Suppose $a$ and $b$ are positive integers for which $8a^ab^b = 27a^bb^a$. Find $a^2 + b^2$.

Points $C$, $A$, $D$, $M$, $E$, $B$, $F$ lie on a line in that order such that $CA = AD = EB = BF = 1$ and $M$ is the midpoint of $DB$. Let $X$ be a point such that a quarter circle arc exists with center $D$ and endpoints $C$, $X$. Suppose that line $XM$ is tangent to the unit circle centered at $B$. Compute $AB$. [i]2020 CCA Math Bonanza Individual Round #11[/i]

Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur. Proposed by: G.Galyapin

Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that $$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$ Determine the minimum value of $x_1+x_2+x_3+...+x_n$. [i]Proposed by Loh Kwong Weng[/i]

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve? [asy] size(170); defaultpen(fontsize(6pt)); dotfactor=4; label("$\circ$",(0,1)); label("$\circ$",(0.865,0.5)); label("$\circ$",(-0.865,0.5)); label("$\circ$",(0.865,-0.5)); label("$\circ$",(-0.865,-0.5)); label("$\circ$",(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy] $ \textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt3 \qquad\textbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3 $

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

Let $\gamma$ a circle and $P$ a point who lies outside the circle. Two arbitrary lines $l$ and $l'$ which pass through $P$ intersect the circle at the points $X$, $Y$ , respectively $X'$, $Y'$ , such that $X$ lies between $P$ and $Y$ and $X'$ lies between $P$ and $Y'$. Prove that the line determined by the circumcentres of the triangles $PXY'$ and $PX'Y$ passes through a fixed point.

Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ($a<b$) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$. Find $a,b$.

Some math team members decide to study at Cary Library after school. They walk $6$ blocks north, then $8$ blocks west to get there. If they walk $n$ blocks east from the library, they can buy boba from CoCo's. If CoCo's is the same distance from school as it is from the library, find $n$.

What is the sum of the first $2011$ integers closest in value to $0,$ including $0$ itself?

Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns. Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.

Segments $AC$ and $BD$ meet at $P$, and $|PA| = |PD|$, $|PB| = |PC|$. $O$ is the circumcenter of the triangle $PAB$. Show that $OP$ and $CD$ are perpendicular.

Let $m$ be the number of ordered solutions $(a,b,c,d,e)$ satisfying: $1)$ $a,b,c,d,e\in \mathbb{Z}^{+}$; $2)$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}=1$; Prove that $m$ is odd.

Find all pairs of functions $ f, g: \mathbb R \to \mathbb R$ such that [list] (i) if $ x < y$, then $ f(x) < f(y)$; (ii) $ f(xy) \equal{} g(y)f(x) \plus{} f(y)$ for all $ x, y \in \mathbb R$. [/list]

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess $\text{L}$-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible location, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?

In the expression $(x + 100) (x + 99) ... (x-99) (x-100)$, the brackets were expanded and similar terms were given. The expression $x^{201} + ...+ ax^2 + bx + c$ turned out. Find the numbers $a$ and $c$.