In triangle $ABC$, points $A'$, $B'$, and $C'$ are chosen with $A'$ on segment $AB$, $B'$ on segment $BC$, and $C'$ on segment $CA$ so that triangle $A'B'C'$ is directly similar to $ABC$. The incenters of $ABC$ and $A'B'C'$ are $I$ and $I'$ respectively. Lines $BC$, $A'C'$, and $II'$ are parallel. If $AB=100$ and $AC=120$, what is the length of $BC$?

Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars? [asy] draw((0,0)--(36,0)--(36,24)--(0,24)--cycle); draw((0,4)--(36,4)); draw((0,8)--(36,8)); draw((0,12)--(36,12)); draw((0,16)--(36,16)); draw((0,20)--(36,20)); fill((4,0)--(8,0)--(8,20)--(4,20)--cycle, black); fill((12,0)--(16,0)--(16,12)--(12,12)--cycle, black); fill((20,0)--(24,0)--(24,8)--(20,8)--cycle, black); fill((28,0)--(32,0)--(32,24)--(28,24)--cycle, black); label("\$120", (0,24), W); label("\$80", (0,16), W); label("\$40", (0,8), W); label("Jan", (6,0), S); label("Feb", (14,0), S); label("Mar", (22,0), S); label("Apr", (30,0), S); [/asy] $ \textbf{(A)}\ 60\qquad\textbf{(B)}\ 70\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 85 $

let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.

Given a triangle $ABC$. Point $M$ moves along the side $BA$ and point $N$ moves along the side $AC$ beyond point $C$ such that $BM=CN$. Find the geometric locus of the centers of the circles circumscribed around the triangle $AMN$.

On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals. [i]A. Tereshin[/i]

Let $A$ be an infinite set of real numbers containing at least one irrational number. Prove that for every natural number $n > 1$ there exists a subset $S$ of $A$ with n elements such that the sum of the elements of $S$ is an irrational number.

Let $ABC$ be an acute-angled triangle in which $AB <AC$. On the side $BC$ mark a point $D$ such that $AD = AB$, and on the side $AB$ mark a point $E$ such that the segment $DE$ passes through the orthocenter of triangle $ABC$. Prove that the center of the circumcircle of triangle $ADE$ lies on the segment $AC$.

Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.

The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$, while the other two multiply to $30$. What is the sum of the ages of Jonie's four cousins? $\textbf{(A) }21 \qquad \textbf{(B) }22 \qquad \textbf{(C) }23 \qquad \textbf{(D) }24 \qquad \textbf{(E) }25$

Let us define [i]cutting[/i] a convex polygon with $n$ sides by choosing a pair of consecutive sides $AB$ and $BC$ and substituting them by three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, the triangle $MBN$ is removed and a convex polygon with $n+1$ sides is obtained. Let $P_6$ be a regular hexagon with area $1$. $P_6$ is [i]cut[/i] and the polygon $P_7$ is obtained. Then $P_7$ is cut in one of seven ways and polygon $P_8$ is obtained, and so on. Prove that, regardless of how the cuts are made, the area of $P_n$ is always greater than $2/3$.

Find the largest possible value of the expression $x+y+z$, $x,y, z \in Z$, given that the equation $10x^3 +20y^3+2006xyz = 2007z^3$ holds.

Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$. A) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$, B) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$

Two cells in a $20 \times 20$ board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a $20 \times 20$ board such that every cell is adjacent to at most one marked cell?

A point $P$ lies inside $\usepackage{gensymb} \angle ABC(<90 \degree)$. Show that there exists a point $Q$ inside $\angle ABC$ satisfying the following condition: [center]For any two points $X$ and $Y$ on the rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ respectively satisfying $\angle XPY = \angle ABC$, it holds that $\usepackage{gensymb} \angle XQY = 180 \degree - 2 \angle ABC.$[/center]

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

Determine all solutions $(n, k)$ of the equation $n!+A
n = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.

Range of function $f(x)=\arctan x+\frac{1}{2}\arcsin x$ is $\text{(A)}(-\pi,\pi)\qquad\text{(B)}[-\frac{3}{4}\pi,\frac{3}{4}\pi]\qquad\text{(C)}(-\frac{3}{4}\pi,\frac{3}{4}\pi)\qquad\text{(D)}[-\frac{1}{2}\pi,\frac{1}{2}\pi]$

Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and \[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\] for all $x,y\in\mathbb{R}$. [i] Proposed by usjl[/i]

The orthocenter $H$ of a triangle $ABC$ is not on the sides of the triangle and the distance $AH$ equals the circumradius of the triangle. Find the measure of $\angle A$.

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

Let $I$ be an open interval of length $\frac{1}{n}$, where $n$ is a positive integer. Find the maximum possible number of rational numbers of the form $\frac{a}{b}$ where $1 \le b \le n$ that lie in $I$.

Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

Akshar is reading a $500$ page book, with odd numbered pages on the left, and even numbered pages on the right. Multiple times in the book, the sum of the digits of the two opened pages are $18$. Find the sum of the page numbers of the last time this occurs. Proposed by Minseok Eli Park (wolfpack)

A society created in the help to the police contains $100$ men exactly. Every evening $3$ men are on duty. Prove that you can not organise duties in such a way, that every couple will meet on duty once exactly.

Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.