Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and \[(A^2+B^2)(A^4+B^4)=A^5+B^5.\] Find all possible values of $\det(AB)$ for the given $n$.

Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$. Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle. [color=blue]Should the first sentence read: Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]

8) A train, originally of mass $M$, is traveling on a frictionless straight horizontal track with constant speed $v$. Snow starts to fall vertically and sticks to the train at a rate of $\rho$, where $\rho$ has units of kilograms per second. The train's engine keeps the train moving at constant speed $v$ as snow accumulates on the train. What is the minimum power required from the engine to keep the train traveling at a constant speed v? A) $0$ B) $Mgv$ C) $\frac{1}{2}Mv^2$ D) $\frac{1}{2}pv^2$ E) $\rho v^2$

Twelve $1$'s and ten $-1$'s are written on a chalkboard. You select 10 of the numbers and compute their product, then add up these products for every way of choosing 10 numbers from the 22 that are written on the chalkboard. What sum do you get?

Find all natural numbers $n$ for which the following $5$ conditions hold: $(1)$ $n$ is not divisible by any perfect square bigger than $1$. $(2)$ $n$ has exactly one prime divisor of the form $4k+3$, $k\in \mathbb{N}_0$. $(3)$ Denote by $S(n)$ the sum of digits of $n$ and $d(n)$ as the number of positive divisors of $n$. Then we have that $S(n)+2=d(n)$. $(4)$ $n+3$ is a perfect square. $(5)$ $n$ does not have a prime divisor which has $4$ or more digits.

There are two circles with centers $O_1,O_2$ lie inside of circle $\omega$ and are tangent to it. Chord $AB$ of $\omega$ is tangent to these two circles such that they lie on opposite sides of this chord. Prove that $\angle O_1AO_2 + \angle O_1BO_2 > 90^\circ$. [i]Proposed by Iman Maghsoudi[/i]

Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR, APQ$ and $PQCR$. Find the minimum possible value of $M$.

3. Plane is divided with horizontal and vertical lines into unit squares. Into each square we write a positive integer so that each positive integer appears exactly once. Determine whether it is possible to write numbers in such a way, that each written number is a divisor of a sum of its four neighbours.

All the $5$-digit numbers from $11111$ to $99999$ are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting $444445$-digit number is not a power of two.

From a point $P$, two tangent lines are drawn to a circle $\Gamma$, which touch it at points $A$ and $B$. A circle $\Phi$ is drawn with center at $P$ and passes through $A$ and $B$ and is taken a point $R$ that is on the circumference $\Phi$ and in the interior of $\Gamma$. The straight line $PR$ intersects $\Gamma$ at the points $S$ and $Q$. The straight lines $AR$ and $BR$ meet $\Gamma$ again at points $C$ and $D$, respectively. Prove that $CD$ passes through the midpoint of $SQ$.

Show that there are infinitely many odd composite numbers in the sequence $1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...$ .

Let $ABC$ be a triangle. Prove that \[\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab} \ge 4\left(\sin^2\frac{A}{2}+\sin^2\frac{B}{2}+\sin^2\frac{C}{2}\right).\]

Triangle $ABC$ has circumcenter $O$ and incenter $I$. The incircle is tangent to $AC, AB$ at $E, F$, respectively. $H$ is the orthocenter of $\triangle BIC$. $\odot(AEF)$ and $\odot(ABC)$ intersects again at $S$. $BC, AH$ intersects $OI$ again at $J, K$, respectively. Prove that $H, K, J, S$ are concyclic. [i]Proposed by chengbilly[/i]

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.

Triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ and $E$ be points on the respective sides $AB$ and $AC$ so that $DE$ is perpendicular to $AO$. Show that the four points $B,D,E$ and $C$ lie on a circle.

Let $a_0,a_1,\dots,a_n$ be positive real numbers such that $a_{k+1}-a_k \geq 1$ for all $k=0,1,\dots,n-1.$ Prove that \[1+\frac{1}{a_0} \left( 1+\frac1{a_1-a_0}\right)\cdots\left(1+\frac1{a_n-a_0}\right)\leq \left(1+\frac1{a_0}\right) \left(1+\frac1{a_1}\right)\cdots \left(1+\frac1{a_n}\right).\]

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$. [i]Alex Zhu.[/i]

Can both $(x^2+y)$ and $(y^2+x)$ be exact squares for natural $x$ and $y$?

The center cell of a $5 \times 5$ square grid is removed. Determine the number of ways to color the remaining 24 cells one of four colors (cyan, magenta, yellow, and black) such that any $2 \times 2$ square of cells not containing the center cell contains cells of all four colors.

Inside a quadrilateral $ABCD$ there exists a point $P$ such that $AP$ is perpendicular to $AD$ and the line $BP$ is perpendicular to $DC$. Besides, $AB = 7$, $AP = 3$, $BP = 6$, $AD = 5 $ and $CD = 10$. Calculate the area of the triangle $ABC$.

Each square of a $33\times 33$ square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the minimal number of separating edges in the grid.

Let $P$ be an interior point of a circle and $A_1,A_2...,A_{10}$ be points on the circle such that $\angle A_1PA_2 = \angle A_2PA_3 = ... = \angle A_{10}PA_1 = 36^o$. Prove that $PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}$.

Let $n$ be a positive integer. Suppose that we have disks of radii $1, 2, . . . , n.$ Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions: $i)$ Of each size exactly one disk lies in the stack. $ii)$ If we look at the stack from directly above, we can see the edges of all of the $n$ disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.) Determine the number of distinct stacks of disks satisfying these conditions. (Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)

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 $\sum_{k=0}^nk p_n(k)=n!$.