Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30% of the speed of light here and back, and that their engines allowed them to instantaneously hop to this speed. He also says that Caloprians can only reproduce on their home planet. Call the minimum life span, in years, of a Caloprian, assuming some had to reach Earth to wipe out the dinosaurs, $T$. Assume that, once a Caloprian reaches Earth, they instantaneously wipe out the dinosaurs. Then, $T$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime. Find $m+n$. [i](B. Dejean, 6 points)[/i]

Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively. Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$. [i](M. Karpuk)[/i]

Find all positive integers $l$ for which the equation \[ a^3+b^3+ab=(lab+1)(a+b) \] has a solution over positive integers $a,b$.

Let $n$ be a positive integer. Find the maximal real number $k$, such that the following holds: For any $n$ real numbers $x_1,x_2,...,x_n$, we have $\sqrt{x_1^2+x_2^2+\dots+x_n^2}\geq k\cdot\min(|x_1-x_2|,|x_2-x_3|,...,|x_{n-1}-x_n|,|x_n-x_1|)$

Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$

Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$

Let $O$ be the circumcenter,$R$ be the circumradius, and $k$ be the circumcircle of a triangle $ABC$ . Let $k_1$ be a circle tangent to the rays $AB$ and $AC$, and also internally tangent to $k$. Let $k_2$ be a circle tangent to the rays $AB$ and $AC$ , and also externally tangent to $k$. Let $A_1$ and $A_2$ denote the respective centers of $k_1$ and $k_2$. Prove that: $(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.$

A function $f:\mathbb N\to\mathbb N$, where $\mathbb N$ is the set of positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m>n$) the number $f(m)-f(n)$ is divisible by $m-n$. Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n)=p(n)$ for all positive integers $n$?) [i](Folklore)[/i]

$S$ is a circle with $AB$ a diameter and $t$ is the tangent line to $S$ at $B$. Consider the two points $C$ and $D$ on $t$ such that $B$ is between $C$ and $D$. Suppose $E$ and $F$ are the intersections of $S$ with $AC$ and $AD$ and $G$ and $H$ are the intersections of $S$ with $CF$ and $DE$. Show that $AH=AG$.

Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.

Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$.

The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and \[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \] for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $.

Let x be a real number such that $(\sqrt{6})^x -3^x = 2^{x-2}$. Evaluate $\frac{4^{x+1}}{9^{x-1}}$ .

Let $a_1, a_2, \dots$ be an infinite sequence of positive integers such that, for all positive integers $m$ and $n,$ we have that $a_{m+n}$ divides $a_ma_n-1.$ Prove that there exists an integer $C$ such that, for all positive integers $k>C,$ we have $a_k=1.$

Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition: The difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive integers such that $f(K)$ is the biggest element of $K \in F$. Determine the least value of $f(K)$.

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$

Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation $k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.

Compute $\int^1_0\left((e-1)\sqrt{\ln(1+ex-x)}+e^{x^2}\right)dx$.

There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone) such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)

Prove that any integer has a multiple consisting of all ten digits $\{0,1,2,3,4,5,6,7,8,9\}$. \\ [i]Note: Any digit can be repeated any number of times[/i]

Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

The following problem is open in the sense that the answer to part (b) is not currently known. Let $n$ be an odd positive integer and let \[ f_n(x,y,z) = x^n + y^n + z^n + (x+y+z)^n. \] $(a)$ Prove that there exist infinitely many values of $n$ such that \[ f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2}, \] for some integer polynomials $g_n(x,y,z)$ and $h_n(x,y,z)$, neither of which is constant modulo 2. $(b)$ Determine all values of $n$ such that \[ f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2}, \] for some integer polynomials $g_n(x,y,z)$ and $h_n(x,y,z)$, neither of which is constant modulo 2. (Two integer polynomials are $\emph{congruent modulo 2}$ if every coefficient of their difference is even. A polynomial is $\emph{constant modulo 2}$ if it is congruent to a constant polynomial modulo 2.)

For each $n\in \mathbb{Z}_{>0}$ let $a_n$ be the closest integer to $\sqrt{n}$. Compute the general term of the sequence: $(b_n)_{n\geqslant 1}$ with \[b_n=\sum_{k=1}^{n^2} a_k.\] [i]Pall Dalyay[/i]

let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $ let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $ suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $ now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $