$10000$ 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 $2013$.

$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD$. The crease is $EF$, where $E$ is on $AB$ and $F$is on $CD$. The dimensions $AE=8$, $BE=17$, and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]

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 $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

Let $n$ be a given positive integer. Prove that there exist infinitely many integer triples $(x,y,z)$ such that \[nx^2+y^3=z^4,\ \gcd (x,y)=\gcd (y,z)=\gcd (z,x)=1.\]

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.) [i]Proposed by Alexander S. Golovanov, Russia[/i]

When $12{}^1{}^8$ is divided by $18{}^1{}^2$, the result is $(\tfrac{m}{n})^3$, where $m$ and $n$ are relatively prime integers. Find $m-n$.

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]

Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$ [asy] size(250); path p=rotate(45)*polygon(4); int i; for(i=0; i<5; i=i+1) { draw(shift(2-(1/2)^(i-1),0)*scale((1/2)^i)*p); } label("$S_1$", (0,-0.75)); label("$S_2$", (1,-0.75)); label("$S_3$", (3/2,-0.75)); label("$\cdots$", (7/4, -3/4)); label("$\cdots$", (2.25, 0));[/asy]

Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$. [i]A. Kovaldji, V. Senderov[/i]

In $\triangle RED, RD =1, \angle DRE = 75^\circ$ and $\angle RED = 45^\circ$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC} \perp \overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA = AR$. Then $AE = \tfrac{a-\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.

Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?

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

Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]

Let $ \mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $ |8 \minus{} x| \plus{} y \le 10$ and $ 3y \minus{} x \ge 15$. When $ \mathcal{R}$ is revolved around the line whose equation is $ 3y \minus{} x \equal{} 15$, the volume of the resulting solid is $ \frac {m\pi}{n\sqrt {p}}$, where $ m$, $ n$, and $ p$ are positive integers, $ m$ and $ n$ are relatively prime, and $ p$ is not divisible by the square of any prime. Find $ m \plus{} n \plus{} p$.

Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \le f(n) \le n+2012$ for all $n$. Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.

For positive integers $a>b>1$, define \[x_n = \frac {a^n-1}{b^n-1}\] Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers. [i]V. Senderov[/i]

Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.

If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.