Real numbers $(x,y)$ satisfy the following equations: $$(x + 3)(y + 1) + y^2 = 3y$$ $$-x + x(y + x) = - 2x - 3.$$ Find the sum of all possible values of $x$. [i]Team #1[/i]

Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and \[ (x,y) = \left( \frac{r^2-1}{r^2+1}, \frac{2r}{r^2+1} \right) \] where $r$ runs through the elements of $F$ such that $r^2\neq -1$.

A game is played with two heaps of $p$ and $q$ stones. Two players alternate playing, with $A$ starting. A player in turn takes away one heap and divides the other heap into two smaller ones. A player who cannot perform a legal move loses the game. For which values of $p$ and $q$ can $A$ force a victory?

Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order? [i]Roland Hablutzel, Venezuela[/i] [hide="Remark"]The original question was: Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?[/hide]

Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.

Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function, with a continous derivative. Given that $f(0)=0$ and $0\leqslant f'(x)\leqslant 1$ for every $x>0$ prove that\[\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,\]for any positive integer $n{}$ and real number $a>0.$

Point $M$ lies inside triangle $ABC$. Prove that for any other point $N$ lying inside the triangle $ABC$, at least one of the following three inequalities is fulfilled: $AN>AM, BN>BM, CN>CM$.

Let $ f(x)$ be a polynomial of second degree the roots of which are contained in the interval $ [\minus{}1,\plus{}1]$ and let there be a point $ x_0\in [\minus{}1.\plus{}1]$ such that $ |f(x_0)|\equal{}1$. Prove that for every $ \alpha \in [0,1]$, there exists a $ \zeta \in [\minus{}1,\plus{}1]$ such that $ |f'(\zeta)|\equal{}\alpha$ and that this statement is not true if $ \alpha>1$.

Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ . Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

Given right hexagon $H$ with side length $1$. On the sides of $H$ points $A_1$,$A_2$,$\ldots$,$A_k$ such that at least one of them is the midpoint of some side and for every $1 \leq i \leq k$ lines $A_{i-1}A_i$ and $A_iA_{i+1}$ form equal angles with the side, that contains the point $A_i$ (let $A_0=A_k$ and $A_{k+1}=A_1$. It is known that the length of broken line $A_1A_2\ldots A_kA_1$ is a positive integer Prove that $n$ is divisible by $3$ [i]M. Zorka[/i]

Right $ \triangle ABC$ has $ AB \equal{} 3$, $ BC \equal{} 4$, and $ AC \equal{} 5$. Square $ XYZW$ is inscribed in $ \triangle ABC$ with $ X$ and $ Y$ on $ \overline{AC}$, $ W$ on $ \overline{AB}$, and $ Z$ on $ \overline{BC}$. What is the side length of the square? [asy]size(200);defaultpen(fontsize(10pt)+linewidth(.8pt)); real s = (60/37); pair A = (0,0); pair C = (5,0); pair B = dir(60)*3; pair W = waypoint(B--A,(1/3)); pair X = foot(W,A,C); pair Y = (X.x + s, X.y); pair Z = (W.x + s, W.y); label("$A$",A,SW); label("$B$",B,NW); label("$C$",C,SE); label("$W$",W,NW); label("$X$",X,S); label("$Y$",Y,S); label("$Z$",Z,NE); draw(A--B--C--cycle); draw(X--W--Z--Y);[/asy] $ \textbf{(A)}\ \frac {3}{2}\qquad \textbf{(B)}\ \frac {60}{37}\qquad \textbf{(C)}\ \frac {12}{7}\qquad \textbf{(D)}\ \frac {23}{13}\qquad \textbf{(E)}\ 2$

Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$. The angles $\angle BCD$ and $\angle CDA$ are acute. The lines $BC$ and $DA$ are cut at a point $E$. It is known that $AE = 2$, $AC = 6$, $CD =\sqrt{72}$ and area $( \vartriangle BCD)= 18$. (a) Find the height of the trapezoid $ABCD$. (b) Find the area of $\vartriangle ABC$.

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $CDX$ and $ABY$ meet line $XY$ again at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Evan Chang (squareman), USA[/i]

A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

In triangle $ABC$, let $I_a$ $,I_b$, $I_c$ be the centers of the excircles tangent to sides $BC$, $CA$, $AB$, respectively. Let $P$ and $Q$ be the tangency points of the excircle of center $I_a$ with lines $AB$ and $AC$. Line $PQ$ intersects $I_aB$ and $I_aC$ at $D$ and $E$. Let $A_1$ be the intersection of $DC$ and $BE$. In an analogous way we define points $B_1$ and $C_1$. Prove that $AA_1$, $BB_1$ , $CC_1$ are concurrent.

Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions: [b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$ [b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint. [b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length. Prove that $ R$ has at least one side of integral length. [i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.

A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $\text{150,000,000 km}$ from the Sun. Which of the following is the closest for the acceleration of the Earth in this orbit? (A) $\text{exactly 0 m/s}^2$ (B) $\text{0.006 m/s}^2$ (C) $\text{0.6 m/s}^2$ (D) $\text{6 m/s}^2$ (E) $\text{10 m/s}^2$

Patrick tosses four four-sided dice, each numbered $1$ through $4$. What's the probability their product is a multiple of four?

Find $n$ such that $\dfrac1{2!9!}+\dfrac1{3!8!}+\dfrac1{4!7!}+\dfrac1{5!6!}=\dfrac n{10!}$.

Some figures stand in certain cells of a chess board. It is known that a figure stands on each row, and that different rows have a different number of figures. Prove that it is possible to mark $8$ figures so that on each row and column stands exactly one marked figure.

A fair die is rolled six times. The probability of rolling at least a five at least five times is $ \textbf{(A)}\ \frac{13}{729} \qquad\textbf{(B)}\ \frac{12}{729} \qquad\textbf{(C)}\ \frac{2}{729} \qquad\textbf{(D)}\ \frac{3}{729} \qquad\textbf{(E)}\ \text{none of these} $

Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$.

In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$. [i](Proposed by Anton Trygub)[/i]