For each positive integer $n\leq 49$ we define the numbers $a_n = 3n+\sqrt{n^2-1}$ and $b_n=2(\sqrt{n^2+n}+\sqrt{n^2-n})$. Prove that there exist two integers $A,B$ such that \[ \sqrt{a_1-b_1}+\sqrt{a_2-b_2} + \cdots + \sqrt{a_{49}-b_{49}} = A+B\sqrt2. \]

Circle $ A$ has radius $ 100$. Circle $ B$ has an integer radius $ r<100$ and remains internally tangent to circle $ A$ as it rolls once around the circumference of circle $ A$. The two circles have the same points of tangency at the beginning and end of circle $ B$'s trip. How many possible values can $ r$ have? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 90$

Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.

Leta,b,c are postive real numbers,proof that $ \frac{a}{b\plus{}2c}\plus{}\frac{b}{c\plus{}2a}\plus{}\frac{c}{a\plus{}2b}\geq1$

Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find $u+v.$

Let $ a,b,c,d$ be nonnegative real numbers. Prove that \[ a^4\plus{}b^4\plus{}c^4\plus{}d^4\plus{}2abcd \ge a^2b^2\plus{}a^2c^2\plus{}a^2d^2\plus{}b^2c^2\plus{}b^2d^2\plus{}c^2d^2.\]

Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.

Consider a circle $(O)$ and two fixed points $B,C$ on $(O)$ such that $BC$ is not the diameter of $(O)$. $A$ is an arbitrary point on $(O)$, distinct from $B,C$. Let $D,J,K$ be the midpoints of $BC,CA,AB$, respectively, $E,M,N$ be the feet of perpendiculars from $A$ to $BC$, $B$ to $DJ$, $C$ to $DK$, respectively. The two tangents at $M,N$ to the circumcircle of triangle $EMN$ meet at $T$. Prove that $T$ is a fixed point (as $A$ moves on $(O)$).

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.

Find all positive integers $n$ such that: \[ \dfrac{n^3+3}{n^2+7} \] is a positive integer.

Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum $S=13,689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$? ${ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 $

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have? $\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$

Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation. [i]Proposed by Evan Chen[/i]

Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then \[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]

Define a search algorithm called $\texttt{powSearch}$. Throughout, assume $A$ is a 1-indexed sorted array of distinct integers. To search for an integer $b$ in this array, we search the indices $2^0,2^1,\ldots$ until we either reach the end of the array or $A[2^k] > b$. If at any point we get $A[2^k] = b$ we stop and return $2^k$. Once we have $A[2^k] > b > A[2^{k-1}]$, we throw away the first $2^{k-1}$ elements of $A$, and recursively search in the same fashion. For example, for an integer which is at position $3$ we will search the locations $1, 2, 4, 3$. Define $g(x)$ to be a function which returns how many (not necessarily distinct) indices we look at when calling $\texttt{powSearch}$ with an integer $b$ at position $x$ in $A$. For example, $g(3) = 4$. If $A$ has length $64$, find \[g(1) + g(2) + \ldots + g(64).\]

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

Let $\{a_n\}$ be a sequence of nondecreasing positive integers such that $\textstyle\frac{r}{a_r} = k+1$ for some positive integers $k$ and $r$. Prove that there exists a positive integer $s$ such that $\textstyle\frac{s}{a_s} = k$.

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? $ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$

Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds \[ \frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1. \]

For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ For how many values of $n$ is $n + S(n) + S(S(n)) = 2007?$ $\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$

$a,b,c$ are positive real numbers. prove the following inequality: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$ (20 points)

Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.