Prove that for a nowhere dense, compact set $K\subset \mathbb{R}^2$ the following are equivalent: (i) $K=\bigcup_{i=1}^{\infty}K_n$ where $K_n$ is a compact set with connected complement for all $n$. (ii) $K$ does not have a nonempty closed subset $S\subseteq K$ such that any neighborhood of any point in $S$ contains a connected component of $\mathbb{R}^2 \setminus S$.

Let $ABC$ be an equilateral triangle of side length $15.$ Let $A_b$ and $B_a$ be points on side $AB,$ $A_c$ and $C_a$ be points on $AC,$ and $B_c$ and $C_b$ be points on $BC$ such that $\triangle{AA_bA_c}, \triangle{BB_cB_a},$ and $\triangle{CC_aC_b}$ are equilateral triangles with side lengths $3,4,$ and $5,$ respectively. Compute the radius of the circle tangent to segments $\overline{A_bA_c}, \overline{B_aB_c},$ and $\overline{C_aC_b}.$

Determine the number of ordered pairs of positive integers $(m,n)$ with $1\leq m\leq 100$ and $1\leq n\leq 100$ such that \[ \gcd(m+1,n+1) = 10\gcd(m,n). \]

Given is a grid 8x8. Every square is colored in black or white, so that in every 3x3, the number of white squares is even. What is the minimum number of black squares

An acute triangle $\bigtriangleup ABC$ is given which $BC>CA>AB$. $I$ is the interior and the incircle of $\bigtriangleup ABC$ meets $BC, CA, AB$ at $D,E,F$. $AD$ and $BE$ meet at $P$. Let $l_{1}$ be a tangent from D to the circumcircle of $\bigtriangleup DIP$, and define $l_{2}$ and $l_{3}$ on $E$ and $F$, respectively. Prove $l_{1},l_{2},l_{3}$ meet at one point.

Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $\big(a_1, a_2, a_3, a_4\big)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial $$q(x) = c_3x^3+c_2x^2+c_1x+c_0$$such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$ $(\textbf{A})\: {-}6\qquad(\textbf{B}) \: {-}1\qquad(\textbf{C}) \: 4\qquad(\textbf{D}) \: 6\qquad(\textbf{E}) \: 11$

Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.

Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.

Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum?

A merganser mates every 7th day, a scaup mates every 11th day, and a gadwall mates every 13th day. A merganser, scaup, and gadwall all mate on Day 0. On Days N, N+1, and N+2 the merganser, scaup, and gadwall mate in some order with no two birds mating on the same day. Determine the smallest possible value of N. [i]2022 CCA Math Bonanza Lightning Round 3.4[/i]

Let us define the sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,...$. with the following conditions $a_1 = 3, b_1 = 1$ and $a_{n +1} =\frac{a_n^2+b_n^2}{2}$ and $b_{n + 1}= a_n \cdot b_n$ for each $n = 1, 2,...$. Find all different prime factors οf the number $a_{2000} + b_{2000}$.

Consider a charged capacitor made with two square plates of side length $L$, uniformly charged, and separated by a very small distance $d$. The EMF across the capacitor is $\xi$. One of the plates is now rotated by a very small angle $\theta$ to the original axis of the capacitor. Find an expression for the difference in charge between the two plates of the capacitor, in terms of (if necessary) $d$, $\theta$, $\xi$, and $L$. Also, approximate your expression by transforming it to algebraic form: i.e. without any non-algebraic functions. For example, logarithms and trigonometric functions are considered non-algebraic. Assume $ d << L $ and $ \theta \approx 0 $. $\emph{Hint}$: You may assume that $ \frac {\theta L}{d} $ is also very small. [i]Problem proposed by Trung Phan[/i] [hide="Clarification"] There are two possible ways to rotate the capacitor. Both were equally scored but this is what was meant: [asy]size(6cm); real h = 7; real w = 2; draw((-w,0)--(-w,h)); draw((0,0)--(0,h), dashed); draw((0,0)--h*dir(64)); draw(arc((0,0),2,64,90)); label("$\theta$", 2*dir(77), dir(77)); [/asy] [/hide]

If $ AD$ is the altitude, $ BE$ the angle bisector, and $ CF$ the median of a triangle $ ABC$, prove that $ AD,BE,$ and $ CF$ are concurrent if and only if: $ a^2(a\minus{}c)\equal{}(b^2\minus{}c^2)(a\plus{}c),$ where $ a,b,c$ are the lengths of the sides $ BC,CA,AB$, respectively.

Suppose function $f:[0,\infty)\to[0,\infty)$ satisfies (1)$\forall x,y \geq 0,$ we have $f(x)f(y)\leq y^2f(\frac{x}{2})+x^2f(\frac{y}{2})$; (2)$\forall 0 \leq x \leq 1, f(x) \leq 2016$. Prove that $f(x)\leq x^2$ for all $x\geq 0$.

Given are a circle and an ellipse lying inside it with focus $C$. Find the locus of the circumcenters of triangles $ABC$, where $AB$ is a chord of the circle touching the ellipse.

Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that $MN = BM + ND$. [i](5 points)[/i]

Determine the number of ordered quintuples $(a,b,c,d,e)$ of integers with $0\leq a<$ $b<$ $c<$ $d<$ $e\leq 30$ for which there exist polynomials $Q(x)$ and $R(x)$ with integer coefficients such that \[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\] [i]Proposed by Michael Ren[/i]

Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor +\lfloor 2.4 \rfloor +\lfloor 3.1 \rfloor +\cdots+\lfloor 99 \rfloor$. [i]Proposed by Jack Cornish[/i]

For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2+\{\alpha\}=21$$ can be expressed in the form $$\frac{\sqrt{a}-\sqrt{b}}{c}-d$$ where $a, b, c,$ and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d.$

How many ordered pairs of integers $(x, y)$ satisfy \[8(x^3+x^2y+xy^2+y^3) = 15(x^2+y^2+xy+1)?\]

Let $M$ be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains $\emptyset \subset A \subset B \subset C \subset D \subset M$ of six different set, beginning with the empty set and ending with the $M$, are there?

Cat and Claire are having a discussion about their favorite positive two-digit numbers. [b]Cat:[/b] My number has a $1$ in its tens digit. Is it possible that your number is a multiple of my number? [b]Claire:[/b] No, however, my number is not prime. Additionally, if I told you the two digits of my number, you still wouldn't know my number. [b]Cat:[/b] Aha, my number and your number aren't relatively prime! [b]Claire:[/b] Then our numbers must share the same ones digit! What is the product of Cat and Claire's numbers?

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }8$