Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ meet at $P$. Let the area of triangle $APB$ be $24$ and let the area of triangle $CPD$ be $25$. What is the minimum possible area of quadrilateral $ABCD$?

A polynomial $P (x)$ with real coefficients and of degree $n \ge 3$ has $n$ real roots $x_1 <x_2 < \cdots < x_n$ such that \[x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1} \] Prove that the maximum value of $|P (x)|$ on the interval $[x_1 , x_n ]$ is attained in the interval $[x_{n-1} , x_n ]$.

Let $f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0$ be a strictly decreasing function. (a) Be $a_n$ a sequence of strictly positive reals so that $\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1})$ Prove that $a_n$ is ascending, that $\displaystyle\lim_{k\rightarrow +\infty} f(a_k)$ = 0and that $\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty$ (b) Prove that there exist such a sequence ($a_n$) in $\mathbb{R}^+_0$ if you know $\displaystyle\lim_{x\rightarrow +\infty} f(x)=0$.

Evaluate \[S=\sum_{k=1}^n k(k+1)\ldots (k+p), \] where $n$ and $p$ are positive integers.

How many pairs of integers $(x,y)$ are there such that $\frac{x}{y+7}+\frac{y}{x+7}=1$? $ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 11 $

Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.

Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\]

Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.

2) Let $a, b, c$ be real numbers such that $$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$. b) Prove that $|a|+|b|+|c|< 5$

Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. [i]Budi Surodjo, Jogjakarta[/i]

For a given positive integer $n$, you may perform a series of steps. At each step, you may apply an operation: you may increase your number by one, or if your number is divisible by 2, you may divide your number by 2. Let $\ell(n)$ be the minimum number of operations needed to transform the number $n$ to 1 (for example, $\ell(1) = 0$ and $\ell(7) = 4$). How many positive integers $n$ are there such that $\ell(n) \leq 12$?

The graphs of the lines $$y=x+2, \quad y=3x+4, \quad y=5x+6,\quad y=7x+8,\quad y=9x+10,\quad y=11x+12$$ are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into.

$21$ numbers are written in a row. $u,v,w$ are three consecutive numbers so $v=\frac{2uw}{u+w}$ . The first number is $\frac{1}{100}$ , the last one is $\frac{1}{101}$ . Find the $15$th number.

Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.

If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.

Rhombus $ ABCD$ is similar to rhombus $ BFDE$. The area of rhombus $ ABCD$ is 24, and $ \angle BAD \equal{} 60^\circ$. What is the area of rhombus $ BFDE$? [asy] size(180); defaultpen(linewidth(0.7)+fontsize(11)); pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3)); pair point=(3/2, sqrt(3)/2); draw(B--C--D--A--B--F--D--E--B); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F));[/asy] $ \textbf{(A) } 6 \qquad \textbf{(B) } 4\sqrt {3} \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 6\sqrt {3}$