Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

A telephone number has the form $ ABC \minus{} DEF \minus{} GHIJ$, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, $ A > B > C$, $ D > E > F$, and $ G > H > I > J$. Furthermore, $ D$, $ E$, and $ F$ are consecutive even digits; $ G$, $ H$, $ I$, and $ J$ are consecutive odd digits; and $ A \plus{} B \plus{} C \equal{} 9$. Find $ A$. $ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8$

Let $a, b, c$ be integers satisfying $ab + bc + ca = 1.$ Prove that $(1+ a^2 )(1+ b^2 )(1+ c^2 )$ is a perfect square.

Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let \[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1. \] Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that: \[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2 \]

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

A phone company starts a new type of service. A new customer can choose $k$ phone numbers in this network which are call-free, whether that number is calling or is being called. A group of $n$ students want to use the service. (a) If $n\geq 2k+2$, show that there exist 2 students who will be charged when speaking. (b) It $n=2k+1$, show that there is a way to arrange the free calls so that everybody can speak free to anybody else in the group. [i]Valentin Vornicu[/i]

Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.

Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$. A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.

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: $(y - 1)^2 \geq x(x-1)$.