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$

Prove that For all positive integers $m$ and $n$ , one has $| n \sqrt{2005} - m | > \frac{1}{90n}$

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

Does there exist such infinite set of positive integers $S$ that satisfies the condition below? *for all $a,b$ in $S$, there exists an odd integer $k$ that $a$ divides $b^k+1$.

Given $m,n\geq 2$.Paint each cell of a $m\times n$ board $S$ red or blue so that:for any two red cells in a row,one of the two columns they belong to is all red,and the other column has at least one blue cell in it.Find the number of ways to paint $S$ like this.

Construct a right triangle given the hypotenuse and the median drawn to the leg.

Given circle $\Omega_1$ and $\Omega_2$ interesting at $P$ and $Q$. $X$ and $Y$ on line $PQ$ such that $X, P, Q, Y$ in that order. Point $A$ and $B$ on $\Omega_1$ and $\Omega_2$ respectively such that the intersections of $\Omega_1$ with $AX$ and $AY$, intersections of $\Omega_2$ with $BX$ and $BY$ are all in one line. $l$. Prove that $AB, l$ and perpendicular bisector of $PQ$ are concurrent.

In an $n$-by-$m$ grid, $1$ row and $1$ column are colored blue, the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n,m)$

A [i]Nim-style game[/i] is defined as follows. Two positive integers $k$ and $n$ are specified, along with a finite set $S$ of $k$-tuples of integers (not necessarily positive). At the start of the game, the $k$-tuple $(n, 0, 0, ..., 0)$ is written on the blackboard. A legal move consists of erasing the tuple $(a_1,a_2,...,a_k)$ which is written on the blackboard and replacing it with $(a_1+b_1, a_2+b_2, ..., a_k+b_k)$, where $(b_1, b_2, ..., b_k)$ is an element of the set $S$. Two players take turns making legal moves, and the first to write a negative integer loses. In the event that neither player is ever forced to write a negative integer, the game is a draw. Prove that there is a choice of $k$ and $S$ with the following property: the first player has a winning strategy if $n$ is a power of 2, and otherwise the second player has a winning strategy. [i]Proposed by Linus Hamilton[/i]

Every two residents of a city have an even number of common friends in the city. One day, some of the people sent postcards to some of their friends. Each resident with odd number of friends sent exactly one postcard, and every other - no more than one. Every resident received no more than one postcard. Prove that the number of ways the cards could be sent is odd.

Let $p{}$ be a prime number. Prove that the equation has $x^2-x+3-ps=0$ with $x,s\in\mathbb{Z}$ has solutions if and only if the equation $y^2-y+25-pt=0$ with $y,t\in\mathbb{Z}$ has solutions.

The sequence $(x_n)$ is defined as follows: $$x_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n$$ for every non-negative integer $n$. a. For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors b. Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$.

One day I went for a walk in the morning at $x$ minutes past $5'O$ clock, where $x$ is a 2 digit number. When I returned, it was $y$ minutes past $6'O$ clock, and I noticed that (i) I walked for exactly $x$ minutes and (ii) $y$ was a 2 digit number obtained by reversing the digits of $x$. How many minutes did I walk?

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i] [b][i]Original formulation [/i][/b] Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove: (a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$ (b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $ [i]Proposed by Germany, FR[/i]

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ with center $O$. Let $P$ be the intersection of two diagonals $AC$ and $BD$. Let $Q$ be a point lying on the segment $OP$. Let $E$ and $F$ be the orthogonal projections of $Q$ on the lines $AD$ and $BC$, respectively. The points $M$ and $N$ lie on the circumcircle of triangle $QEF$ such that $QM \parallel AC$ and $QN \parallel BD$. Prove that the two lines $ME$ and $NF$ meet on the perpendicular bisector of segment $CD$. [i]Proposed by Tran Quang Hung, Vietnam[/i]

Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees. [i]Proposed by Raina Yang[/i]

A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.

The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$. Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere.