An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$. [i]Author: Alex Zhu[/i]

Find all polynomials with real coefficients $P$ such that \[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\] for every $x\in\mathbb{R}$.

A regular octahedron has side length $ 1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $ \frac {a\sqrt {b}}{c}$, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ b$ is not divisible by the square of any prime. What is $ a \plus{} b \plus{} c$? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14$

Find all positive integers $n, n>1$ for wich holds : If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$.

Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties: [list] [*] $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$ [*] $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$[/list]

The diagram below shows four regular hexagons each with side length $1$ meter attached to the sides of a square. This figure is drawn onto a thin sheet of metal and cut out. The hexagons are then bent upward along the sides of the square so that $A_1$ meets $A_2$, $B_1$ meets $B_2$, $C_1$ meets $C_2$, and $D_1$ meets $D_2$. If the resulting dish is filled with water, the water will rise to the height of the corner where the $A_1$ and $A_2$ meet. there are relatively prime positive integers $m$ and $n$ so that the number of cubic meters of water the dish will hold is $\sqrt{\frac{m}{n}}$. Find $m+n$. [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(7cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.3, xmax = 14.52, ymin = -8.3, ymax = 6.3; /* image dimensions */ draw((0,1)--(0,0)--(1,0)--(1,1)--cycle); draw((1,1)--(1,0)--(1.87,-0.5)--(2.73,0)--(2.73,1)--(1.87,1.5)--cycle); draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); draw((0,0)--(1,0)--(1.5,-0.87)--(1,-1.73)--(0,-1.73)--(-0.5,-0.87)--cycle); draw((0,1)--(0,0)--(-0.87,-0.5)--(-1.73,0)--(-1.73,1)--(-0.87,1.5)--cycle); /* draw figures */ draw((0,1)--(0,0)); draw((0,0)--(1,0)); draw((1,0)--(1,1)); draw((1,1)--(0,1)); draw((1,1)--(1,0)); draw((1,0)--(1.87,-0.5)); draw((1.87,-0.5)--(2.73,0)); draw((2.73,0)--(2.73,1)); draw((2.73,1)--(1.87,1.5)); draw((1.87,1.5)--(1,1)); draw((0,1)--(1,1)); draw((1,1)--(1.5,1.87)); draw((1.5,1.87)--(1,2.73)); draw((1,2.73)--(0,2.73)); draw((0,2.73)--(-0.5,1.87)); draw((-0.5,1.87)--(0,1)); /* dots and labels */ dot((1.87,-0.5),dotstyle); label("$C_1$", (1.72,-0.1), NE * labelscalefactor); dot((1.87,1.5),dotstyle); label("$B_2$", (1.76,1.04), NE * labelscalefactor); dot((1.5,1.87),dotstyle); label("$B_1$", (0.96,1.8), NE * labelscalefactor); dot((-0.5,1.87),dotstyle); label("$A_2$", (-0.26,1.78), NE * labelscalefactor); dot((-0.87,1.5),dotstyle); label("$A_1$", (-0.96,1.08), NE * labelscalefactor); dot((-0.87,-0.5),dotstyle); label("$D_2$", (-1.02,-0.18), NE * labelscalefactor); dot((-0.5,-0.87),dotstyle); label("$D_1$", (-0.22,-0.96), NE * labelscalefactor); dot((1.5,-0.87),dotstyle); label("$C_2$", (0.9,-0.94), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

For a positive integer $n$, an [i]$n$-branch[/i] $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to [i]appear[/i] in $B$ if it is an element of the last set $S_m$. Define an [i]$n$-plant[/i] to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it [i]perfect[/i] if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches. Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Yang Liu[/i]

Compute the number of positive integers $n < 2012$ that share exactly two positive factors with 2012. [i]Proposed by Aaron Lin[/i]

Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

$\tfrac11+\tfrac13+\tfrac15=\tfrac12+\tfrac14+\tfrac16+\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)$ equal to $12$? $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$ $\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$ $\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$ $\textbf{(J) }9\hspace{14.2em}\textbf{(K) }10\hspace{13.5em}\textbf{(L) }11$ $\textbf{(M) }12\hspace{13.3em}\textbf{(N) }13$

15 positive integers, all less than 1998(and no one equal to 1), are relatively prime (no pair has a common factor > 1). Show that at least one of them must be prime.

The positive integers $a,b,c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^2+b^2=c^2$. Prove that $b+c$ is the square of an integer.

Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on). [i]Victor Wang[/i]

A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.

If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$.

Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$. Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.

Find the smallest integer $k$ for which the conditions $(1)$ $a_1, a_2, a_3, \ldots$ is a nondecreasing sequence of positive integers $(2)$ $a_n=a_{n-1}+a_{n-2}$ for all $n>2$ $(3)$ $a_9=k$ are satisfied by more than one sequence.

Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \{1,2,...,50 \}$ contains two distinct elements $ a,b$ such that $ a\plus{}b$ divides $ ab$.

Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

There are $n$ distinct points on a circumference. Choose one of the points. Connect this point and the $m$th point from the chosen point counterclockwise with a segment. Connect this $m$th point and the $m$th point from this $m$th point counterclockwise with a segment. Repeat such steps until no new segment is constructed. From the intersections of the segments, let the number of the intersections - which are in the circle - be $I$. Answer the following questions ($m$ and $n$ are positive integers that are relatively prime and they satisfy $6 \leq 2m < n$). 1) When the $n$ points take different positions, express the maximum value of $I$ in terms of $m$ and $n$. 2) Prove that $I \geq n$. Prove that there is a case, which is $I=n$, when $m=3$ and $n$ is arbitrary even number that satisfies the condition.

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: [i](i)[/i] $i$ and $n - i$ always receive the same color, and [i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD=2$ and $DB=5,$ point $E$ is on $\overline{AC}$ with $CE=1$ and $EA=3,$ $AB=8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$