Sara wrote on the board an integer with less than thirty digits and ending in $2$. Celia erases the $2$ from the end and writes it at the beginning. The number that remains written is equal to twice the number that Sara had written. What number did Sara write?

Given $10$ positive integers with a sum equal to $1000$. The product of their factorials is a $10$-th power of an integer. Prove that all these numbers are equal.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

The letters $AAABBCC$ are arranged in random order. The probability no two adjacent letters will be the same is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $n$ be a positive integer, and let $a_0,\,a_1,\dots,\,a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that \[ \sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}. \] [i]Note:[/i] $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$.

Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$. [i]Author: Charles Leytem, Luxembourg[/i]

Let $a, b, c$ be positive real numbers. Prove the inequality $(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c$. [i]Proposed by Tajikistan[/i]

The Serpent Gorynych has $1976$ heads. The fabulous hero can cut down $33, 21, 17$ or $1$ head with one blow of the sword, but at the same time, the Serpent grows, respectively, $48, 0, 14$ or $349$ heads. If all the heads are cut off, then no new heads will grow. Will the hero be able to defeat the Serpent?

In a handball tournament with $n$ teams, each team played against other teams exactly once. In each game, the winner got $2$ points, the loser got $0$ point, and each team got $1$ point if there was a tie. After the tournament ended, each team had different score than the others, and the last team defeated the first three teams. What is the least possible value of $n$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of above} $

Given 10 light switches, each can be in two states: on and off. For each pair of switches there is a light bulb which is on if and only if when both switches are on (45 bulbs in total). The bulbs and the switches are unmarked so it is unclear which switches correspond to which bulb. In the beginning all switches are off. How many flips are needed to find out regarding all bulbs which switches are connected to it? On each step you can flip precisely one switch

Determine all positive integers expressible, for every integer $ n \geq 3 $, in the form \begin{align*} \frac{(a_1 + 1)(a_2 + 1) \ldots (a_n + 1) - 1}{a_1a_2 \ldots a_n}, \end{align*} where $ a_1, a_2, \ldots, a_n $ are pairwise distinct positive integers.

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n}.\] Let $L_n$ denote the least common multiple of the numbers $1, 2, 3,\cdots, n$. For how many integers $n$ with $1 \le n \le 22$ is $k_n<L_n$? $\textbf{(A)} ~0 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~10 $

On the plane we consider $70$ points $A_1,A_2,...,A_{70}$ with integer coodinates. Suppose each pooints has weight $1$ and the centers of gravity of the triangles $ A_1A_2A_3$, $A_2A_3A_4$, $..$., $A_{68}A_{69}A_{70}$, $A_{69}A_{70}A_{1}$, $A_{70}A_{1}A_{2}$ have integer coodinates. Prove that the centers of gravity of any triple $A_i,A_j,...,A_{k}$ has integer coodinates.

Chris and Michael play a game on a $5 \times 5$ board, initially containing some black and white counters as shown below: [img]https://cdn.artofproblemsolving.com/attachments/8/0/42e1a64b3524a0db722c007b8d6b8eddf2d9e5.png[/img] Chris begins by removing any black counter, and sliding a white counter from an adjacent square onto the empty square. From that point on, the players take turns. Michael slides a black counter onto an adjacent empty square, and Chris does the same with white counters (no more counters are removed). If a player has no legal move, then he loses. (a) Show that, even if Chris and Michael play cooperatively, the game will come to an end. (b) Which player has a winning strategy?

In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that \[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]

Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides. [i]A. Berzin’sh, O. Musin[/i]

Let the function $f:N^*\to N^*$ such that [b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$; [b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$ Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$. (High School Affiliated to Nanjing Normal University )

For each integer $ n\geq 2 $, let $ S_n $ be the sum of all products $ jk $, where $ j $ and $ k $ are integers and $ 1\leq j<k\leq n $. What is the sum of the 10 least values of $ n $ such that $ S_n $ is divisible by $ 3 $? $\textbf{(A) }196\qquad\textbf{(B) }197\qquad\textbf{(C) }198\qquad\textbf{(D) }199\qquad\textbf{(E) }200$

A square $2n\times 2n$ grid is given. Let us consider all possible paths along grid lines, going from the centre of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its centre at the grid centre is passed through only once. (a) Prove that the number of all such paths is equal to $4\prod_{i=2}^n(16i-9)$. (b) Find the number of pairs of such paths that divide the grid into two congruent figures. (c) How many quadruples of such paths are there that divide the grid into four congruent parts?

The polynomial $ W(x) = x^4 + ax^3 + bx + cx + d $ is given. Prove that if the equation $ W(x) = 0 $ has four real roots, then for there to exist $ m $ such that $ W(x+m) = x^4+px^2+q $, it is necessary and it is enough that the sum of certain two roots of the equation $ W(x) = 0 $ equals the sum of the remaining ones.

Consider an $n\times n$ matrix whose entry at the intersection of the $i$-th row and the $j-$th column equals $i+ j -1$. What is the largest possible value of the product of $n$ entries of the matrix, no two of which are in the same row or column?

Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that $$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$ Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$.

In trapezoid $ABCD$ the sides $BC$ and $AD$ are parallel, $AC = BC + AD$, and the angle between the diagonals is equal to $ 60^o$. Prove that $AB = CD$. (Stanislav Smirnov, St Petersburg)