A circle inscribed within quadrilateral $ABCD$ is tangent to $AB$ at $E$, to $BC$ at $F$, to $CD$ at $G$, and to $DA$ at $H$. Suppose that $AE = 6$, $EB = 30$, $CG = 10$, and $GD = 2$. Compute $EF^2 + F G^2 + GH^2 + HE^2$. .

$p,q$ are nonnegative integers.Given two conditions: A: $p^3-q^3$ is an even number. B: $p+q$ is an even number. Then, which one of the followings are true? $(\text{A})$A is sufficient but unnecessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient and necessary condition of B. $(\text{D})$A is insufficient and unnecessary condition of B.

Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

A small chunk of ice falls from rest down a frictionless parabolic ice sheet shown in the figure. At the point labeled $\mathbf{A}$ in the diagram, the ice sheet becomes a steady, rough incline of angle $30^\circ$ with respect to the horizontal and friction coefficient $\mu_k$. This incline is of length $\frac{3}{2}h$ and ends at a cliff. The chunk of ice comes to a rest precisely at the end of the incline. What is the coefficient of friction $\mu_k$? [asy] size(200); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(sqrt(3),0)--(0,1)); draw(anglemark((0,1),(sqrt(3),0),(0,0))); label("$30^\circ$",(1.5,0.03),NW); label("A", (0,1),NE); dot((0,1)); label("rough incline",(0.4,0.4)); draw((0.4,0.5)--(0.5,0.6),EndArrow); dot((-0.2,4/3)); label("parabolic ice sheet",(0.6,4/3)); draw((0.05,1.3)--(-0.05,1.2),EndArrow); label("ice chunk",(-0.5,1.6)); draw((-0.3,1.5)--(-0.25,1.4),EndArrow); draw((-0.2,4/3)--(-0.19, 1.30083)--(-0.18,1.27)--(-0.17,1.240833)--(-0.16,1.21333)--(-0.15,1.1875)--(-0.14,1.16333)--(-0.13,1.140833)--(-0.12,1.12)--(-0.11,1.100833)--(-0.10,1.08333)--(-0.09,1.0675)--(-0.08,1.05333)--(-0.07,1.040833)--(-0.06,1.03)--(-0.05,1.020833)--(-0.04,1.01333)--(-0.03,1.0075)--(-0.02,1.00333)--(-0.01,1.000833)--(0,1)); draw((-0.6,0)--(-0.6,4/3),dashed,EndArrow,BeginArrow); label("$h$",(-0.6,2/3),W); draw((0.2,1.2)--(sqrt(3)+0.2,0.2),dashed,EndArrow,BeginArrow); label("$\frac{3}{2}h$",(sqrt(3)/2+0.2,0.7),NE); [/asy] $ \textbf{(A)}\ 0.866\qquad\textbf{(B)}\ 0.770\qquad\textbf{(C)}\ 0.667\qquad\textbf{(D)}\ 0.385\qquad\textbf{(E)}\ 0.333 $

If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

Let $A$ be a $8\times 8$ array with entries from the set $\{-1,1\}$ such that any $2\times 2$ sub-square of the array has the absolute value of the sum of its element equal with 2. Prove that the array must have at least two identical lines.

Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by James Lin[/i]

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$. Let $P$ be a point on side $BC$, and let $\omega$ be the circle with diameter $CP$. Suppose that $\omega$ intersects $AC $and $AP$ again at $Q$ and $R$, respectively. Show that $CP^2 = AC \cdot CQ - AP \cdot P R$.

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$

Find $$\prod^{2017}_{k=1} e^{\pi ik/2017}2 cos \left( \frac{\pi k}{2017} \right)$$

Let $ABCD$ be a cyclic quadrilateral such that $AB=AD$. Let $E$ and $F$ be points on the sides $BC$ and $CD$, respectively, such that $BE+DF=EF$. Prove that $\angle BAD = 2 \angle EAF$.

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

Show that there is a perfect square (a number which is a square of an integer) such that sum of its digits is $2011.$

Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$, $B'$, $C'$ the projections of the point $M$ on the sides $BC$, $CA$ and $AB$ respectively. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.

$A_1,A_2,\cdots,A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.

Compute $1+2\cdot3^4$. [i]Proposed by Evan Chen[/i]

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles but ends up with one sheet left over each time. How many piles does he need?

Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

What is the sum of distinct real numbers $x$ such that $(2x^2+5x+9)^2=56(x^3+1)$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ \dfrac{7}{4} \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \dfrac{9}{2} \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ABC$ be an acute triangle with orthocenter $H$ and let $M$ be the midpoint of $\overline{BC}$. (The [i]orthocenter[/i] is the point at the intersection of the three altitudes.) Denote by $\omega_B$ the circle passing through $B$, $H$, and $M$, and denote by $\omega_C$ the circle passing through $C$, $H$, and $M$. Lines $AB$ and $AC$ meet $\omega_B$ and $\omega_C$ again at $P$ and $Q$, respectively. Rays $PH$ and $QH$ meet $\omega_C$ and $\omega_B$ again at $R$ and $S$, respectively. Show that $\triangle BRS$ and $\triangle CRS$ have the same area. [i]Proposed by Aaron Lin[/i]