Eight people are posing together in a straight line for a photo. Alice and Bob must stand next to each other, and Claire and Derek must stand next to each other. How many different ways can the eight people pose for their photo?

The minimum value of $ x(x \plus{} 4)(x \plus{} 8)(x \plus{} 12)$ in real numbers is ? $\textbf{(A)}\ \minus{} 240 \qquad\textbf{(B)}\ \minus{} 252 \qquad\textbf{(C)}\ \minus{} 256 \qquad\textbf{(D)}\ \minus{} 260 \qquad\textbf{(E)}\ \minus{} 280$

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Let a [i]lattice tetrahedron[/i] denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a [i]move[/i] consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves

$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$ [i]A. Golovanov[/i]

Let $f(x)=x^{2002}-x^{2001}+1$. Prove that for every positive integer $m$, the numbers $m,f(m),f(f(m)),\ldots$ are pairwise coprime.

Let $O$ be the circumcenter of a triangle $\vartriangle ABC$. It is given that $\angle ABC = 70^o$, $\angle ACB =50^o$. Let the angle bisector of $\angle BAC$ intersect the circumcircle of $\vartriangle ABC$ again at $D$. Compute $\angle ADO$.

An acute-angled triangle $ABC$ is fixed on a plane with largest side $BC$. Let $PQ$ be an arbitrary diameter of its circumscribed circle, and the point $P$ lies on the smaller arc $AB$, and the point $Q$ is on the smaller arc $AC$. Points $X, Y, Z$ are feet of perpendiculars dropped from point $P$ to the line $AB$, from point $Q$ to the line $AC$ and from point $A$ to line $PQ$. Prove that the center of the circumscribed circle of triangle $XYZ$ lies on a fixed circle.

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n). \]

Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.

Compute the number of functions $f\colon\{1, \dots, 15\} \to \{1, \dots, 15\}$ such that, for all $x \in \{1, \dots, 15\}$, \[ \frac{f(f(x)) - 2f(x) + x}{15} \]is an integer. [i]Proposed by Ankan Bhattacharya[/i]

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

[asy]draw((0,0)--(10,0)--(5,5sqrt(3))--cycle); draw(Circle((0,0),0.75)); fill(Circle((0,0),0.75),white); draw(Circle((5,0),0.75)); fill(Circle((5,0),0.75),white); draw(Circle((10,0),0.75)); fill(Circle((10,0),0.75),white); draw(Circle((5,5sqrt(3)),0.75)); fill(Circle((5,5sqrt(3)),0.75),white); draw(Circle((2.5,2.5sqrt(3)),0.75)); fill(Circle((2.5,2.5sqrt(3)),0.75),white); draw(Circle((7.5,2.5sqrt(3)),0.75)); fill(Circle((7.5,2.5sqrt(3)),0.75),white);[/asy] In a magic triangle, each of the six whole numbers $ 10\minus{}15$ is placed in one of the circles so that the sum, $ S$, of the three numbers on each side of the triangle is the same. The largest possible value for $ S$ is \[ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 38 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 40 \]

Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.

$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.