If $a_1,a_2,\cdots,a_{2013}\in[-2,2]$ and $a_1+a_2+\cdots+a_{2013}=0$ , find the maximum of $a^3_1+a^3_2+\cdots+a^3_{2013}$.

Can one place positive integers at all vertices of a cube in such a way that for every pair of numbers connected by an edge, one will be divisible by the other , and there are no other pairs of numbers with this property? (A Shapovalov)

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $ 100$ cans, how many rows does it contain? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 11$

Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.

Find all positive integers $n$ for which there exists a polynomial $P(x) \in \mathbb{Z}[x]$ such that for every positive integer $m\geq 1$, the numbers $P^m(1), \ldots, P^m(n)$ leave exactly $\lceil n/2^m\rceil$ distinct remainders when divided by $n$. (Here, $P^m$ means $P$ applied $m$ times.) [i]Proposed by Carl Schildkraut, USA[/i]

An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used? $\textbf{(A)}\ 190 \qquad\textbf{(B)}\ 191 \qquad\textbf{(C)}\ 192 \qquad\textbf{(D)}\ 195 \qquad\textbf{(E)}\ 196$

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

let $a_1,a_2,...a_n$ a sequence of real numbers such that $a_1+....+a_n=0$. define $b_i=a_1+a_2+....a_i$ for all $1 \leq i \leq n$ .suppose $b_i(a_{j+1}-a_{i+1}) \geq 0$ for all $1 \leq i \leq j \leq n-1$. Show that $$\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|$$

Let $n$ and $k$ be positive integers. A function $f : \{1, 2, 3, 4, \dots , kn - 1, kn\} \to \{1, \cdots , 5\}$ is [i]good[/i] if $f(j + k) - f(j)$ is multiple of $k$ for every $j = 1, 2. \cdots , kn - k$. [b](a)[/b] Prove that, if $k = 2$, then the number of good functions is a perfect square for every positive integer $n$. [b](b)[/b] Prove that, if $k = 3$, then the number of good functions is a perfect cube for every positive integer $n$.

In quadrilateral $ABCD$ sides $BC$ and $AD$ are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles $A$ and $B$ intersect in point $P$, those of angles $B$ and $C$ intersect in point $Q$, those of angles $C$ and $D$ intersect in point $R$, and those of angles $D$ and $A$ intersect in point S. Suppose that $PS$ is parallel to $QR$. Prove that $|AB| =|CD|$. [asy] unitsize(1.2 cm); pair A, B, C, D, P, Q, R, S; A = (0,0); D = (3,0); B = (0.8,1.5); C = (3.2,1.5); S = extension(A, incenter(A,B,D), D, incenter(A,C,D)); Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A); P = extension(A, S, B, Q); R = extension(D, S, C, Q); draw(A--D--C--B--cycle); draw(B--Q--C); draw(A--S--D); dot("$A$", A, SW); dot("$B$", B, NW); dot("$C$", C, NE); dot("$D$", D, SE); dot("$P$", P, dir(90)); dot("$Q$", Q, dir(270)); dot("$R$", R, dir(90)); dot("$S$", S, dir(90)); [/asy] Attention: the figure is not drawn to scale.

In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.

Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$. Trần Quang Hùng

A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born? $ \textbf{(A)}\ \text{Friday} \qquad\textbf{(B)}\ \text{Saturday} \qquad\textbf{(C)}\ \text{Sunday} \qquad\textbf{(D)}\ \text{Monday} \qquad\textbf{(E)}\ \text{Tuesday} $

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

We say that a set $S$ of natural numbers is [i]synchronous [/i] provided that the digits of $a^2$ are the same (in occurence and numbers, if differently ordered) for all numbers $a$ in $S$. For example, $\{13, 14, 31\}$ is synchronous, since we find $\{13^2, 14^2, 31^2\} = \{169, 196, 961\}$. But $\{119, 121\}$ is not synchronous, for even though $119^2 = 14161$ and $121^2 = 14641$ have the same digits, they occur in different numbers. Show that there exists a synchronous set containing $2021$ different natural numbers.

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$. by Iman Maghsoudi

One rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005$ th digit that you write?

Let $ a$, $ b$, $ c$ be real numbers such that for every two of the equations \[ x^2\plus{}ax\plus{}b\equal{}0, \quad x^2\plus{}bx\plus{}c\equal{}0, \quad x^2\plus{}cx\plus{}a\equal{}0\] there is exactly one real number satisfying both of them. Determine all possible values of $ a^2\plus{}b^2\plus{}c^2$.

If the product $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9$, what is the sum of $a$ and $b$? $\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 35 \qquad \textbf{(E)}\ 37$

Let $ABC$ be an acute triangle and $H$ is orthocenter. Let $D$ be the intersection of $BH$ and $AC$ and $E$ be the intersection of $CH$ and $AB$. The circumcircle of $ADE$ cuts the circumcircle of $ABC$ at $F \neq A$. Prove that the angle bisectors of $\angle BFC$ and $\angle BHC$ concur at a point on $BC.$

A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\tfrac{p}{2}$ such that $\tfrac{ab-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.

Let $P_0$, $P_1$, $\dots$, $P_8 = P_0$ be successive points on a circle and $Q$ be a point inside the polygon $P_0 P_1 \dotsb P_7$ such that $\angle P_{i - 1} QP_i = 45^\circ$ for $i = 1$, $\dots$, 8. Prove that the sum \[\sum_{i = 1}^8 P_{i - 1} P_i^2\] is minimal if and only if $Q$ is the centre of the circle.

Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$

Let $ a$, $ b$ and $ m$ be positive integers. Prove that there exists a positive integer $ n$ such that $ (a^n \minus{} 1)b$ is divisible by $ m$ if and only if $ \gcd (ab, m) \equal{} \gcd (b, m)$.

A tree with $n\geq 2$ vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers $x_1,x_2,\dots,x_n$ associated with them. Each edge is associated with the product of the two numbers corresponding to the vertices it connects. Let $S$ be a sum of number across all edges. Prove that \[\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.\] (Author: V. Dolnikov)