Given a convex, $n$-sided polygon $P$, form a $2n$-sided polygon $\text{clip}(P)$ by cutting off each corner of $P$ at the edges' trisection points. In other words, $\text{clip}(P)$ is the polygon whose vertices are the $2n$ edge trisection points of $P$, connected in order around the boundary of $P$. Let $P_1$ be an isosceles trapezoid with side lengths $13,13,13,$ and $3$, and for each $i\geq 2$, let $P_i=\text{clip}(P_{i-1}).$ This iterative clipping process approaches a limiting shape $P_\infty=\lim_{i\to\infty}P_i$. If the difference of the areas of $P_{10}$ and $P_\infty$ is written as a fraction $\tfrac xy$ in lowest terms, calculate the number of positive integer factors of $x\cdot y$.

On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.

Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

Find all triples of real numbers $(a, b, c)$ satisfying $$a+b+c=14, \quad a^2+b^2+c^2=84,\quad a^3+b^3+c^3=584.$$

Let $\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. The greatest common divisor of $b$ and $81$ is $\text{(A) }1\qquad\text{(B) }3\qquad\text{(C) }9\qquad\text{(D) }27\qquad\text{(E) }81$

Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

Three integers $A, B, C$ are written on a whiteboard. Every move, Mr. Doba can either subtract $1$ from all numbers on the board, or choose two numbers on the board and subtract $1$ from both of them whilst leaving the third untouched. For how many ordered triples $(A, B, C)$ with $1 \le A < B < C\le 20$ is it possible for Mr. Doba to turn all three of the numbers on the board to $0$?

Given $25$ different positive numbers. Prove that you can choose two of them such, that none of the other numbers equals neither to the sum nor to the difference between the chosen numbers.

For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

We call $ A_1, A_2, \ldots, A_n$ an $ n$-division of $ A$ if (i) $ A_1 \cap A_2 \cap \cdots \cap A_n \equal{} A$, (ii) $ A_i \cap A_j \neq \emptyset$. Find the smallest positive integer $ m$ such that for any $ 14$-division $ A_1, A_2, \ldots, A_{14}$ of $ A \equal{} \{1, 2, \ldots, m\}$, there exists a set $ A_i$ ($ 1 \leq i \leq 14$) such that there are two elements $ a, b$ of $ A_i$ such that $ b < a \leq \frac {4}{3}b$.

A quadrilateral $ABCD$ is inscribed into a circle with center $O.$ Points $P$ and $Q$ are opposite to $C$ and $D$ respectively. Two tangents drawn to that circle at these points meet the line $AB$ in points $E$ and $F.$ ($A$ is between $E$ and $B$, $B$ is between $A$ and $F$). The line $EO$ meets $AC$ and $BC$ in points $X$ and $Y$ respectively, and the line $FO$ meets $AD$ and $BD$ in points $U$ and $V$ respectively. Prove that $XV=YU.$

Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively. A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$ Prove that: $ X,\,Y,\,Z$ are collinear.

$ABCD$ is a cyclic quadrilateral inscribed in circle $O$. Let $O_1$ be the $A$-excenter of $ABC$ and $O_2$ the $A$-excenter of $ABD$. Show that $A, B, O_1, O_2$ is concyclic, and $O$ passes through the center of $(ABO_1O_2)$. Recall that for concyclic $X, Y, Z, W$, the notation $(XYZW)$ denotes the circumcircle of $XYZW$.

Each cell of the 4x4 board has a grasshopper. When a grasshopper jumps, it moves to one of the adjacent cells (down, up, right, or left). The grasshopper cannot move diagonally or go off the board. At most how many cells can remain empty after each grasshopper jumps once?

Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be [i]Muirhead-able[/i] if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$, the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$. For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing? [i]Proposed by Evan Chen[/i]

A survey was taken in Ms. Susan's class to see what grades the class received: [center][img width=35]https://cdn.artofproblemsolving.com/attachments/5/c/e96cb42de6d5e1b100f37bbb71768d399842cb.png[/img][/center] What percent of the class received an "A"? $\textbf{(A) }3\%\qquad\textbf{(B) }5\%\qquad\textbf{(C) }10\%\qquad\textbf{(D) }15\%\qquad\textbf{(E) }27\%$

100 unit squares of an infinite squared plane form a $ 10\times 10$ square. Unit segments forming these squares are coloured in several colours. It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original $ 10\times 10$ square.) What maximum number of colours may appear in this colouring? [i]Author: S. Berlov[/i]

Find all triples $(a, b, c)$ of positive integers for which $a + (a, b) = b + (b, c) = c + (c, a)$. Here $(a, b)$ denotes the greatest common divisor of integers $a, b$. [i](Proposed by Mykhailo Shtandenko)[/i]

Let $K$, in square units, be the area of a trapezoid such that the shorter base, the altitude, and the longer base, in that order, are in arithmetic progression. Then: $\textbf{(A)}\ K \; \text{must be an integer} \qquad \textbf{(B)}\ K \; \text{must be a rational fraction} \\ \textbf{(C)}\ K \; \text{must be an irrational number} \qquad \textbf{(D)}\ K\; \text{must be an integer or a rational fraction} \qquad$ $\textbf{(E)}\ \text{taken alone neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is true}$

$17.$ Let $a,$ $b,$ and $c.$ be such that $a+b+c=0$ and $$P=\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}$$ is defined. What is the value of $P ?$

For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6) = 2$ and $b(23)=5$. If $S = \textstyle\sum_{p=1}^{2007}b(p)$, find the remainder when S is divided by 1000.

Consider the four lines $y = mx-k^2$ for different integer $k$. Let $(x_i,y_i)$, $i = 1,2,3,4$ be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat $x_1\leq x_2\leq x_3\leq x_4$. Show that $x_1 + x_4 =x_2+x_3$ and $y_1y_4 =y_2y_3$.

Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.