In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.

In a quadratic table with $ 2007$ rows and $ 2007$ columns is an odd number written in each field. For $ 1\leq i\leq2007$ is $ Z_i$ the sum of the numbers in the $ i$-th row and for $ 1\leq j\leq2007$ is $ S_j$ the sum of the numbers in the $ j$-th column. $ A$ is the product of all $ Z_i$ and $ B$ the product of all $ S_j$. Show that $ A\plus{}B\neq0$

In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.

A [i]string of parentheses[/i] is any word that can be composed by the following rules. 1) () is a string of parentheses. 2) If $s$ is a string of parentheses then $(s)$ is a string of parentheses. 3) If $s$ and t are strings of parentheses then $st$ is a string of parentheses. The [i]midcode [/i] of a string of parentheses is the tuple of natural numbers obtained by finding, for all pairs of opening and its corresponding closing parenthesis, the number of characters remaining to the left from the medium position between these parentheses, and writing all these numbers in non-decreasing order. For example, the midcode of $(())$ is $(2,2)$ and the midcode of ()() is $(1,3)$. Prove that midcodes of arbitrary two different strings of parentheses are different.

How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}$

5) f twice cont diff, $|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1$. prove $\lim_{x\rightarrow +\infty} f(x) = 0$

Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.

(F.Nilov) Two arcs with equal angular measure are constructed on the medians $ AA'$ and $ BB'$ of triangle $ ABC$ towards vertex $ C$. Prove that the common chord of the respective circles passes through $ C$.

In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$. Compute $EC - EB$.

[u]Round 1 [/u] [b]p1.[/b] Pirate Jim had $8$ boxes with gun powder weighing $1, 2, 3, 4, 5, 6, 7$, and $8$ pounds (the weight is printed on top of every box). Pirate Bob hid a $1$-pound gold bar in one of these boxes. Pirate Jim has a balance scale that he can use, but he cannot open any of the boxes. Help him find the box with the gold bar using two weighings on the balance scale. [b]p2.[/b] James Bond will spend one day at Dr. Evil's mansion to try to determine the answers to two questions: a) Is Dr. Evil at home? b) Does Dr. Evil have an army of ninjas? The parlor in Dr. Evil's mansion has three windows. At noon, Mr. Bond will sneak into the parlor and use open or closed windows to signal his answers. When he enters the parlor, some windows may already be opened, and Mr. Bond will only have time to open or close one window (or leave them all as they are). Help Mr. Bond and Moneypenny design a code that will tell Moneypenny the answers to both questions when she drives by later that night and looks at the windows. Note that Moneypenny will not have any way to know which window Mr. Bond opened or closed. [b]p3.[/b] Suppose that you have a triangle in which all three side lengths and all three heights are integers. Prove that if these six lengths are all different, there cannot be four prime numbers among them. p4. Fred and George have designed the Amazing Maze, a $5\times 5$ grid of rooms, with Adorable Doors in each wall between rooms. If you pass through a door in one direction, you gain a gold coin. If you pass through the same door in the opposite direction, you lose a gold coin. The brothers designed the maze so that if you ever come back to the room in which you started, you will find that your money has not changed. Ron entered the northwest corner of the maze with no money. After walking through the maze for a while, he had $8$ shiny gold coins in his pocket, at which point he magically teleported himself out of the maze. Knowing this, can you determine whether you will gain or lose a coin when you leave the central room through the north door? [b]p5.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? [u]Round 2 [/u] [b]p6.[/b] $1000$ non-zero numbers are written around a circle and every other number is underlined. It happens that each underlined number is equal to the sum of its two neighbors and that each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? [b]p7.[/b] A grasshopper is sitting at the edge of a circle of radius $3$ inches. He can hop exactly $4$ inches in any direction, as long as he stays within the circle. Which points inside the circle can the grasshopper reach if he can make as many jumps as he likes? [img]https://cdn.artofproblemsolving.com/attachments/1/d/39b34b2b4afe607c1232f4ce9dec040a34b0c8.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct? [asy]unitsize(3mm); fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; path divider=(-2,2)--(-3,3)--cycle; fill(onewhite,white); fill(rotate(90)*onewhite,white); fill(rotate(180)*onewhite,white); fill(rotate(270)*onewhite,white);[/asy] $ \textbf{(A)}\ B \equal{} W \qquad \textbf{(B)}\ W \equal{} R \qquad \textbf{(C)}\ B \equal{} R \qquad \textbf{(D)}\ 3B \equal{} 2R \qquad \textbf{(E)}\ 2R \equal{} W$

Show that among all quadrilaterals of a given perimeter the square has the largest area.

Let $p$ be a prime number and let $a_1,a_2,\dots,a_k$ be distinct integers chosen from $1,2,\dots,p-1$. For $1\le i \le k$, let $f_i^{(n)}$ denote the remainder of the integer $na_1$ upon division by $p$, so $0\le f_i^{(n)}<p$. Define $S=\{n:1\le n \le p-1,f_1^{(n)}<\dots<f_k^{(n)}\}$ Show that $S$ has less than $\frac{2p}{k+1}$ elements.

Let $ABC$ be an acute triangle. A line, parallel to $BC$, intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At which placement of points $M$ and $P$, is the radius of the circumcircle of the triangle $BMP$ is the smallest?

Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having $0$ as a left-hand endpoint, such that for every positive $x\in I$ the average of $f$ over the closed interval $[0,x]$ is equal to $\sqrt{ f(0) f(x)}.$

Find all integer coefficient polynomials $Q$ such that [list] [*] $Q(n)\ge 1$ $\forall n\in \mathbb{Z}_+$. [*] $Q(mn)$ and $Q(m)Q(n)$ have the same number of prime divisors $\forall m,n\in\mathbb{Z}_+$. [/list]

We are given a finite number of functions of the form $y = c2^{-|x-d|}$. In each case $c$ and $d$ are parameters with $c > 0$. The function $f(x)$ is defined on the interval $[a, b]$ as follows: For each $x$ in $[a, b]$, $f(x)$ is the maximum value taken by any of the given functions $y$ (defined above) at that point $x$. It is known that $f(a) = f(b)$. Prove that the total length of the intervals in which the function $f$ is increasing is equal to the total length of the intervals in which it is decreasing (that is, both are equal to $(b- a)/2$ ). (NB Vasiliev)

Prove that the only way to cover a square of side $1$ with a finite number of circles that do not overlap, it is with only one circle of radius greater than or equal to $\frac{1}{\sqrt2}$. Circles can occupy part of the outside of the square and be of different radii.

A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$. Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$

Find the smallest positive integer $n$ such that $n^4 + (n+1)^4$ is composite.

Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out? [i]A. Golovanov[/i]

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

If $a=\log_8225$ and $b=\log_215$, then $\textbf{(A) }a=\frac{1}{2}b\qquad\textbf{(B) }a=\frac{2b}{3}\qquad\textbf{(C) }a=b\qquad\textbf{(D) }b=\frac{1}{2}a\qquad \textbf{(E) }a=\frac{3b}{2}$

A rectangular array has $ n$ rows and $ 6$ columns, where $ n \geq 2$. In each cell there is written either $ 0$ or $ 1$. All rows in the array are different from each other. For each two rows $ (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$ and $ (y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})$, the row $ (x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6})$ can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.

Prove that for each coloring of the points inside or on the boundary of a square with $1391$ colors, there exists a monochromatic regular hexagon.