Given a trapezoid with bases $AB$ and $CD$, there exists a point $E$ on $CD$ such that drawing the segments $AE$ and $BE$ partitions the trapezoid into $3$ similar isosceles triangles, each with long side twice the short side. What is the sum of all possible values of $\frac{CD}{AB}$? [i]Proposed by Adam Bertelli[/i]

For positive integer numbers $a$, $b$ and $c$ it is known that $a^2+b^2+c^2$ and $a^3+b^3+c^3$ are both divisible by $a+b+c$. In addition, $gcd(a+b+c, 6) = 1$. Prove that $a^5+b^5+c^5$ is divisible by $(a+b+c)^2$. [i] A. Antropov [/i]

Let $a$, $b$, $c$ be three positive reals such that $(a+b)(b+c)(c+a)=1$. Prove that the following inequality holds: \[ ab+bc+ca \leq \frac 34 . \] [i]Cezar Lupu[/i]

Let $m,n,k$ be positive integers with $m\ge2$ and $k\ge\log_2(m-1)$. Prove that $$\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.$$

For which values of $n$ it is possible to cover the side wall of staircase of n steps (for $n = 6$ in the figure) with plates of shown shape? The width and height of each step is $1$ dm, the dimensions of plate are $2 \times 2$ dm and from the corner there is cut out a piece with dimensions $1\times 1$ dm. [img]https://cdn.artofproblemsolving.com/attachments/e/e/ac7a52f3dd40480f82024794708c5a449e0c2b.png[/img]

Find all positive integers n with the following property: For every positive divisor $d$ of $n$, $d+1$ divides $n+1$.

Let $\bigtriangleup ABC$ be a triangle and $D$ be a point in line $BC$ such that $AD$ bisects $\angle BAC$. Furthermore, let $F$ and $G$ be points on the circumcircle of $\bigtriangleup ABC$ and $E\neq D$ point in line $BC$ such that $AF=AE=AD=AG$. If $X$ and $Y$ are the feet of perpendiculars from $D$ to $EF$ and $EG,$ respectively. Prove that $XY\parallel AD$.

Find all $n\in \mathbb{N}$, $n>1$ with the following property: All divisors of $n$ can be put in a rectangular table in such way that the sums of the numbers by rows are equal and the sums of the numbers by columns are also equal.

A cyclic quadrilateral $ABCD$ has circumcircle $\Gamma$, and $AB+BC=AD+DC$. Let $E$ be the midpoint of arc $BCD$, and $F (\neq C)$ be the antipode of $A$ [i]wrt[/i] $\Gamma$. Let $I,J,K$ be the incenter of $\triangle ABC$, the $A$-excenter of $\triangle ABC$, the incenter of $\triangle BCD$, respectively. Suppose that a point $P$ satisfies $\triangle BIC \stackrel{+}{\sim} \triangle KPJ$. Prove that $EK$ and $PF$ intersect on $\Gamma.$

For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights?

Square $ABCD$ with side length $2$ begins in position $1$ with side $AD$ horizontal and vertex $A$ in the lower right corner. The square is rotated $90^o$ clockwise about vertex $ A$ into position $2$ so that vertex $D$ ends up where vertex $B$ was in position $1$. Then the square is rotated $90^o$ clockwise about vertex $C$ into position $3$ so that vertex $B$ ends up where vertex $D$ was in position $2$ and vertex $B$ was in position $1$, as shown below. The area of the region of points in the plane that were covered by the square at some time during its rotations can be written $\frac{p\pi + \sqrt{q} + r}{s}$, where $p, q, r,$ and $s$ are positive integers, and $p$ and $s$ are relatively prime. Find $p + q + r + s$. [img]https://cdn.artofproblemsolving.com/attachments/9/2/cb15769c30018545abfa82a9f922201c4ae830.png[/img]

Let $AB$ be a diameter of a circle with centre $O$. We choose a point $C$ on the circumference of the circle such that $OC$ and $AB$ are perpendicular to each other. Let $P$ be an arbitrary point on the (smaller) arc $BC$ and let the lines $CP$ and $AB$ meet at $Q$. We choose $R$ on $AP$ so that $RQ$ and $AB$ are perpendicular to each other. Show that $BQ =QR$.

Isabella's house has $ 3$ bedrooms. Each bedroom is $ 12$ feet long, $ 10$ feet wide, and $ 8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $ 60$ square feet in each bedroom. How many square feet of walls must be painted? $ \textbf{(A)}\ 678 \qquad \textbf{(B)}\ 768 \qquad \textbf{(C)}\ 786 \qquad \textbf{(D)}\ 867 \qquad \textbf{(E)}\ 876$

There are $80$ peoples, one of them is murderer, and other one is witness of crime. Every day detective interrogates some peoples from this group. Witness will says about crime only if murderer will not be in interrogatory with him. It is enough $12$ days to find murderer ?

Let $a,b,c$ be positive real numbers. Prove that \[\left((3a^2+1)^2+2\left(1+\frac{3}{b}\right)^2\right)\left((3b^2+1)^2+2\left(1+\frac{3}{c}\right)^2\right)\left((3c^2+1)^2+2\left(1+\frac{3}{a}\right)^2\right)\geq 48^3\]

In a triangle $ABC$, $AD$ is the altitude from $A$, and $H$ is the orthocentre. Let $K$ be the centre of the circle passing through $D$ and tangent to $BH$ at $H$. Prove that the line $DK$ bisects $AC$.

$ C$ and $ D$ are points on a semicircle. The tangent at $ C$ meets the extended diameter of the semicircle at $ B$, and the tangent at $ D$ meets it at $ A$, so that $ A$ and $ B$ are on opposite sides of the center. The lines $ AC$ and $ BD$ meet at $ E$. $ F$ is the foot of the perpendicular from $ E$ to $ AB$. Show that $ EF$ bisects angle $ CFD$

Let be eight real numbers $ 1\le a_1< a_2< a_3< a_4,x_1<x_2<x_3<x_4. $ Prove that $$ \begin{vmatrix}a_1^{x_1} & a_1^{x_2} & a_1^{x_3} & a_1^{x_4} \\ a_2^{x_1} & a_2^{x_2} & a_2^{x_3} & a_2^{x_4} \\ a_3^{x_1} & a_3^{x_2} & a_3^{x_3} & a_3^{x_4} \\ a_4^{x_1} & a_4^{x_2} & a_4^{x_3} & a_4^{x_4} \\ \end{vmatrix} >0. $$ [i]Marian Andronache, Ion Savu[/i]

Suppose $a$ and $b$ be positive integers not exceeding $100$ such that $$ab=\left(\frac{\text{lcm}(a,b)}{\gcd(a,b)}\right)^2.$$ Compute the largest possible value of $a+b.$

Let $X{}$ be a set of integers which can be partitioned into $N{}$ disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into $N{}$ progressions unique for every such $X{}$ if a) $N = 2{}$ and b) $N = 3$? [i]Viktor Kleptsyn[/i]

The great Zagier and his assistant performed a magic trick, which was a natural one number $n$ involved. To do this, Zagier distributes $n$ identical coins (with heads and tails on the sides) and then leave the hall. The audience now arranges the coins in a row, you can choose the top side of each coin as you like, and then tell the assistant one natural number $k$ from $1$ to $n$. This then turns over exactly one coin, whereupon Zagier is brought in and placed in front of said coins. To the surprise of the non-mathematical, he can name the number $k$ to the public. For which natural numbers $n$ can this trick be carried out, if the two are allowed to coordinate only before the show, but during the show they do not use any magic tricks?

Consider the set of natural numbers $ U = \{1,2,3, ..., 6024 \} $ Prove that for any partition of the $ U $ in three subsets with $ 2008 $ elements each, we can choose a number in each subset so that one of the numbers is the sum of the other two numbers.

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) $\rho v^2$