In triangle $ABC$, the incircle touches the side $AC$ at point $B_1$ and one excircle is touching the same side at point $B_2$. It is known that the segments $BB_1$ and $BB_2$ are equal. Is it true that $\vartriangle ABC$ is isosceles?

Let $(X, <)$ be an arbitrary ordered set. Show that the elements of $X$ can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour. (translated by L. Erdős)

If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that \[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]

Let $a_1,a_2,\ldots ,a_9$ be any non-negative numbers such that $a_1=a_9=0$ and at least one of the numbers is non-zero. Prove that for some $i$, $2\le i\le 8$, the inequality $a_{i-1}+a_{i+1}<2a_i$ holds. Will the statement remain true if we change the number $2$ in the last inequality to $1.9$?

If the real numbers $x, y, z$ are such that $x^2 + 4y^2 + 16z^2 = 48$ and $xy + 4yz + 2zx = 24$, what is the value of $x^2 + y^2 + z^2$?

In the system of base $n^2 + 1$ find a number $N$ with $n$ different digits such that: [b](i)[/b] $N$ is a multiple of $n$. Let $N = nN'.$ [b](ii)[/b] The number $N$ and $N'$ have the same number $n$ of different digits in base $n^2 + 1$, none of them being zero. [b] (iii)[/b] If $s(C)$ denotes the number in base $n^2 + 1$ obtained by applying the permutation $s$ to the $n$ digits of the number $C$, then for each permutation $s, s(N) = ns(N').$

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get, if his initial score is $0$? (Maxim Didin)

Which of the following fractions has the largest value? $\text{(A)}\ \frac{3}{7} \qquad \text{(B)}\ \frac{4}{9} \qquad \text{(C)}\ \frac{17}{35} \qquad \text{(D)}\ \frac{100}{201} \qquad \text{(E)}\ \frac{151}{301}$

Consider a table consisting of $2\times 7$ squares. Each little square is surrounded by walls (each internal wall belongs to two squares). We would like to remove some internal walls to make it possible to get from any square to any other one without crossing walls. How many ways can we do this while removing the minimal possible number of internal walls? [i]The figure shows a possible configuration, the remaining walls are marked in red, the removed ones are marked in light pink. Two configurations are considered the same if the same walls are removed.[/i] [img]https://cdn.artofproblemsolving.com/attachments/d/c/1a3d9ab0d0971929e6d484a970d4b1f36f0031.png[/img]

Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer. Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.

Is there a scalene triangle $ABC$ similar to triangle $IHO$, where $I$, $H$, and $O$ are the incenter, orthocenter, and circumcenter, respectively, of triangle $ABC$? [i]Proposed by Li4 and usjl.[/i]

Jared has 3 distinguishable Rolexes. Each day, he selects a subset of his Rolexes and wears them on his arm (the order he wears them does not matter). However, he does not want to wear the same Rolex 2 days in a row. How many ways can he wear his Rolexes during a 6 day period?

Lance, Sally, Joy and Fred are chosen for the team. In how many ways can the three starters be chosen? $\textbf{(A)} 2 \qquad\textbf{(B)} 4 \qquad\textbf{(C)} 6 \qquad\textbf{(D)} 8 \qquad\textbf{(E)} 10$

Triangle $ ABC$ is a right triangle with $ \angle ACB$ as its right angle, $ m\angle ABC \equal{} 60^\circ$, and $ AB \equal{} 10$. Let $ P$ be randomly chosen inside $ \triangle ABC$, and extend $ \overline{BP}$ to meet $ \overline{AC}$ at $ D$. What is the probability that $ BD > 5\sqrt2$? [asy]import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P); draw(A--C--B--cycle); draw(B--D); dot(P); label("A",A,S); label("D",D,S); label("C",C,S); label("P",P,NE); label("B",B,N);[/asy] $ \textbf{(A)}\ \frac {2 \minus{} \sqrt2}{2} \qquad \textbf{(B)}\ \frac {1}{3} \qquad \textbf{(C)}\ \frac {3 \minus{} \sqrt3}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {5 \minus{} \sqrt5}{5}$

$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.

Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever $A$ and $B$ are players in the tournament for which $A$ won (strictly) more matches than $B$ over the course of the tournament, it is also the case that $A$ won the match against $B$ during the tournament. In how many ways could the tournament have gone?

The equation $ x^2 \plus{} ax \plus{} b \equal{} 0$ has two distinct real roots. Prove that the equation $ x^4 \plus{} ax^3 \plus{} (b \minus{} 2)x^2 \minus{} ax \plus{} 1 \equal{} 0$ has four distinct real roots.

A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters. (a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?

Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\ (1) $f$ has periodic $2017$\\ (2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\ Proposed by William Chao

We are given $\Delta ABC$, for which the excircle to side $BC$ is tangent to the continuations of $AB$ and $AC$ in points $E$ and $F$ respectively. Let $D$ be the reflection of $A$ in line $EF$. If it is known that $\angle BAC=2\angle BDC$, then determine $\angle BAC$.

Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$1.43$. Sharona bought some of the same pencils and paid $\$1.87$. How many more pencils did Sharona buy than Jamar? $\textbf{(A)}\hspace{.05in}2 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}4 \qquad \textbf{(D)}\hspace{.05in}5 \qquad \textbf{(E)}\hspace{.05in}6 $

Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$

A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side. [img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]