Given an equilateral triangle, all points on its sides are colored in one of two given colors. Prove that the is a right-angled triangle such that its three vertices are in the same color and on the sides of the equilateral triangle. [i]Alhaji Akbar, Jakarta[/i]

What is the value of $$3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}?$$ $\textbf{(A) } \frac{31}{10} \qquad \textbf{(B) } \frac{49}{15} \qquad \textbf{(C) } \frac{33}{10} \qquad \textbf{(D) } \frac{109}{33} \qquad \textbf{(E) } \frac{15}{4}$

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]

Let $ABC$ be a triangle with incenter $I$ such that $AB=20$ and $AC=19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA=IP=IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Ankit Bisain[/i]

How many subsets of $\{1, 2, \cdots, 10\}$ are there that don't contain $2$ consecutive integers?

Let $a_1, a_2, \ldots, a_{100}$ be positive integers, satisfying $$\frac{a_1^2+a_2^2+\ldots+a_{100}^2} {a_1+a_2+\ldots+a_{100}}=100.$$ What is the maximal value of $a_1$?

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $P$. The circumscribed circles of triangles $APD$ and $BPC$ intersect the line $AB$ at points $E, F$ correspondingly. $Q$ and $R$ are the projections of $P$ onto the lines $FC, DE$ correspondingly. Show that $AB \parallel QR$. [i](Proposed by Mykhailo Shtandenko)[/i]

Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.

Let $f$ be the function defined by $f(x)=ax^2-\sqrt2$ for some positive $a$. If $f(f(\sqrt2 ))=-\sqrt 2$, then $a=$ $\text{(A)} \ \frac{2-\sqrt2}{2} \qquad \text{(B)} \ \frac12 \qquad \text{(C)} \ 2-\sqrt2 \qquad \text{(D)} \ \frac{\sqrt{2}}{2} \qquad \text{(E)} \ \frac{2+\sqrt2}{2}$

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A triangle $ABC$ is inscribed in the circle $(O,R)$. A circle $(O',R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. $P$ is a point on the circle $(O)$. Rays $PA, PB, PC$ meet $(O')$ at $A_1,B_1,C_1$. Let $A_2B_2C_2$ be the triangle formed by the intersections of the line symmetric to $B_1C_1$ about $BC$, the line symmetric to $C_1A_1$ about $CA$ and the line symmetric to $A_1B_1$ about $AB$. Prove that the circumcircle of $A_2B_2C_2$ is tangent to $(O)$. Nguyễn Văn Linh

Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.

Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.

Find all solutions of the equation $$p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101$$ with $ p$, $q$ and $r$ positive prime numbers and $k$ a positive integer.

We call a set $A\subset \mathbb{R}$ [i]free of arithmetic progressions[/i] if for all distinct $a,b,c\in A$ we have $a+b\neq 2c.$ Prove that the set $\{0,1,2,\ldots 3^8-1\}$ has a subset $A$ which is free of arithmetic progressions and has at least $256$ elements.

Set $P$ as a polynomial function by $p_n(x)=\sum_{k=0}^{n-1} x^k$. a) Prove that for $m,n\in N$, when dividing $p_n(x)$ by $p_m(x)$, the remainder is $$p_i(x),\forall i=0,1,...,m-1.$$ b) Find all the positive integers $i,j,k$ that make $$p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x).$$ [i](square1zoa)[/i]