Let $f(n,k)$ with $n,k\in\mathbb Z_{\geq 2}$ be defined such that $\frac{(kn)!}{(n!)^{f(n,k)}}\in\mathbb Z$ and $\frac{(kn)!}{(n!)^{f(n,k)+1}}\not\in\mathbb Z$ Define $m(k)$ such that for all $k$, $n\geq m(k)\implies f(n,k)=k$. Show that $m(k)$ exists and furthermore that $m(k)\leq \mathcal{O}\left(k^2\right)$

How many $n\times n$ matrices $A$, with all entries from the set $\{0, 1, 2\}$, are there, such that for all $i=1,2,\cdots,n$ $A_{ii} > \displaystyle{\sum \limits_{j=1 j\neq i}^n} A_{ij}$ [Where $A_{ij}$ is the $(i,j)$th element of the matrix $A$]

Prove that there is no function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $f(f(n))=n+1.$ Here $\mathbb{N}$ is the positive integers $\{1,2,3,\dots\}.$

One can holds $ 12$ ounces of soda. What is the minimum number of cans to provide a gallon ($ 128$ ounces) of soda? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$

A series of figures is shown in the picture below, each one of them created by following a secret rule. If the leftmost figure is considered the first figure, how many squares will the 21st figure have? [img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img] Note: only the little squares are to be counted (i.e., the $2 \times 2$ squares, $3 \times 3$ squares, $\dots$ should not be counted) Extra (not part of the original problem): How many squares will the 21st figure have, if we consider all $1 \times 1$ squares, all $2 \times 2$ squares, all $3 \times 3$ squares, and so on?.

Let $ f:\mathbb{R} \to \mathbb{R} $ be an increasing function satisfying the following conditions: 1. $ f(x+1) = f(x) + 1 $ for each $ x \in \mathbb{R} $, 2. there exists an integer p such that $ f(f(f(O))) = p $. Prove that for every real number $ x $ $$ \lim_{n\to \infty} \frac{x_n}{n} = \frac{p}{3}.$$ where $ x_1 = x $ and $ x_n =f(x_{n-1}) $ for $ n = 2, 3, \ldots $.

Assume that the polynomial $\left(x+1\right)^n-1$ is divisible by some polynomial $P\left(x\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0$, whose degree $k$ is even and whose coefficients $c_{k-1}$, $c_{k-2}$, ..., $c_1$, $c_0$ all are odd integers. Show that $k+1\mid n$.

Find the largest positive integer $m$ which makes it possible to color several cells of a $70\times 70$ table red such that [list] [*] There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells; [*] There are two rows with exactly $m$ red cells each. [/list]

The hexagonal pattern constructed below has two smaller hexagons per side and has a total of $30$ edges. A similar figure is constructed with $20$ smaller hexagons per side. Compute the number of edges in this larger figure. [Insert Diagram] [i]Proposed by Ezra Erives[/i]

A jar contains $5$ different colors of gumdrops. $30\%$ are blue, $20\%$ are brown, $15\%$ red, $10\%$ yellow, and the other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown? $ \textbf{(A)}\ 35 \qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64 $

In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] pair A, B, C, D, E, F; A = (0,3); B=(0,0); C=(11,0); D=(11,3); E=foot(C, A, (9/4,0)); F=foot(A, C, (35/4,3)); draw(A--B--C--D--cycle); draw(A--E--C--F--cycle); filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray); dot(A^^B^^C^^D^^E^^F); label("$A$", A, W); label("$B$", B, W); label("$C$", C, (1,0)); label("$D$", D, (1,0)); label("$F$", F, N); label("$E$", E, S); [/asy]

Given quadrilateral $ABCD$ inscribed into a circle with diagonal $AC$ as diameter. Let $E$ be a point on segment $BC$ s.t. $\sphericalangle DAC=\sphericalangle EAB$. Point $M$ is midpoint of $CE$. Prove that $BM=DM$.

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

On a blackboard there are $2010$ natural nonzero numbers. We define a "move" by erasing $x$ and $y$ with $y\neq0$ and replacing them with $2x+1$ and $y-1$, or we can choose to replace them by $2x+1$ and $\frac{y-1}{4}$ if $y-1$ is divisible by 4. Knowing that in the beginning the numbers $2006$ and $2008$ have been erased, show that the original set of numbers cannot be attained again by any sequence of moves.

Let a tetrahedron $ABCD$ and its inner point $O$ be given. For any edge $e$ of $ABCD$ consider the segment $f(e)$ containing $O$ such that $f(e)\parallel e$ and the endpoints of $f(e)$ lie on the faces of the tetrahedron. Show that \[\sum_{e\text{ edge}}\,\frac{\,f(e)\,}{e}=3.\]

$x \geq 0$ and $y$ are real numbers for which $y^2 \geq x(x + 1)$. Prove that: $(y - 1)^2 \geq x(x-1)$.

Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$. [i]Proposed by Ankan Bhattacharya[/i]

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.