Given $P$ a real polynomial with degree greater than $ 1$. Find all pairs $(f,Q)$ with function $f : R \to R$ and the real polynomial $Q$ satisfying the following two conditions: i) for all $x, y \in R$, we have $f(P(x) + f(y)) = y + Q(f(x))$. ii) there exists $x_0 \in R$ such that $f(P(x_0)) = Q(f(x_0))$.

Establish the maximum and minimum values that the sum $|a| + |b| + |c|$ can have if $a, b, c$ are real numbers such that the maximum value of $|ax^2 + bx + c|$ is $1$ for $-1 \leq x \leq 1.$

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

Compute the sum of the digits of $1001^{10}$

Consider $\vartriangle ABC$ such that $CA + AB = 3BC$. Let the incircle $\omega$ touch segments $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$, respectively, and define $P$ and $Q$ such that segments $\overline{P E}$ and $\overline{QF}$ are diameters of $\omega$. Define the function $D$ of a point $K$ to be the sum of the distances from $K$ to $P$ and $Q$ (i.e. $D(K) = KP + KQ$). Let $W, X, Y$ , and $Z$ be points chosen on lines $\overleftrightarrow {BC}$, $\overleftrightarrow {CE}$, $\overleftrightarrow {EF}$, and $\overleftrightarrow {F B}$, respectively. Given that $BC =\sqrt{133}$ and the inradius of $\vartriangle ABC$ is $\sqrt{14}$, compute the minimum value of $D(W) + D(X) + D(Y ) + D(Z)$.

Prove that $$\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}$$ for all real numbers $a, b, c > 0$

Find the number of integers $n$ for which $n^2+10n<2008$.

Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? ${{ \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72}\qquad\textbf{(E)}\ 80} $

Consider a square painting of size $1 \times 1$. A rectangular sheet of paper of area $2$ is called its “envelope” if one can wrap the painting with it without cutting the paper. (For instance, a $2 \times 1$ rectangle and a square with side $\sqrt2$ are envelopes.) a) Show that there exist other envelopes. (4) b) Show that there exist infinitely many envelopes. (3)

Compute the number of non-negative integers $k < 2^{20}$ such that $\binom{5k}{k}$ is odd. [i]Proposed by David Tang[/i]

The circle of center $I$ is inscribed in the convex quadrilateral $ABCD$. Let $M$ and $N$ be points on the segments $AI$ and $CI$, respectively, such that $\angle MBN = \frac 12 \angle ABC$. Prove that $\angle MDN = \frac 12 \angle ADC$.

For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$? (NB Vassiliev)

An equilateral triangle is cut as shown in figure 1 and the parts are used to form figure 2. What is the shape of figure 2?

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Colin has a peculiar $12$-sided dice: it is made up of two regular hexagonal pyramids. Colin wants to paint each face one of three colors so that no two adjacent faces on the same pyramid have the same color. How many ways can he do this? Two paintings are considered identical if there is a way to rotate or flip the dice to go from one to the other. Faces are adjacent if they share an edge. [center][img]https://cdn.artofproblemsolving.com/attachments/b/2/074e9a4bc404d45546661a5ae269248d20ed5a.png[/img][/center]

A1,A2,...,Am are subsets of X and we have |Ai|=mk (m,k natural numbers) prove that we can separate X into k sets such that every set has at least one member of each Ai.

A square-shaped field is divided into $n$ rectangular farms whose sides are parallel to the sides of the field. What is the greatest value of $n$, if the sum of the perimeters of the farms is equal to $100$ times of the perimeter of the field? $ \textbf{(A)}\ 10000 \qquad\textbf{(B)}\ 20000 \qquad\textbf{(C)}\ 50000 \qquad\textbf{(D)}\ 100000 \qquad\textbf{(E)}\ 200000 $

Given two positive integers $a \ne b$, let $f(a, b)$ be the smallest integer that divides exactly one of $a, b$, but not both. Determine the number of pairs of positive integers $(x, y)$, where $x \ne y$, $1\le x, y, \le 100$ and $\gcd(f(x, y), \gcd(x, y)) = 2$.

Let $P$ be a convex n-gon in the plane with vertices labeled $V_1,...,V_n$ in counterclockwise order. A point $Q$ not outside $P$ is called a balancing point of $P$ if, when the triangles the blue and green regions are the same. Suppose $P$ has exactly one balancing point/ Show that the balancing point must be a vertex of $P$

Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ . $$f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$$ . [hide=Original]$$f(x+f(y))+f(x+y)=2x+f(y)+y$$[/hide]

A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $ Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $

Let $\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\omega+c\omega^2,$ where $0\leq a \leq 1,0\leq b\leq 1,$ and $0\leq c\leq 1.$ What is the area of $S$? $\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi$

The roots of $x^3 + a x^2 + b x + c = 0$ are $\alpha, \beta$ and $\gamma.$ Find the cubic whose roots are $\alpha^3, \beta^3, \gamma^3.$