For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.

Find all natural numbers $n < 1978$ with the following property: If $m$ is a natural number, $1 < m < n$, and $(m, n) = 1$ (i.e., $m$ and $n$ are relatively prime), then $m$ is a prime number.

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime positive integers such that $ \frac{m}{n}$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $ m\plus{}n$.

Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy \[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B$. The probability that team $A$ finishes with more points than team $B$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Given natural $n$. Consider all the fractions $1/(pq)$, where $p$ and $q$ are relatively prime, $0<p<q\le n , p+q>n$. Prove that the sum of all such a fractions equals to $1/2$.

$10000$ nonzero digits are written in a $100$-by-$100$ table, one digit per cell. From left to right, each row forms a $100$-digit integer. From top to bottom, each column forms a $100$-digit integer. So the rows and columns form $200$ integers (each with $100$ digits), not necessarily distinct. Prove that if at least $199$ of these $200$ numbers are divisible by $2013$, then all of them are divisible by $2013$.

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

Let $R=(8,6)$. The lines whose equations are $8y=15x$ and $10y=3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

Suppose we wish to pick a random integer between $1$ and $N$ inclusive by flipping a fair coin. One way we can do this is through generating a random binary decimal between $0$ and $1$, then multiplying the result by $N$ and taking the ceiling. However, this would take an infinite amount of time. We therefore stopping the flipping process after we have enough flips to determine the ceiling of the number. For instance, if $N=3$, we could conclude that the number is $2$ after flipping $.011_2$, but $.010_2$ is inconclusive. Suppose $N=2014$. The expected number of flips for such a process is $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $100m+n$. [i]Proposed by Lewis Chen[/i]

For every pair of reals $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \begin{align*} x_n & = (1 - a) x_{n - 1} + a y_{n - 1}, \\ y_n & = (1 - b) x_{n - 1} + b y_{n - 1}. \end{align*} The [i]supermean[/i] of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$, the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\textstyle\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$. [i]Proposed by Lewis Chen[/i]

We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality \[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\] holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i]. [b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.

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 $m+n$.

A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$ Two runners start at point A at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point A. At that time each runner will have run a distance of $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n. $

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that \[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]

If $n$ is a positive integer, let $\phi(n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Compute the value of the infinite sum \[ \sum_{n=1}^\infty \frac{\phi(n) 2^n}{9^n - 2^n} \, . \]

For any positive integer $n$, the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ is written in the lowest form $\frac{p_n}{q_n}$; that is, $p_n$ and $q_n$ are relatively prime positive integers. Find all $n$ such that $p_n$ is 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]

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

How many positive integers less that $200$ are relatively prime to either $15$ or $24$?