For positive integer $k$, define $x_k=3k+\sqrt{k^2-1}-2(\sqrt{k^2-k}+\sqrt{k^2+k})$. Then $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_{1681}}=\sqrt{m}-n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

If $\sqrt{\log_bn}=\log_b\sqrt n$ and $b\log_bn=\log_bbn,$ then the value of $n$ is equal to $\frac jk,$ where $j$ and $k$ are relatively prime. What is $j+k$?

Pete has an instrument which can locate the midpoint of a line segment, and also the point which divides the line segment into two segments whose lengths are in a ratio of $n : (n + 1)$, where $n$ is any positive integer. Pete claims that with this instrument, he can locate the point which divides a line segment into two segments whose lengths are at any given rational ratio. Is Pete right?

Let $p$ be an odd prime. Prove that \[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]

Kolya found several pairwise relatively prime integers, each of which is less than the square of any other. Prove that the sum of reciprocals of these numbers is less than $2$.

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 $m+n$

Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000.

Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$. [i]Proposed by Lewis Chen[/i]

Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$, satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \quad \forall i, j$ from $\{1, 2,\ldots , 2004\}$, where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$. Find the least positive integer $k$ for which in every $k$-subset of $S$, having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.

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 $x^2+m.$

Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as $\frac{m}{n}$, where $m$ and $n$ Are relatively prime positive integers. Find $m+n$.

Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.

Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $ [asy] size(170); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair C=origin, A=(0,3), B=(4,0), D=(7.2,12.6), E=(7.2,0); draw(A--C--B--A--D--B--E--D); label("$A$",A,W); label("$B$",B,S); label("$C$",C,SW); label("$D$",D,NE); label("$E$",E,SE); [/asy] $ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 128\qquad\textbf{(C)}\ 153\qquad\textbf{(D)}\ 243\qquad\textbf{(E)}\ 256 $

Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n}.\] Let $L_n$ denote the least common multiple of the numbers $1, 2, 3,\cdots, n$. For how many integers $n$ with $1 \le n \le 22$ is $k_n<L_n$? $\textbf{(A)} ~0 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~10 $

The decimal representation of $m/n$, where $m$ and $n$ are relatively prime positive integers and $m < n$, contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of $n$ for which this is possible.