On an infinite chessboard are placed $2009 \ n \times n$ cardboard pieces such that each of them covers exactly $n^2$ cells of the chessboard. Prove that the number of cells of the chessboard which are covered by odd numbers of cardboard pieces is at least $n^2.$ [i](9 points)[/i]

Let $ABC$ be a triangle. Its excircles touch sides $BC, CA, AB$ at $D, E, F$, respectively. Prove that the perimeter of triangle $ABC$ is at most twice that of triangle $DEF$.

Let $V$ be the volume of the octahedron $ABCDEF$ with $A$ and $F$ opposite, $B$ and $E$ opposite, and $C$ and $D$ opposite, such that $AB = AE = EF = BF = 13$, $BC = DE = BD = CE = 14$, and $CF = CA = AD = FD = 15$. If $V = a\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime, find $a + b$.

Natalia and Marcela count $1$ by $1$ starting together at $1$, but Marcela's speed is triple that of Natalia (when Natalia says her second number, Marcela says the fourth number). When the difference of the numbers that they say in unison is any of the multiples of $ 29$, between $500$ and $600$, Natalia continues counting normally and Marcela begins to count downwards in such a way that, at one point, the two say in unison the same number. What is said number?

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally? $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36$

Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.

Prove that for every four positive numbers $a,b,c,d$ the following inequality is true: $$\sqrt{\frac{a^2+b^2+c^2+d^2}4}\ge\sqrt[3]{\frac{abc+abd+acd+bcd}4}.$$

Given an isosceles triangle $\triangle ABC$, $AB=AC$. A line passes through $M$, the midpoint of $BC$, and intersects segment $AB$ and ray $CA$ at $D$ and $E$, respectively. Let $F$ be a point of $ME$ such that $EF=DM$, and $K$ be a point on $MD$. Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$. $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$. Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$. Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$, then $BF$ is perpendicular to $BC$.

On a circle there are given $ n\plus{}3$ distinct points,from which $ n$ are colored red, two yellow, and one blue. Determine the number of polygons which have a) the vertices of the same color b) the vertices of two colors c) the vertices of three colors.

How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ? (A): $16$, (B): $17$, (C): $18$, (D): $19$, (E) None of the above.

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

Let $ F:\mathbb{R}\longrightarrow\mathbb{R} $ be a primitive with $ F(0)=0 $ of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined by $ f(x)=\frac{x}{1+e^x} , $ and let be a sequence $ \left( x_n \right)_{n\ge 0} $ such that $ x_0>0 $ and defined as $ x_n=F\left( x_{n-1} \right) . $ Calculate $ \lim_{n\to\infty } \frac{1}{n}\sum_{k=1}^n \frac{x_k}{\sqrt{x_{k+1}}} $ [i]Florian Dumitrel[/i]

In the diagram, $ABDF$ is a trapezoid with $AF$ parallel to $BD$ and $AB$ perpendicular to $BD.$ The circle with center $B$ and radius $AB$ meets $BD$ at $C$ and is tangent to $DF$ at $E.$ Suppose that $x$ is equal to the area of the region inside quadrilateral $ABEF$ but outside the circle, that y is equal to the area of the region inside $\triangle EBD$ but outside the circle, and that $\alpha = \angle EBC.$ Prove that there is exactly one measure $\alpha,$ with $0^\circ \leq \alpha \leq 90^\circ,$ for which $x = y$ and that this value of $\frac 12 < \sin \alpha < \frac{1}{\sqrt 2}.$ [asy] import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen fftttt = rgb(1,0.2,0.2); draw(circle((6.04,2.8),1.78),qqttff); draw((6.02,4.58)--(6.04,2.8),fftttt); draw((6.02,4.58)--(6.98,4.56),fftttt); draw((6.04,2.8)--(8.13,2.88),fftttt); draw((6.98,4.56)--(8.13,2.88),fftttt); dot((6.04,2.8),ds); label("$B$", (5.74,2.46), NE*lsf); dot((6.02,4.58),ds); label("$A$", (5.88,4.7), NE*lsf); dot((6.98,4.56),ds); label("$F$", (7.06,4.6), NE*lsf); dot((7.39,3.96),ds); label("$E$", (7.6,3.88), NE*lsf); dot((8.13,2.88),ds); label("$D$", (8.34,2.56), NE*lsf); dot((7.82,2.86),ds); label("$C$", (7.5,2.46), NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy]

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

Factor the polynomial $bc (b+c) +ca (c-a)-ab(a + b)$.

There are $n$ sheep and a wolf in sheep's clothing . Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. First, the wolf chooses some sheep to make friend's with. In each of the following days, the wolf eats one of its friends. Whenever the wolf eats a sheep $A$: (a) If a friend of $A$ is originally a friend of the wolf, it un-friends the wolf. (b) If a friend of $A$ is originally not a friend of the wolf, it becomes a friend of the wolf. Repeat the procedure until the wolf has no friend left. Find the largest integer $m$ in terms of $n$ satisfying the following: There exists an initial friendsheep structure such that the wolf has $m$ different ways of choosing initial sheep to become friends, so that the wolf has a way to eat all of the sheep.

There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|$. $\textbf{(A) }0\qquad\textbf{(B) }\dfrac1{2002}\qquad\textbf{(C) }\dfrac1{2001}\qquad\textbf{(D) }\dfrac2{2001}\qquad\textbf{(E) }\dfrac1{1000}$

The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$.

Given a hexagon $ABCDEF$ such that $AB=BC$, $CD=DE$ , $EF=FA$ and $\angle A = \angle C = \angle E $ Prove that $AD, BE, CF$ are concurrent.

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!) (a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$. (b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$. [i]Proposed by Ray Li[/i]

The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of [asy] draw((-3,0)..(0,3)..(3,0)); draw((-3.5,0)--(-2.5,0)); draw((0,2.5)--(0,3.5)); draw((2.5,0)--(3.5,0)); draw((1.8,1.8)--(2.5,2.5)); draw((-1.8,1.8)--(-2.5,2.5)); draw((0,0)--3*dir(120),EndArrow); label("$10$",(-2.6,0),E); label("$11$",(2.6,0),W);[/asy] $ \text{(A)}\ 10.05\qquad\text{(B)}\ 10.15\qquad\text{(C)}\ 10.25\qquad\text{(D)}\ 10.3\qquad\text{(E)}\ 10.6 $

Let a triangle $ABC$ and the cevians $AL, BN , CM$ such that $AL$ is the bisector of angle $A$. If $\angle ALB = \angle ANM$, prove that $\angle MNL = 90$.

The number of geese in a flock increases so that the difference between the populations in year $ n \plus{} 2$ and year $ n$ is directly proportional to the population in year $ n \plus{} 1$. If the populations in the years $ 1994$, $ 1995$, and $ 1997$ were $ 39$, $ 60$, and $ 123$, respectively, then the population in $ 1996$ was $ \textbf{(A)}\ 81\qquad \textbf{(B)}\ 84\qquad \textbf{(C)}\ 87\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ 102$