago, far, far away there existed the Old Republic Galaxy with a large number of stars. It was known that for any four stars in the galaxy there existed a point in space such that the distance from that point to any of these four stars was less than or equal to $R$. Master Yoda asked Luke Skywalker the following question: Must there exist a point $P$ in the galaxy such that all stars in the galaxy are within a distance $R$ of the point $P$? Give a justified argument that will help Like answer Master Yoda’s question. [b]p6.[/b] The Old Republic contained an odd number of inhabited planets. Some pairs of planets were connected to each other by space flights of the Trade Federation, and some pairs of planets were not connected. Every inhabited planet had at least one connections to some other inhabited planet. Luke knew that if two planets had a common connection (they are connected to the same planet), then they have a different number of total connections. Master Yoda asked Luke if there must exist a planet that has exactly two connections. Give a justified argument that will help Luke answer Master Yoda’s question. PS. You should use hide for answers.

Show that if $a$ and $b$ are positive integers, then \[\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}\] is an integer for only finitely many positive integer $n$.

What is the largest prime factor of 8091?

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

Integers a, b, c, d, and e satisfy the following three properties: (i) $2 \le a < b <c <d <e <100$ (ii)$ \gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence. What is the value of c?

The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$. Suppose that $ABC$ is an equilateral triangle of side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. Then the area of $AMNPQ$ is $n-p\sqrt{q}$, where $n, p, q$ are positive integers and $q$ is not divisible by the square of a prime. Compute $100n+10p+q$. [i]Proposed by Michael Ren[/i]

Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$. [i]Proposed by Mr.Etesami[/i]