The point $ P$ is a fixed point on a circle and $ Q$ is a fixed point on a line. The point $ R$ is a variable point on the circle such that $ P,Q,$ and $ R$ are not collinear. The circle through $ P,Q,$ and $ R$ meets the line again at $ V$. Show that the line $ VR$ passes through a fixed point.

Let $P$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have positive imaginary part, and suppose that $P=r(\cos \theta^\circ+i\sin \theta^\circ),$ where $0<r$ and $0\le \theta <360.$ Find $\theta.$

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals, such that $\angle BAC = \angle ADB$, $\angle CBD = \angle DCA$, $AB = 15$, $CD = 8$. Show that $ABCD$ is cyclic and find the distance between its circumcenter and the intersection point of its diagonals.

The ratio of Mary's age to Alice's age is $ 3: 5$. Alice is $ 30$ years old. How old is Mary? $ \textbf{(A) } 15\qquad \textbf{(B) } 18\qquad \textbf{(C) } 20\qquad \textbf{(D) } 24\qquad \textbf{(E) } 50$

Let $\Gamma_1$ and $\Gamma_2$ be two circles with radii $r_1$ and $r_2,$ respectively, where $r_1>r_2.$ Suppose $\Gamma_1$ and $\Gamma_2$ intersect at two distinct points $A$ and $B.$ A point $C$ is selected on ray $\overrightarrow{AB},$ past $B,$ and the tangents to $\Gamma_1$ and $\Gamma_2$ from $C$ are marked as points $P$ and $Q,$ respectively. Suppose that $\Gamma_2$ passes through the center of $\Gamma_1$ and that points $P, B, Q$ are collinear in that order, with $PB=3$ and $QB=2.$ What is the length of $AB?$ [i]Proposed by Kyle Lee[/i]

A permutation $(a_1, a_2, a_3, \dots, a_{100})$ of $(1, 2, 3, \dots, 100)$ is chosen at random. Denote by $p$ the probability that $a_{2i} > a_{2i - 1}$ for all $i \in \{1, 2, 3, \dots, 50\}$. Compute the number of ordered pairs of positive integers $(a, b)$ satisfying $\textstyle\frac{1}{a^b} = p$. [i]Proposed by Aaron Lin[/i]

Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$. Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$

Circle $\omega$ lies in the interior of circle $\Omega$, on which a point $X$ moves. The tangents from $X$ to $\omega$ intersect $\Omega$ for the second time at points $A\neq X$ and $B\neq X$. Prove that the lines $AB$ are either all tangent to a fixed circle, or they all pass through a point.

For each $i=1,2,..N$, let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$.

In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that: \[AL+BM+CN \leq 3(AD+BE+CF)\] When does equality occur?

The figure shown is called a [i]trefoil[/i] and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $ 2$? [asy]unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(12pt)); pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180); pair E=B+C; draw(D--E--B--O--C--B--A,linetype("4 4")); draw(Arc(O,1,0,60),linewidth(1.2pt)); draw(Arc(O,1,120,180),linewidth(1.2pt)); draw(Arc(C,1,0,60),linewidth(1.2pt)); draw(Arc(B,1,120,180),linewidth(1.2pt)); draw(A--D,linewidth(1.2pt)); draw(O--dir(40),EndArrow(HookHead,4)); draw(O--dir(140),EndArrow(HookHead,4)); draw(C--C+dir(40),EndArrow(HookHead,4)); draw(B--B+dir(140),EndArrow(HookHead,4)); label("2",O,S); draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar); draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar);[/asy]$ \textbf{(A)}\ \frac13\pi\plus{}\frac{\sqrt3}{2} \qquad \textbf{(B)}\ \frac23\pi \qquad \textbf{(C)}\ \frac23\pi\plus{}\frac{\sqrt3}{4} \qquad \textbf{(D)}\ \frac23\pi\plus{}\frac{\sqrt3}{3} \qquad \textbf{(E)}\ \frac23\pi\plus{}\frac{\sqrt3}{2}$

The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can be written as $f(x)$, with $f$ a polynomial. Find $f(-1)$.

$ 101$ positive integers are written on a line. Prove that we can write signs $ \plus{}$, signs $ \times$ and parenthesis between them, without changing the order of the numbers, in such a way that the resulting expression makes sense and the result is divisible by $ 16!$.

Let $O$ be the circumcenter of a triangle $ABC$, and let $l$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$. Determine the value of $\angle BAC$ if the line $l$ goes through the midpoint of the line segment $AO$.

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically? $\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\ $\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\ $\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\ $\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\ $\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.

A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures? $ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $

The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.

there are some identical squares with sides parallel, in a plane. Among any $k+1$ of them, there are two with a point in common. Prove they can be divided into $2k-1$ sets, such that all the squares in one set aint pairwise disjoint.

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

In the coordinate plane, denote by $ S(a)$ the area of the region bounded by the line passing through the point $ (1,\ 2)$ with the slope $ a$ and the parabola $ y\equal{}x^2$. When $ a$ varies in the range of $ 0\leq a\leq 6$, find the value of $ a$ such that $ S(a)$ is minimized.

Let $a$ be the least positive integer with $20$ positive divisors and $b$ be the least positive integer with $16$ positive divisors. What is $a+b$? (Note that for any integer $n$, both $1$ and $n$ are considered divisors of $n$.)

Let $a,b,c>0$ be real numbers with sum 1. Prove that \[ \frac{a^2}b + \frac{b^2}c + \frac{c^2} a \geq 3(a^2+b^2+c^2) . \]

In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of them also goes to spanish language. $10$ students who study spanish also study german. $12$ students who study german also do sports. Only $4$ students go to all three activities. How many of them does only one of the activities, and how much of them do not go to any activity?

Let $n$ and $k$ be two natural numbers such that $k$ is even and for each prime $p$ if $p|n$ then $p-1|k$. let $\{a_1,....,a_{\phi(n)}\}$ be all the numbers coprime to $n$. What's the remainder of the number $a_1^k+.....+a_{\phi(n)}^k$ when it's divided by $n$? [i]proposed by Yahya Motevassel[/i]