Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$? $ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $

An angle $< 45^o$ is given in the plane of the drawing. Furthermore, the projection $P_1$ of a point $P$ lying above the plane of the drawing and the distance from $P$ to $P_1$ are given. $P_1$ lies within the given angle. On the legs of the given angle, construct points $A$ and $B$, respectively, such that the triangle $PAB$ has a minimal perimeter.

Given a positive integer $n$ with prime factorization $p_1^{e_1}p_2^{e_2}... p_k^{e_k}$ , we define $f(n)$ to be $\sum^k_{i=1}p_ie_i$. In other words, $f(n)$ is the sum of the prime divisors of $n$, counted with multiplicities. Let $M$ be the largest odd integer such that $f(M) = 2023$, and $m$ the smallest odd integer so that $f(m) = 2023$. Suppose that $\frac{M}{m}$ equals $p_1^{e_1}p_2^{e_2}... p_l^{e_l}$ , where the $e_i$ are all nonzero integers and the $p_i$ are primes. Find $\left| \sum^l_{i=1} (p_i + e_i) \right|$.

Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$? [asy] unitsize(12); pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A; real theta = 41.5; pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1; filldraw(P1--P2--P3--cycle, gray(0.9)); draw(Circle(A, 4)); draw(Circle(B, 4)); draw(Circle(C, 4)); dot(P1); dot(P2); dot(P3); defaultpen(fontsize(10pt)); label("$P_1$", P1, E*1.5); label("$P_2$", P2, SW*1.5); label("$P_3$", P3, N); label("$\omega_1$", A, W*17); label("$\omega_2$", B, E*17); label("$\omega_3$", C, W*17); [/asy] $\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554$

Let $ \Gamma_1,\Gamma_2$ and $ \Gamma_3$ be three non-intersecting circles,which are tangent to the circle $ \Gamma$ at points $ A_1,B_1,C_1$,respectively.Suppose that common tangent lines to $ (\Gamma_2,\Gamma_3)$,$ (\Gamma_1,\Gamma_3)$,$ (\Gamma_2,\Gamma_1)$ intersect in points $ A,B,C$. Prove that lines $ AA_1,BB_1,CC_1$ are concurrent.

$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.

Let $ABC$ be a triangle with $AB=AC$. Let point $Q$ be on plane such that $AQ \parallel BC$ and $AQ = AB$. Now let the $P$ be the foot of perpendicular from $Q$ to $BC$. Show that the circle with diameter $PQ$ is tangent to the circumcircle of triangle $ABC$.

A circle $K$ centered at $(0,0)$ is given. Prove that for every vector $(a_1,a_2)$ there is a positive integer $n$ such that the circle $K$ translated by the vector $n(a_1,a_2)$ contains a lattice point (i.e., a point both of whose coordinates are integers).

Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the $3$-by-$3$ array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players has $3$ of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over? $\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 112 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 148 \qquad\textbf{(E)}\ 160$

A round robin tournament is held with $2016$ participants. Each player plays each other player once and exactly one game results in a tie. Let $W$ be the sum of the squares of each team's win total and let $L$ be the sum of the squares of each team's loss total. Find the maximum possible value of $W-L$. [i]Proposed by Matthew Weiss

On a $5 \times 5$ board, pieces made up of $4$ squares are placed, as seen in the figure, each covering exactly $4$ squares of the board. The pieces can be rotated or turned over. They can also overlap, but they cannot protrude from the board. Suppose that each square on the board is covered by at most two pieces. Determine the maximum number of squares on the board that can be covered (by one or two pieces). [asy] size(3cm); draw((0,0)--(0,1)--(1,1)--(1,0)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,2)--(2,2)--(2,1)--(3,1)--(3,2)--(2,2)); [/asy]

(1) Evaluate $ \int_0^1 xe^{x^2}dx$. (2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.

For all integers $ n$ greater than $ 1$, define $ a_n \equal{} \frac {1}{\log_n 2002}$. Let $ b \equal{} a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5$ and $ c \equal{} a_{10} \plus{} a_{11} \plus{} a_{12} \plus{} a_{13} \plus{} a_{14}$. Then $ b \minus{} c$ equals $ \textbf{(A)}\ \minus{} 2 \qquad \textbf{(B)}\ \minus{} 1 \qquad \textbf{(C)}\ \frac {1}{2002} \qquad \textbf{(D)}\ \frac {1}{1001} \qquad \textbf{(E)}\ \frac {1}{2}$

Cyclic pentagon $ ABCDE$ has a right angle $ \angle{ABC} \equal{} 90^{\circ}$ and side lengths $ AB \equal{} 15$ and $ BC \equal{} 20$. Supposing that $ AB \equal{} DE \equal{} EA$, find $ CD$.

If the point $(x,-4)$ lies on the straight line joining the points $(0,8)$ and $(-4,0)$ in the xy-plane, then $x$ is equal to $\textbf{(A) }-2\qquad\textbf{(B) }2\qquad\textbf{(C) }-8\qquad\textbf{(D) }6\qquad \textbf{(E) }-6$

Compute $$\sum_{n=1}^{\infty} \frac{2^{n+1}}{8 \cdot 4^n - 6 \cdot 2^n +1}$$

We want to write down as many distinct positive integers as possible, so that no two numbers on our list have a sum or a difference divisible by $2019$. At most how many integers can appear on such a list?

How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease?

Circle $\omega$ has radius 10 with center $O$. Let $P$ be a point such that $PO=6$. Let the midpoints of all chords of $\omega$ through $P$ bound a region of area $R$. Find the value of $\lfloor 10R \rfloor$. [i]Proposed by Andrew Zhao[/i]

A mixture is prepared by adding $50.0$ mL of $0.200$ M $\ce{NaOH}$ to $75.0$ mL of $0.100$ M $\ce{NaOH}$. What is the $\[[OH^-]$ in the mixture? $ \textbf{(A) }\text{0.0600 M}\qquad\textbf{(B) }\text{0.0800 M}\qquad\textbf{(C) }\text{0.140 M}\qquad\textbf{(D) }\text{0.233 M}\qquad$

In a triangle $ABC$ the bisectors $BB'$ and $CC'$ are drawn. After that, the whole picture except the points $A, B'$, and $C'$ is erased. Restore the triangle using a compass and a ruler. (A.Karlyuchenko)

When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as $$\frac{n}{6^7},$$where $n$ is a positive integer. What is $n$? $\textbf{(A) } 42 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 84 $

A polynomial $P(x,y)$ with integer coefficients satisfies two following conditions: 1. for every integer $a$ there exists exactly one integer $y$, such that $P(a,y)=0$ 2. for every integer $b$ there exists exactly one integer $x$, such that $P(x,b)=0$ a) Prove that if the degree of $P$ is $2$, then it is divisible by either $x-y+C$ for some integer $C$, or $x+y+C$ for some integer $C$ b) Is there a polynomial $P$ that isn't divisible by any of $x-y+C$ or $x+y+C$ for integers $C$?

How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20}$= 0.05 has two digits after the decimal point, so $20$ is a valid N. [i]Proposed by Hari Desikan (HariDesikan)[/i]

Positive numbers $x, y, z$ satisfy the condition $$xy + yz + zx + 2xyz = 1.$$ Prove that $4x + y + z \ge 2.$ [i]A. Khrabrov[/i]