A convex quadrilateral $ABCD$ is said to be [i]dividable[/i] if for every internal point $P$, the area of $\triangle PAB$ plus the area of $\triangle PCD$ is equal to the area of $\triangle PBC$ plus the area of $\triangle PDA$. Characterize all quadrilaterals which are dividable.

Find $x$ if\[\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.\]

Find the remainder when $19^{92}$ is divided by 92.

Let $S$ be a set of 100 integers. Suppose that for all positive integers $x$ and $y$ (possibly equal) such that $x + y$ is in $S$, either $x$ or $y$ (or both) is in $S$. Prove that the sum of the numbers in $S$ is at most 10,000.

A triangle $ABC$ with $\beta= 45^o$ is given. There is a point $P$ on side $BC$, where $BP : PC =1 : 2$ (inner division) and $\angle APC = 60^o$. Someone claims that you can do it with elementary geometric theorems alone without using the plane trigonometry, the size of the angle $\gamma$ determine.

In base $10$, the product $31\times33$ does not equal $1243$. In what base does $31\times33=1243$?

A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes. What is the maximum number of coins that Martha can take away?

With positive $ a,b,c, $ prove: $$ \frac{a}{8a^2+5b^2+3c^2} +\frac{b}{8b^2+5c^2+3a^2} +\frac{c}{8c^2+5a^2+3b^2}\le\frac{1}{16}\left( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} \right) $$ [i]Titu Zvonaru[/i]

Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]

In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.

Let $a\leqslant b\leqslant c$ be non-negative integers. A triangle on a checkered plane with vertices in the nodes of the grid is called an $(a,b,c)$[i]-triangle[/i] if there are exactly $a{}$ nodes on one side of it (not counting vertices), exactly $b{}$ nodes on the second side, and exactly $c{}$ nodes on the third side. [list] [*]Does there exist a $(9,10,11)$-triangle? [*]Find all triples of non-negative integers $a\leqslant b\leqslant c$ for which there exists an $(a,b,c)$-triangle. [*]For each such triple, find the minimum possible area of the $(a,b,c)$-triangle. [/list] [i]Proposed by P. Kozhevnikov[/i]

If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2 C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$ prove that the triangle must have a right angle.

Let $x_1, x_2, x_3$ be the solutions to $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of $$\sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right].$$

Let $ABC$ be an acute non isosceles triangle. The angle bisector of angle $A$ meets again the circumcircle of the triangle $ABC$ in $D$. Let $O$ be the circumcenter of the triangle $ABC$. The angle bisectors of $\angle AOB$, and $\angle AOC$ meet the circle $\gamma$ of diameter $AD$ in $P$ and $Q$ respectively. The line $PQ$ meets the perpendicular bisector of $AD$ in $R$. Prove that $AR // BC$.

Let $D$ be the foot of the altitude towards $BC$ in the triangle $ABC$. Let $E$ be the intersection of $AB$ with the bisector of angle $\angle C$. Assume that the angle $\angle AEC = 45^o$ . Determine the angle $\angle EDB$.

Given triangle $ABC$ and point $D$ on the $AC$ side. Let $r_1, r_2$ and $r$ denote the radii of the incircle of the triangles $ABD, BCD$, and $ABC$, respectively. Prove that $r_1 + r_2> r$.

Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O.$ Suppose that $AB = 15$, $AC = 14$, and $P$ is a point in the interior of $\triangle ABC$ such that $AP = \frac{13}{2}$, $BP^2 = \frac{409}{4}$, and $P$ is closer to $\overline{AC}$ than to $\overline{AB}$. Let $E$, $F$ be the points where $\overline{BP}$, $\overline{CP}$ intersect $\omega$ again, and let $Q$ be the intersection of $\overline{EF}$ with the tangent to $\omega$ at $A.$ Given that $AQOP$ is cyclic and that $CP^2$ is expressible in the form $\frac{a}{b} - c \sqrt{d}$ for positive integers $a$, $b$, $c$, $d$ such that $\gcd(a, b) = 1$ and $d$ is not divisible by the square of any prime, compute $1000a+100b+10c+d.$ [i]Proposed by Edward Wan[/i]

The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of $\triangle ABC?$ $\textbf{(A) } \frac{1}{6}\sqrt{6} \qquad \textbf{(B) } \frac{3}{2}\sqrt{2}-\frac{3}{2} \qquad \textbf{(C) } 2\sqrt3-3\sqrt2 \qquad \textbf{(D) } \frac{1}{2}\sqrt{2} \qquad \textbf{(E) } \sqrt 3-1$

In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half.

Given a circle and Points $P,B,A$ on it.Point $Q$ is Interior of this circle such that: $1)$ $\angle PAQ=90$. $ 2)PQ=BQ$. Prove that $\angle AQB - \angle PQA=\stackrel{\frown}{AB}$. proposed by Davoud Vakili, Iran.

Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$.

Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?

Let $ ABC $ be a right angle in $ A, $ and $ M $ be the mid of $ BC. $ On the perpendicular of $ AM $ through $ A $ choose a point $ D $ so that $ DM $ meets $ AB $ at a point, namely $ P. $ Let $ E $ be the projection of $ D $ on $ BC. $ Show that $ \angle BPM =\angle EAC. $

Find the greatest exponent $k$ for which $2001^{k}$ divides $2000^{2001^{2002}}+2002^{2001^{2000}}$.

Circles $w_1$ and $w_2$ with centers at points $O_1$ and $O_2$ respectively, intersect at points $A$ and $B$. A line passing through point $B$, intersects the circles $w_1$ and $w_2$ at points $C$ and $D$ other than $B$. Tangents to the circles $w_1$ and $w_2$ at points $C$ and $D$ intersect at point $E$. Line $EA$ intersects the circumscribed circle $w$ of triangle $AO_1O_2$ at point $F$. Prove that the length of the segment is $EF$ is equal to the diameter of the circle $w$. (Vovchenko V., Plotnikov M.)