Let ABC be a triangle with angles $\alpha, \beta, \gamma$ commensurable with $\pi$. Starting from a point $P$ interior to the triangle, a ball reflects on the sides of $ABC$, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices $A,B,C$, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment $0$ to infinity consists of segments parallel to a finite set of lines.

Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.

In the plane there are $n$ $(n\geq4)$ marked points. There are at least $n+1$ pairs of marked points such that the distance between each pair of points is $1$. Find the greatest integer $k$ such that there is a marked point that is the center of the circle with radius $1$ on which there are at least $k$ of the marked points.

The three row sums and the three column sums of the array \[\begin{bmatrix} 4 & 9 & 2 \\ 8 & 1 & 6 \\ 3 & 5 & 7 \end{bmatrix} \]are the same. What is the least number of entries that must be altered to make all six sums different from one another? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

Determine all sets of natural numbers $ A $ that have at least two elements, and satisfying the following proposition: $$ \forall x,y\in A\quad x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A. $$ [i]Marius Perianu[/i]

Prove that there is no square with its four vertices on four concentric circles whose radii form an arithmetic progression.

$ ABC$ is a triangle which is not isosceles. Let the circumcenter and orthocenter of $ ABC$ be $ O$, $ H$, respectively, and the altitudes of $ ABC$ be $ AD$, $ BC$, $ CF$. Let $ K\neq A$ be the intersection of $ AD$ and circumcircle of triangle $ ABC$, $ L$ be the intersection of $ OK$ and $ BC$, $ M$ be the midpoint of $ BC$, $ P$ be the intersection of $ AM$ and the line that passes $ L$ and perpendicular to $ BC$, $ Q$ be the intersection of $ AD$ and the line that passes $ P$ and parallel to $ MH$, $ R$ be the intersection of line $ EQ$ and $ AB$, $ S$ be the intersection of $ FD$ and $ BE$. If $ OL \equal{} KL$, then prove that two lines $ OH$ and $ RS$ are orthogonal.

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $\omega_0$ be a circle tangent to chord $AB$ and arc $ACB$. For each $i = 1, 2$, let $\omega_i$ be a circle tangent to $AB$ at $T_i$ , to $\omega_0$ at $S_i$ , and to arc $ACB$. Suppose $\omega_1 \ne \omega_2$. Prove that there is a circle passing through $S_1, S_2, T_1$, and $T_2$, and tangent to $\Gamma$ if and only if $\angle ACB = 90^o$. .

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.

The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.

Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?

On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.

If $K$, $L$, $M$ denote the midpoints of the sides $AB$, $BC$, $CA$ in triangle $\triangle ABC$, then for all $P$ in the plane of triangle $\triangle ABC$, we have $$\frac{AB}{PK}+\frac{BC}{PL}+\frac{CA}{PM} \geq \frac{AB\cdot BC \cdot CA}{4\cdot PK\cdot PL\cdot PM}$$

$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.

Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?

Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.

$ABC$ is an equilateral triangle. $D$ and$ D'$ are points on opposite sides of the plane $ABC$ such that the two tetrahedra $ABCD$ and $ABCD'$ are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices $A, B, C, D, D'$ is such that the angle between any two adjacent faces is the same, find $DD'/AB$ .

A trapezoid has bases with lengths equal to $5$ and $15$ and legs with lengths equal to $13$ and $13.$ Determine the area of the trapezoid.

In a pile you have 100 stones. A partition of the pile in $ k$ piles is [i]good[/i] if: 1) the small piles have different numbers of stones; 2) for any partition of one of the small piles in 2 smaller piles, among the $ k \plus{} 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone). Find the maximum and minimal values of $ k$ for which this is possible.

Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is: ${A/5 = B/10 = C/13}$.

If $57a + 88b + 125c \geq 1148$, where $a,b,c > 0$, what is the minimum value of \[ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? \]

Let $k \ge 0$ and $a, b, c$ be three positive real numbers such that $$\frac{a}{b}+\frac{b}{c}+ \frac{c}{a}= (k + 1)^2 + \frac{2}{k+ 1}.$$ Prove that $$a^2 + b^2 + c^2 \le (k^2 + 1)(ab + bc + ca).$$

The schoolboy wrote a homework essay on the topic “How I spent my summer.” Two of his comrades from a neighboring school decided not to bother themselves with work and rewrote his essay. But while rewriting they made several mistakes - each their own. Before submitting their work, both students gave their essays to four other friends to rewrite (each gave them to two acquaintances). These four schoolchildren do the same, and so on. With each rewrite, all previous mistakes are saved and, possibly, new ones are made. It is known that on some day each new essay contained at least $10$ errors. Prove that there was a day when at least $11$ new mistakes were made in total.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

[u]Round 5[/u] [b]5.1.[/b] In a triangle $\vartriangle ABC$ with $AB = 48$, let the angle bisectors of $\angle BAC$ and $\angle BCA$ meet at $I$. Given $\frac{[ABI]}{[BCI]}=\frac{24}{7}$ and $\frac{[ACI]}{[ABI]}=\frac{25}{24}$ , find the area of $\vartriangle ABC$. [b]5.2.[/b] At a dinner party, $9$ people are to be seated at a round table. If person $A$ cannot be seated next to person $B$ and person $C$ cannot be next to person $D$, how many ways can the $9$ people be seated? Rotations of the table are indistinguishable. [b]5.3.[/b] Let $f(x)$ be a monic cubic polynomial such that $f(1) = f(7) = f(10) = a$ and $f(2) = f(5) = f(11) = b$. Find $|a - b|$. [u]Round 6[/u] [b]6.1.[/b] If $N$ has $16$ positive integer divisors and the sum of all divisors of $N$ that are multiples of $3$ is $39$ times the sum of divisors of $N$ that are not multiples of $3$, what is the smallest value of $N$? [b]6.2.[/b] In the two parabolas $y = x^2/16$ and $x = y^2/16$, the single line tangent to both parabolas intersects the parabolas at $A$ and $B$. If the parabolas intersect each other at $C$ which is not the origin, find the area of $\vartriangle ABC$. [b]6.3.[/b] Five distinguishable noncollinear points are drawn. How many ways are there to draw segments connecting the points, such that there are exactly two disjoint groups of connected points? Note that a single point can be considered a connected group of points. [u]Round 7[/u] [b]7.1.[/b] Let $a, b$ be positive integers, and $1 = d_1 < d_2 < d_3 < ... < d_n = a$ be the divisors of $a$, and $1 = e_1 < e_2 <e_3 < ... < e_m = b$ be the divisors of b. Given $gcd(a, b) = d_2 = e_6$, find the smallest possible value of $a + b$. [b]7.2.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 2$ and $AC = 3$. Let X be the point on $BC$ such that $m \angle BAX =\frac13 m\angle BAC$. Given that $AX = 1$, the sum of all possible values of $CX^2$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$. [b]7.3.[/b] Bob has a playlist of $6$ different songs in some order, and he listens to his playlist repeatedly. Every time he finishes listening to the third song in the playlist, he randomly shuffles his playlist and listens to the playlist starting with the new first song. The expected number of times Bob shuffles his songs before he listens each one of his $6$ songs at least once can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find $a+b$. [u]Round 8[/u] [b]8.1.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}, \underline{H}, \underline{I}$, and $\underline{J}$ represent distinct digits ($0$ to $9$) in the equation $\underline{FBGA} - \underline{ABAC} = \underline{DCE}$ (where $\underline{ABAC}$ and $\underline{F BGA}$ are four-digit numbers, and $\underline{DCE }$ is a three-digit number). If $\underline{A} < \underline{B} < \underline{C} < \underline{D}$ and $\underline{ABCDEF GHIJ}$ is minimized, find $\underline{ABCD} + \underline{EF G} + \underline{HI} + \underline{J}$. [b]8.2.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}$,,, and $\underline{F}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC} \cdot \underline{C} = \underline{DEA}, \underline{ABC} \cdot \underline{D} = \underline{BAF E}$, and $ \underline{DEA} + \underline{BAF E}0 = \underline{BF ACA}$ (where $\underline{ABC}$ and $\underline{DEA}$ are three-digit numbers, $\underline{BAF E}$ is a four-digit number, and $\underline{BF ACA}$ is a five-digit number). Find $\underline{ABC} + \underline{DE} + \underline{F}$. [b]8.3.[/b] $\underline{A}, \underline{B}, \underline{C}, \underline{D}, \underline{E}, \underline{F}, \underline{G}$, and $\underline{H}$ represent distinct digits ($0$ to $9$) in the equations $\underline{ABC } \cdot \underline{D} = \underline{AF GE}$, $\underline{ABC } \cdot \underline{C} = \underline{GHC}$, $\underline{GHC} + \underline{HF F} = \underline{AEHC}$, and $\underline{AF GE}0 + \underline{AEHC} = \underline{AEABC}$ (where $\underline{ABC}$, $\underline{GHC}$ and $\underline{HF F}$ are three-digit numbers, $\underline{AF GE}$ is a four-digit number, and $\underline{AEABC}$ is a five-digit number). Find $\underline{ABCD} + \underline{EF GH}$. [u]Round 9[/u] Estimate the arithmetic mean of all answers to this question. Only integer answers between $0$ to $100, 000$ will count for credit and count toward the average. Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05|I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3129699p28347299]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].