Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius 1 and passes through the center of $D$. What is the radius of circle $B$? [asy] size(200); defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(Circle(origin, 2)); draw(Circle((-1,0), 1)); draw(Circle((6/9, 8/9), 8/9)); draw(Circle((6/9, -8/9), 8/9)); label("$A$", (-1.2, -0.2), NE); label("$B$", (6/9, 7/9), N); label("$C$", (6/9, -7/9), S); label("$D$", 2*dir(110), dir(110));[/asy] $ \textbf{(A)}\; \frac23\qquad \textbf{(B)}\; \frac{\sqrt{3}}2\qquad \textbf{(C)}\; \frac78\qquad \textbf{(D)}\; \frac89\qquad \textbf{(E)}\; \frac{1+\sqrt3}3 $

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$

Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group. [b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.

Johan is swimming laps in the pool. At $\text{12:17}$ he realized that he had just finished one-third of his workout. By $\text{12:22}$ he had completed eight more laps, and he realized that he had just finished five-elevenths of his workout. After $\text{12:22}$ how many more laps must Johan swim to complete his workout?

Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.

A set consisting of $n$ points of a plane is called a [i]bosonti $n$-point[/i] if any three of its points are located in vertices of an isosceles triangle. Find all positive integers $n$ for which there exists a bosonti $n$-point.

Q. For integers $m,n\geq 1$, Let $A_{m,n}$ , $B_{m,n}$ and $C_{m,n}$ denote the following sets: $A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$ $B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n\}$ given that $\alpha _i \geq 0$ and $\alpha_ i\in \mathbb{Z}$ for all $i$ $C_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1< \alpha_2 < \ldots< \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$ $(a)$ Define a one-one onto map from $A_{m,n}$ onto $B_{m+1,n-1}$. $(b)$ Define a one-one onto map from $A_{m,n}$ onto $C_{m,n+m-1}$. $(c)$ Find the number of elements of the sets $A_{m,n}$ and $B_{m,n}$.

Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$. [i]Tiger Zhang[/i]

Let $a,b,c,d$ to be non negative real numbers satisfying $ab+ac+ad+bc+bd+cd=6$. Prove that \[\dfrac{1}{a^2+1} + \dfrac{1}{b^2+1} + \dfrac{1}{c^2+1} + \dfrac{1}{d^2+1} \ge 2\]

There are two circles with centers $O_1,O_2$ lie inside of circle $\omega$ and are tangent to it. Chord $AB$ of $\omega$ is tangent to these two circles such that they lie on opposite sides of this chord. Prove that $\angle O_1AO_2 + \angle O_1BO_2 > 90^\circ$. [i]Proposed by Iman Maghsoudi[/i]

For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers [i] good [/i] if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good. ([i]In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.[/i]) [i] Lajos Hajdu and Andras Sarkozy, Hungary [/i]

For all positive integers $n$, $k$, let $f(n, 2k)$ be the number of ways an $n \times 2k$ board can be fully covered by $nk$ dominoes of size $2 \times 1$. (For example, $f(2, 2)=2$ and $f(3, 2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2k)$ is odd.

The units digit of $3^{1001}7^{1002}13^{1003}$ is $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $

Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.

Let $n$ be a positive integer and let $G$ be a simple undirected graph on $n$ vertices. Let $d_i$ be the degree of its $i$-th vertex, $i = 1, \dots , n$. Denote $\Delta=\max d_i$. Prove that if \[\sum_{i=1}^n d_i^2>n\Delta(n-\Delta),\] then $G$ contains a triangle.

Let $O$ be the circumcenter of a triangle $ABC$, $P$ be the midpoint of $AO$, and $Q$ be the midpoint of $BC$. If $\angle ABC=4\angle OPQ$ and $\angle ACB=6\angle OPQ$, compute $\angle OPQ$.