An $8\times8$ chessboard is completely tiled by $2\times1$ dominoes. Prove that we can place positive integers in all cells of the table in such a way that the sums of numbers in every domino are equal and the numbers placed in two adjacent cells are coprime if and only if they belong to the same domino. (Two cells are called adjacent if they have a common side.) Well this can belong to number theory as well...

Find all positive integers $n$ for which there exist distinct positive integers $a_1,a_2,\ldots,a_n$ such that \[ \frac 1{a_1} + \frac 2{a_2} + \cdots + \frac n { a_n} = \frac { a_1 + a_2 + \cdots + a_n } n. \]

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Given $2015$ points in the plane, show that if every four of them form a convex quadrilateral then the points are the vertices of a convex $2015-$sided polygon.

On sides $AC$ and $BC$ of acute-angled triangle $ABC$ rectangles with equal areas $ACPQ$ and $BKLC$ were built exterior. Prove that midpoint of $PL$, point $C$ and center of circumcircle are collinear.

Let $ABCD$ be a trapezoid with $AB\parallel CD$. Points $E,F$ lie on segments $AB,CD$ respectively. Segments $CE,BF$ meet at $H$, and segments $ED,AF$ meet at $G$. Show that $S_{EHFG}\le \dfrac{1}{4}S_{ABCD}$. Determine, with proof, if the conclusion still holds when $ABCD$ is just any convex quadrilateral.

A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?

Let $\mathcal{C}$ be the family of circumferences in $\mathbb{R}^2$ that satisfy the following properties: (i) if $C_n$ is the circumference with center $(n,1/2)$ and radius $1/2$, then $C_n\in \mathcal{C}$, for all $n\in \mathbb{Z}$. (ii) if $C$ and $C'$, both in $\mathcal{C}$, are externally tangent, then the circunference externally tangent to $C$ and $C'$ and tanget to $x$-axis also belongs to $\mathcal{C}$. (iii) $\mathcal{C}$ is the least family which these properties. Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of $\mathcal{C}$ and the $x$-axis.

On a chalkboard, Benji draws a square with side length $6.$ He then splits each side into $3$ equal segments using $2$ points for a total of $12$ segments and $8$ points. After trying some shapes, Benji finds that by using a circle, he can connect all $8$ points together. What is the area of this circle? [center] [img] https://cdn.artofproblemsolving.com/attachments/d/f/ead2d0c85f474612d49d3a023a43215e11066b.png [/img] [/center]

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

$ABCD$ is a quadrilateral such that $AB$ is not parallel to $CD$, and $BC$ is not parallel to $AD$. Variable points $P, Q, R, S$ are taken on $AB, BC, CD, DA$ respectively so that $PQRS$ is a parallelogram. Find the locus of its center.

Determine $n \in \Bbb{N}$ such that $n^2 + 2$ divides $2 + 2001n.$

In a system of numeration with base $B$ , there are $n$ one-digit numbers less than $B$ whose cubes have $B-1$ in the units-digits place. Determine the relation between $n$ and $B$

Let $ABC$ be an isosceles triangle with $AB = AC$. Point $D$ lies on side $AC$ such that $BD$ is the angle bisector of $\angle ABC$. Point $E$ lies on side $BC$ between $B$ and $C$ such that $BE = CD$. Prove that $DE$ is parallel to $AB$.

Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

$a^3 + b^3 + 3abc \ge\ c^3$ prove that where a,b and c are sides of triangle.

The distinct reals $x_1, x_2, ... , x_n$ ,($n > 3$) satisfy $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i ^2 = 1$. Show that four of the numbers $a, b, c, d$ must satisfy: \[ a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd \].

Let $n$ be a fixed positive integer. We have a $n\times n$ chessboard. We call a pair of cells [b]good[/b] if they share a common vertex (May be common edge or common vertex). How many [b]good[/b] pairs are there on this chessboard?

The annual incomes of $1000$ families range from $8200$ dollars to $98000$ dollars. In error, the largest income was entered on the computer as $980000$ dollars. The difference between the mean of the incorrect data and the mean of the actual data is $ \text{(A)}\ \text{882 dollars}\qquad\text{(B)}\ \text{980 dollars}\qquad\text{(C)}\ \text{1078 dollars}\qquad\text{(D)}\ \text{482,000 dollars}\qquad\text{(E)}\ \text{882,000 dollars} $

The first $ 2007$ positive integers are each written in base $ 3$. How many of these base-$ 3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.) $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 101 \qquad \textbf{(C)}\ 102 \qquad \textbf{(D)}\ 103 \qquad \textbf{(E)}\ 104$

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

The players Alfred and Bertrand put together a polynomial $x^n + a_{n-1}x^{n- 1} +... + a_0$ with the given degree $n \ge 2$. To do this, they alternately choose the value in $n$ moves one coefficient each, whereby all coefficients must be integers and $a_0 \ne 0$ must apply. Alfred's starts first . Alfred wins if the polynomial has an integer zero at the end. (a) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the right to the left, i.e. for $j = 0, 1,. . . , n - 1$, be determined? (b) For which $n$ can Alfred force victory if the coefficients $a_j$ are from the left to the right, i.e. for $j = n -1, n - 2,. . . , 0$, be determined? (Theresia Eisenkölbl, Clemens Heuberger)

Find all functions $f: \mathbb{N} \rightarrow \mathbb {N}$ such that for all positive integers $m, n$, $f(m+n)\mid f(m)+f(n)$ and $f(m)f(n) \mid f(mn)$.

A $2$-kg rock is suspended by a massless string from one end of a uniform $1$-meter measuring stick. What is the mass of the measuring stick if it is balanced by a support force at the $0.20$-meter mark? [asy] size(250); draw((0,0)--(7.5,0)--(7.5,0.2)--(0,0.2)--cycle); draw((1.5,0)--(1.5,0.2)); draw((3,0)--(3,0.2)); draw((4.5,0)--(4.5,0.2)); draw((6,0)--(6,0.2)); filldraw((1.5,0)--(1.2,-1.25)--(1.8,-1.25)--cycle, gray(.8)); draw((0,0)--(0,-0.4)); filldraw((0,-0.4)--(-0.05,-0.4)--(-0.1,-0.375)--(-0.2,-0.375)--(-0.3,-0.4)--(-0.3,-0.45)--(-0.4,-0.6)--(-0.35,-0.7)--(-0.15,-0.75)--(-0.1,-0.825)--(0.1,-0.84)--(0.15,-0.8)--(0.15,-0.75)--(0.25,-0.7)--(0.25,-0.55)--(0.2,-0.4)--(0.1,-0.35)--cycle, gray(.4)); [/asy] $ \textbf {(A) } 0.20 \, \text{kg} \qquad \textbf {(B) } 1.00 \, \text{kg} \qquad \textbf {(C) } 1.33 \, \text{kg} \qquad \textbf {(D) } 2.00 \, \text{kg} \qquad \textbf {(E) } 3.00 \, \text{kg} $

let $0<a,b<\pi/2$. Show that $\frac{5}{cos^2(a)}+\frac{5}{sin^2(a)sin^2(b)cos^2(b)} \geq 27cos(a)+36sin(a) $