In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$.

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

Bunbury the bunny is hopping on the positive integers. First, he is told a positive integer $n$. Then Bunbury chooses positive integers $a,d$ and hops on all of the spaces $a,a+d,a+2d,\dots,a+2013d$. However, Bunbury must make these choices so that the number of every space that he hops on is less than $n$ and relatively prime to $n$. A positive integer $n$ is called [i]bunny-unfriendly[/i] if, when given that $n$, Bunbury is unable to find positive integers $a,d$ that allow him to perform the hops he wants. Find the maximum bunny-unfriendly integer, or prove that no such maximum exists.

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

Let $f$ be the function of the set of positive integers into itself, defined by $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the number of positive odd integers m such that $f(m) = n$ is equal to the number of positive integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$. [color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]

The number $a = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $$\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$$ is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q?$ $\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$

Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)

The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$. There are relatively prime positive integers $m$ and $n$ so that $\sin^2\theta  = \tfrac{m}{n}$. Find $m + n$. [asy] import graph; size(3.2cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle); draw((-1,3)--(-1,2)); draw((-1,2)--(-0.13,1.5)); draw((-0.13,1.5)--(0.73,2)); draw((0.73,2)--(0.73,3)); draw((0.73,3)--(-0.13,3.5)); draw((-0.13,3.5)--(-1,3)); draw((-1,3.5)--(0.73,3.5)); draw((0.73,3.5)--(0.73,1.5)); draw((-1,1.5)--(0.73,1.5)); draw((-1,3.5)--(-1,1.5)); label("$ A $",(-1.4,3.9),SE*labelscalefactor); label("$ B $",(-1.4,3.28),SE*labelscalefactor); label("$ C $",(-1.4,2.29),SE*labelscalefactor); label("$ D $",(-1.4,1.45),SE*labelscalefactor); label("$ E $",(-0.3,1.4),SE*labelscalefactor); label("$ F $",(0.8,1.45),SE*labelscalefactor); label("$ G $",(0.8,2.24),SE*labelscalefactor); label("$ H $",(0.8,3.26),SE*labelscalefactor); label("$ I $",(0.8,3.9),SE*labelscalefactor); label("$ J $",(-0.25,3.9),SE*labelscalefactor); [/asy]

Prove that the set of integers of the form $2^{k}-3$ ($k=2,3,\cdots$) contains an infinite subset in which every two members are relatively prime.

Let $a$ be any integer. Prove that there are infinitely many primes $p$ such that \[ p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a \] for some integers $n$ and $m$.

Triangle $ABC$ is a right triangle with $AC=7,$ $BC=24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD=BD=15.$ Given that the area of triangle $CDM$ may be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$

Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}

Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The sum $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Show that for any integer $n \ge 2$ the sum of the fractions $\frac{1}{ab}$, where $a$ and $b$ are relatively prime positive integers such that $a < b \le n$ and $a+b > n$, equals $\frac{1}{2}$. (Integers $a$ and $b$ are called relatively prime if the greatest common divisor of $a$ and $b$ is $1$.)

Compute the smallest positive integer $n$ that is a multiple of $29$ with the property that for every positive integer that is relatively prime to $n$, $k^{n}\equiv 1\pmod{n}.$

A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$. [i]Proposed by Ahaan Rungta[/i]

In the diagram below $ \angle CAB, \angle CBD$, and $\angle CDE$ are all right angles with side lengths $AC = 3$, $BC = 5$, $BD = 12$, and $DE = 84$. The distance from point $E$ to the line $AB$ can be expressed as the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$. [asy] size(300); defaultpen(linewidth(0.8)); draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8)); label("$A$",origin,SW); label("$B$",(0,4),dir(160)); label("$C$",(3,0),S); label("$D$",(6,8),dir(80)); label("$E$",(30,-4),E);[/asy]

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4)); [/asy]

Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n}\right)^b$ , where $m, n$ and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: $$\log_2\left({x \over yz}\right) = {1 \over 2}$$ $$\log_2\left({y \over xz}\right) = {1 \over 3}$$ $$\log_2\left({z \over xy}\right) = {1 \over 4}$$ Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$

Let $r,s,t$ positive integers which are relatively prime and $a,b \in G$, $G$ a commutative multiplicative group with unit element $e$, and $a^r=b^s=(ab)^t=e$. (a) Prove that $a=b=e$. (b) Does the same hold for a non-commutative group $G$?