Let $a$ be a positive real number. Collinear points $Z_1, Z_2, Z_3, Z_4$ (in that order) are plotted on the $(x, y)$ Cartesian plane. Suppose that the graph of the equation \[ x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 + \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} \] passes through points $Z_1$ and $Z_4$, and the graph of the equation \[ x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 - \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} \] passes through points $Z_2$ and $Z_3$. If $Z_1Z_2 = 5$, $Z_2Z_3 = 1$, and $Z_3Z_4 = 3$, then $a^2$ can be written as $\frac{m + n\sqrt{p}}{q}$, where $m$, $n$, $p$, and $q$ are positive integers, $m$, $n$, and $q$ are relatively prime, and $p$ is squarefree. Find $m + n + p + q$.

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

Let $n$ be the smallest nonprime integer greater than 1 with no prime factor less than 10. Then A. $100 < n \leq 110$ B. $110 < n \leq 120$ C. $120 < n \leq 130$ D. $130 < n \leq 140$ E. $140 < n \leq 150$

Let $ABC$ be a triangle with $AB \neq AC$, let $I$ be its incenter, $\gamma$ its inscribed circle and $D$ the midpoint of $BC$. The tangent to $\gamma$ from $D$ different to $BC$ touches $\gamma$ in $E$. Prove that $AE$ and $DI$ are parallel.

Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\ldots ,a_n $ whose product is $ 1. $ Prove that the function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} ,\quad f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) , $ is nondecreasing.

Prove that every polygon with perimeter $ 2a $ can be covered by a disk with diameter $ a $.

The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.

Find all real solutions of the following equation: \[ (x + 1995)(x + 1997)(x + 1999)(x + 2001) + 16 = 0. \]

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$. [i]Proposed by Yang Liu[/i]

Find all primes $p$ and positive integers $n$ satisfying \[n \cdot 5^{n-n/p} = p! (p^2+1) + n.\]

Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

Given $n$ numbers different from $0$, ($n \in \mathbb{N}$) which are arranged randomly. We do the following operation: Choose some consecutive numbers in the given order and change their sign (i.e. $x \rightarrow -x$). What is the minimum number of operations needed, in order to make all the numbers positive for any given initial configuration of the $n$ numbers?

Let $ABCDE$ be a cyclic pentagon such that $BC = DE$ and $AB$ is parallel to $DE$. Let $X, Y,$ and $Z$ be the midpoints of $BD, CE,$ and $AE$ respectively. Show that $AE$ is tangent to the circumcircle of the triangle $XYZ$. Proposed by [i]Nikola Velov, Macedonia[/i]

Each day Walter gets $\$3$ for doing his chores or $\$5$ for doing them exceptionally well. After 10 days of doing his chores daily, Walter has received a total of $\$36$. On how many days did Walter do them exceptionally well? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Consider the string of length $ 6$ composed of three characters $ a$, $ b$, $ c$. For each string, if two $ a$s are next to each other, or two $ b$s are next to each other, then replace $ aa$ by $ b$, and replace $ bb$ by $ a$. Also, if $ a$ and $ b$ are next to each other, or two $ c$s are next to each other, remove all two of them (i.e. delete $ ab$, $ ba$, $ cc$). Determine the number of strings that can be reduced to $ c$, the string of length 1, by the reducing processes mentioned above.