Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations: [list=] [*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin; [*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin; [*]$H,$ a reflection across the $x$-axis; and [*]$V,$ a reflection across the $y$-axis. [/list] Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.) $\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$

A point $(X, Y, Z)$ is chosen uniformly at random from the ball of radius $4$ centered at the origin (i.e., the set $\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 4^2\}$). Compute the probability that the inequalities $X^2 \leq 1$ and $X^2 + Y^2 + Z^2 \geq 1$ simultaneously hold.

Determine all points on the straight line which joins $(4, 11)$ to $(16, 1)$ and whose coordinates are positive integers.

A regular hexagon of area $H$ is inscribed in a convex polygon of area $P$. Show that $P \leq \frac{3}{2}H$. When does the equality occur?

Let $a_0,a_1,\ldots$ be a sequence of positive integers such that $a_0=1$, and for all positive integers $n$, $a_n$ is the smallest composite number relatively prime to all of $a_0,a_1,\ldots,a_{n-1}$. Compute $a_{10}$. [i]2020 CCA Math Bonanza Lightning Round #4.2[/i]

Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of $2017$ colors. Magic trick: magicman go away from scene. Then viewers should put on the table $n>1$ cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color. What is minimal $n$ such that magicman and his helper can has strategy to make magic trick successfully?

Let $I$ be the incenter of triangle $ABC$, and let $A'$, $B'$, and $C'$ be the circumcenters of triangles $IBC$, $ICA$, and $IAB$, respectively. Prove that the circumcircles of triangles $ABC$ and $A'B'C'$ are concentric.

Let O be the circumcenter of triangle ABC. Let H be the orthocenter of triangle ABC. the perpendicular bisector of AB meet AC,BC at D,E. the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L. CH meet FG at T,and ABCT is concyclic. Prove that LHBC is concyclic. graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png

Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality \[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]

$5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is [i]properly-connected[/i]. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.

Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of triangle $ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$.

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ Let $A'$ be the midpoint of $BC,$ $M$ be the midpoint of the height $AD$ and $P$ be the intersection of $BM$ and $AA'.$ Prove that $\tan\angle PCB=\sin C\cdot\cos C.$ [i]Daniel Văcărețu[/i]

Given a convex quadrilateral $ABCD$.Let $\ell_A,\ell_B,\ell_C,\ell_D$ be exterior angle bisectors of quadrilateral $ABCD$. Let $\ell_A \cap \ell_B=K,\ell_B \cap \ell_C=L,\ell_C \cap \ell_D=M,\ell_D \cap \ell_A=N$.Prove that if circumcircles of triangles $ABK$ and $CDM$ be externally tangent to each other then circumcircles of the triangles $BCL$ and $DAN$ are externally tangent to each other.(L.Emelyanov)

Let $x$ be the smallest positive integer for which $2x$ is the square of an integer, $3x$ is the third power of an integer, and $5x$ is the fifth power of an integer. Find the prime factorization of $x$. (St. Wagner, Stellenbosch University)

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

Let $f:[0,1] \to \mathbb{R}$ be a continuous function with $f(1)=0.$ Prove that the limit $$\lim_{t \nearrow 1} \left( \frac{1}{1-t} \int\limits_0^1x(f(tx)-f(x)) \mathrm{d}x\right)$$ exists and find its value.

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

For which $n{}$ can we partition a regular $n{}$-gon into finitely many triangles such that no two triangles share a side? [i]Based on a problem of the 2017 Miklós Schweitzer competition[/i]

A ray originating at point $P$ intersects a circle with center $O$ at points $A$ and $B$, with $PB > PA$. Segment $\overline{OP}$ intersects the circle at point $C$. Given that $PA = 31$, $PC = 17$, and $\angle PBO = 60^o$, find the radius of the circle.

The sum of three numbers is $ 20$. The first is $ 4$ times the sum of the other two. The second is seven times the third. What is the product of all three? $ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 400 \qquad \textbf{(E)}\ 800$

The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. Find the smallest possible number of independent deputies after the vacation.

In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$, $I_A$, $I_C$ be the incenter, $A$-excenter, and $C$-excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$, respectively. Prove that $AP=AQ$. [i]Proposed Michal Jan'ik - Czech Republic[/i]

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

Rectangle $ABCD$ has sides in the ratio of $\sqrt2$ to $1$. If $DEC$ is an isosceles right triangle, with $E$ inside the rectangle, find angle $\angle AEB$.

Call a positive integer $n$ [i]tubular [/i] if for any two distinct primes $p$ and $q$ dividing $n, (p + q) | n$. Find the number of tubular numbers less than $100,000$. (Integer powers of primes, including $1, 3$, and $16$, are not considered [i]tubular[/i].)