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 sequence of positive integers is constructed as follows: the first term is $ 1$, the following two terms are $ 2$, $ 4$, the following three terms are $ 5$, $ 7$, $ 9$, the following four terms are $ 10$, $ 12$, $ 14$, $ 16$, etc. Find the $ n$-th term of the sequence.

Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$ a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$ b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$

In the plane there are $2014$ plotted points, such that no $3$ are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.

If the arithmetic mean of two numbers is $6$ and thier geometric mean is $10$, then an equation with the given two numbers as roots is: $\text{(A)} \ x^2+12x+100=0 ~~ \text{(B)} \ x^2+6x+100=0 ~~ \text{(C)} \ x^2-12x-10=0$ $\text{(D)} \ x^2-12x+100=0 \qquad \text{(E)} \ x^2-6x+100=0$

We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$.

Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .

Up to similarity, there is a unique nondegenerate convex equilateral 13-gon whose internal angles have measures that are multiples of 20 degrees. Find it. Give your answer by listing the degree measures of its 13 [i]external[/i] angles in clockwise or counterclockwise order. Start your list with the biggest external angle. You don't need to write the degree symbol $^\circ$.

Find the number of permutations of $\{1,2,...,n\}$ like $\{a_1,...,a_n\}$ st for each $1 \leq i \leq n$: $$a_i | 2i$$

Suppose there are several juice boxes, one of which is poisoned. You have $n$ guinea pigs to test the boxes. The testing happens in the following way: [list] [*] At each round, you can have the guinea pigs taste any number of juice boxes. [*] Conversely, a juice box can be tasted by any number of guinea pigs. [*] After the round ends, any guinea pigs who tasted the poisoned juice die. [/list] Suppose you have to find the poisoned juice box in at most $k$ rounds. What is the maximum number of juice boxes such that it is possible?

Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\omega_{i - 1}$. If \[ S = \sum_{i = 1}^\infty r_i, \] then $S$ can be expressed as $a\sqrt{b} + c$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$. [i]Proposed by Aaron Lin[/i]

The figure below shows a semicircle inscribed in a right triangle. [asy] draw((0, 0) -- (15, 0) -- (0, 8) -- cycle); real r = 120 / 23; real theta = -aTan(8/15); draw(arc((r, r), r, theta + 180, theta + 360)); [/asy] The triangle has legs of length 8 and 15. The semicircle is tangent to the two legs, and its diameter is on the hypotenuse. What is the radius of the semicircle?

Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is$$f(2) + f(3) + f(4) + f(5)+ f(6)?$$ $(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$

The "Sea battle" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.

Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.

Let $R$ be a ring where $\forall a\in R: a^2=0$. Prove that $abc+abc=0$ for all $a,b,c\in R$.

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

Determine all functions $f: \mathbb R \to \mathbb R$, such that $$ f (f (x + y) - f (x - y)) = xy$$ for all real $x$ and $y$.

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.

For prime number $q$ the polynomial $P(x)$ with integer coefficients is said to be factorable if there exist non-constant polynomials $f_q,g_q$ with integer coefficients such that all of the coefficients of the polynomial $Q(x)=P(x)-f_q(x)g_q(x)$ are dividable by $q$ ; and we write: $$P(x)\equiv f_q(x)g_q(x)\pmod{q}$$ For example the polynomials $2x^3+2,x^2+1,x^3+1$ can be factored modulo $2,3,p$ in the following way: $$\left\{\begin{array}{lll} X^2+1\equiv (x+1)(-x+1)\pmod{2}\\ 2x^3+2\equiv (2x-1)^3\pmod{3}\\ X^3+1\equiv (x+1)(x^2-x+1) \end{array}\right.$$ Also the polynomial $x^2-2$ is not factorable modulo $p=8k\pm 3$. a) Find all prime numbers $p$ such that the polynomial $P(x)$ is factorable modulo $p$: $$P(x)=x^4-2x^3+3x^2-2x-5$$ b) Does there exist irreducible polynomial $P(x)$ in $\mathbb{Z}[x]$ with integer coefficients such that for each prime number $p$ , it is factorable modulo $p$?

[b]E[/b]milia wishes to create a basic solution with 7% hydroxide (OH) ions. She has three solutions of different bases available: 10% rubidium hydroxide (Rb(OH)), 8% cesium hydroxide (Cs(OH)), and 5% francium hydroxide (Fr(OH)). (The Rb(OH) solution has both 10% Rb ions and 10% OH ions, and similar for the other solutions.) Since francium is highly radioactive, its concentration in the final solution should not exceed 2%. What is the highest possible concentration of rubidium in her solution?

Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$.

A square of size $2\times2$ with one of its cells occupied by a tower is called a castle. What maximal number of castles one can place on a board of size $7\times7$ so that the castles have no common cells and all the towers stand on the diagonals of the board?

Let be the sequence $ \left( J_n \right)_{n\ge 1} , $ where $ J_n=\int_{(1+n)^2}^{1+(1+n)^2} \sqrt{\frac{x-1-n-n^2}{x-1}} dx. $ [b]a)[/b] Study its monotony. [b]b)[/b] Calculate $ \lim_{n\to\infty } J_n\sqrt{n} . $ [i]Ion Bursuc[/i]

Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly. a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$. b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.