Let $H$ be a subgroup with $h$ elements in a group $G.$ Suppose that $G$ has an element $a$ such that for all $x$ in $H,$ $(xa)^3=1,$ the identity. In $G$, let $P$ be the subset of all products $x_1ax_2a\dots x_na,$ with $n$ a positive integer and the $x_i$ in $H.$ (a) Show that $P$ is a finite set. (b) Show that, in fact, $P$ has no more that $3h^2$ elements.

Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$

Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.

Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by $$f_n(x)=\int^x_0f_{n-1}(t)dt$$ for all integers $n\ge1$. a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$. b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.

Construction Mayhem University has been on a mission to expand and improve its campus! The university has recently adopted a new construction schedule where a new project begins every two days. Each project will take exactly one more day than the previous one to complete (so the first project takes 3, the second takes 4, and so on.) Suppose the new schedule starts on Day 1. On which day will there first be at least $10$ projects in place at the same time?

A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$ at points $A_1,B_1,C_1$, respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$, and $M$ is the midpoint of the segment $PA_1$. Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.

Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called [i]discrete coloring[/i] if there exists a circle which encloses all red points and excludes all white points. Determine the number of [i]discrete colorings[/i] for each set $S$.

Let $a,b$ be positive integers such that $a+b^3$ is divisible by $a^2+3ab+3b^2-1$. Prove that $a^2+3ab+3b^2-1$ is divisible by the cube of an integer greater than 1.

As shown in figure , it is known that $M$ is the midpoint of side $BC$ of $\vartriangle ABC$. $\odot O$ passes through points $A, C$ and is tangent to $AM$. The extension of the segment $BA$ intersects $\odot O$ at point $D$. The lines $CD$ and $MA$ intersect at the point $P$. Prove that $PO \perp BC$. [img]https://cdn.artofproblemsolving.com/attachments/8/a/da3570ec7eb0833c7a396e22ffac2bd8902186.png[/img]

Let $\lambda$ be a real number. Suppose that if \(ABCD\) is any convex cyclic quadrilateral such that \(AC=4\), \(BD=5\), and \(\overline{AB}\perp\overline{CD}\), then the area of $ABCD$ is at least $\lambda$. Then the greatest possible value of $\lambda$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$. [i]Proposed by Eric Shen[/i]

Given positive integer $ n \ge 5 $ and a convex polygon $P$, namely $ A_1A_2...A_n $. No diagonals of $P$ are concurrent. Proof that it is possible to choose a point inside every quadrilateral $ A_iA_jA_kA_l (1\le i<j<k<l\le n) $ not on diagonals of $P$, such that the $ \tbinom{n}{4} $ points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.

How many ordered integer pairs $ \left(x,y\right)$ are there satisfying following equation: \[ y \equal{} \sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1997\plus{}\sqrt{x\plus{}1997\plus{}\ldots \plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}\sqrt{x} } } } } } } }.\] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 1998 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ \text{Infinitely many}$

Let $a_1,a_2,\cdots,a_{22}\in [1,2],$ find the maximum value of $$\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}$$where $a_{23}=a_1.$

What is the minimum possible value of \[\left|x\right|-\left|x-1\right|+\left|x+2\right|-\left|x-3\right|+\left|x+4\right|-\cdots-\left|x-2019\right|\] over all real $x$? [i]2019 CCA Math Bonanza Individual Round #10[/i]

A $10$ digit positive integer is called a $\emph{cute}$ number if its digits are from the set $\{1,2,3\}$ and every two consecutive digits differ by $1$. [list=a] [*]Prove that exactly $5$ digits of a cute number are equal to $2$. [*]Find the total number of cute numbers. [*]Prove that the sum of all cute numbers is divisible by $1408$.[/list]

$AD$ is the angle bisector of the right triangle $ABC$ with $\angle ABC = 60^o$ and $\angle BCA = 90^o$. $E$ is chosen on $\overline{AB}$ so that the line parallel to $\overline{DE}$ through $C$ bisects $\overline{AE}$. Find $\angle EDB$ in degrees.

Find all sets of positive integers $\{x_1, x_2, \dots, x_{20}\}$ such that $$x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1})$$ for $i=1, 2, \dots, 20$ where $x_0=x_{20}, x_{21}=x_1, x_{22}=x_2$.

A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?

Prove that if $ x,y$ are nonnegative real numbers with $ x\plus{}y\equal{}2$, then: $ x^2 y^2 (x^2\plus{}y^2) \le 2$.

In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.

Let $n>3$ be a positive integer satisfying $2^n+1=3p$, where $p$ is a prime. Let $s_0=\frac{2^{n-2}+1}{3}$ and $s_i=s_{i-1}^2-2$ for $i>0$. Show that $p \mid 2s_{n-2}-3$.

We have a grid of $k^2-k+1$ rows and $k^2-k+1$ columns, where $k=p+1$ and $p$ is prime. For each prime $p$, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly $k$ 0's, on each column there are exactly $k$ 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.

The numbers $1, 2, \ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number $N$ remains. Find the remainder when the maximum possible value of $N$ is divided by 1000. [i]Victor Wang.[/i]

An acute triangle $\triangle ABC$ and its incenter $I$, circumcenter $O$ is given. The line that is perpendicular to $AI$ and passes $I$ intersects with $AB$, $AC$ in $D$,$E$. The line that is parallel to $BI$ and passes $D$ and the line that is parallel to $CI$ and passes $E$ intersects in $F$. Denote the circumcircle of $DEF$ as $\omega$, and its center as $K$. $\omega$ and $FI$ intersect in $P$($\neq F$). Prove that $O,K,P$ is collinear.

[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].