How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.

A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.

The smaller angle between the hands of a clock at $ 12: 25$ p.m. is: $ \textbf{(A)}\ 132^\circ 30' \qquad \textbf{(B)}\ 137^\circ 30' \qquad \textbf{(C)}\ 150^\circ \qquad \textbf{(D)}\ 137^\circ 32' \qquad \textbf{(E)}\ 137^\circ$

Prove that among any $1001$ numbers taken from the numbers $1,2,\ldots,1997$ there exist two with the difference $4$.

Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal. Proposed by Mahdi Etesamifard

Find $A$ so that the ratio of $3\frac23$ to $22$ is the same as the ratio of $7\frac56$ to $A$

[u]Round 5[/u] [i]Each of the three problems in this round depends on the answer to two of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [/i] [b]p13.[/b] Let $B$ be the answer to problem $14$, and let $C$ be the answer to problem $15$. A quadratic function $f(x)$ has two real roots that sum to $2^{10} + 4$. After translating the graph of $f(x)$ left by $B$ units and down by $C$ units, the new quadratic function also has two real roots. Find the sum of the two real roots of the new quadratic function. [b]p14.[/b] Let $A$ be the answer to problem $13$, and let $C$ be the answer to problem $15$. In the interior of angle $\angle NOM = 45^o$, there is a point $P$ such that $\angle MOP = A^o$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY)^2$. [b]p15.[/b] Let $A$ be the answer to problem $13$, and let $B$ be the answer to problem $14$. Totoro hides a guava at point $X$ in a flat field and a mango at point $Y$ different from $X$ such that the length $XY$ is $B$. He wants to hide a papaya at point $Z$ such that $Y Z$ has length $A$ and the distance $ZX$ is a nonnegative integer. In how many different locations can he hide the papaya? [u]Round 6[/u] [b]p16.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB = 4$, $CD = 8$, $BC = 5$, and $AD = 6$. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$. [b]p17.[/b] Find the maximum possible value of the greatest common divisor of $\overline{MOO}$ and $\overline{MOOSE}$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.) [b]p18.[/b] Suppose that $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside them are both liars. Suppose that the greatest possible number of liars is $M$ and that the least possible number of liars is $N$. Determine the ordered pair $(M,N)$. [u]Round 7[/u] [b]p19.[/b] Define a [i]lucky [/i] number as a number that only contains $4$s and $7$s in its decimal representation. Find the sum of all three-digit lucky numbers. [b]p20.[/b] Let line segment $AB$ have length $25$ and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC = 15$, $AD = 24$, $BC = 20$, and $BD = 7$. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$. [b]p21.[/b] A $3\times 3$ grid is filled with positive integers and has the property that each integer divides both the integer directly above it and directly to the right of it. Given that the number in the top-right corner is $30$, how many distinct grids are possible? [u]Round 8[/u] [b]p22.[/b] Define a sequence of positive integers $s_1, s_2, ... , s_{10}$ to be [i]terrible [/i] if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$: $\bullet$ $s_i > s_j $ $\bullet$ $j - i + 1$ divides the quantity $s_i + s_{i+1} + ... + s_j$ Determine the minimum possible value of $s_1 + s_2 + ...+ s_{10}$ over all terrible sequences. [b]p23.[/b] The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 - 37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair. [b]p24.[/b] Consider a non-empty set of segments of length $1$ in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3$-[i]amazing [/i] if each endpoint of a segment is the endpoint of exactly three segments in the set. Find the smallest possible size of a $3$-amazing set of segments. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934024p26255963]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. In a similar manner, some number of the particles from the right side of the tube move to the left, at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$. [i]Proposed by Alvin Zou[/i]

A positive integer is called [i]nice[/i] if the sum of its digits in the number system with base $ 3$ is divisible by $ 3$. Calculate the sum of the first $ 2005$ nice positive integers.

A sequence of integers $a_1$, $a_2$, $a_3$, $\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?

We have a $n\times n$ table and we’ve written numbers $0,+1 \ or \ -1$ in each $1\times1$ square such that in every row or column, there is only one $+1$ and one $-1$. Prove that by swapping the rows with each other and the columns with each other finitely, we can swap $+1$s with $-1$s.

In all tetrahedron $ABCD$ holds [list=1] [*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$ [*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$ [/list] for all $t\in [0,1]$

Let $\Gamma$ be a circle of center $S$ and radius $r$ and let be $A$ a point outside the circle. Let $BC$ be a diameter of $\Gamma$ such that $B$ does not belong to the line $AS$ and consider the point $O$ where the perpendicular bisectors of triangle $ABC$ intersect, that is, the circumcenter of $ABC$. Determine all possible locations of point $O$ when $B$ varies in circle $\Gamma$.

Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Find the maximum possible value of \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \]

Determine all function $f$ from the set of real numbers to itself such that for every $x$ and $y$, \[f(x^2-y^2)=(x-y)(f(x)+f(y))\]

The greatest integer that will divide $13,511$, $13,903$, and $14,589$ and leave the same remainder is $\textbf{(A) }28\qquad\textbf{(B) }49\qquad\textbf{(C) }98\qquad$ $\textbf{(D) }\text{an odd multiple of }7\text{ greater than }49\qquad \textbf{(E) }\text{an even multiple of }7\text{ greater than }98$

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that $$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$ Proposed by [i]Petar Filipovski, Macedonia[/i]

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

Given are quadratic trinomials $f$ and $g$ with integral coefficients. For each positive integer $n$ there is an integer $k$ such that \[\frac{f(k)}{g(k)}=\frac{n + 1}{n}. \] Prove that $f$ and $g$ have a common root. [i] Proposed by A. Golovanov [/i]

Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?

Let $n$ be a given positive integer. We say that a positive integer $m$ is [i]$n$-good[/i] if and only if there are at most $2n$ distinct primes $p$ satisfying $p^2\mid m$. (a) Show that if two positive integers $a,b$ are coprime, then there exist positive integers $x,y$ so that $ax^n+by^n$ is $n$-good. (b) Show that for any $k$ positive integers $a_1,\ldots,a_k$ satisfying $\gcd(a_1,\ldots,a_k)=1$, there exist positive integers $x_1,\ldots,x_k$ so that $a_1x_1^n+a_2x_2^n+\cdots+a_kx_k^n$ is $n$-good. (Remark: $a_1,\ldots,a_k$ are not necessarily pairwise distinct) [i]Proposed by usjl.[/i]

Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system \begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*} is divisible by 10.

In a test, $201$ students are trying to solve $6$ problems.We know that for each of $5$ first problems, there are at least $140$ students, who can solve it. Moreover, there is exactly $60$ students, who can solve $6^{th}$ problem. Show that there exist $2$ students, such that two of them combined are able to solve all $6$ question. (For example, number $1$ do $1,2,3,4$ and number $2$ do $3,5,6$)

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]