Let $\omega_1,\omega_2$ be two circles centered at $O_1$ and $O_2$ and lying outside each other. Points $C_1$ and $C_2$ lie on these circles in the same semi plane with respect to $O_1O_2$. The ray $O_1C_1$ meets $\omega _2$ at $A_2,B_2$ and $O_2C_2$ meets $\omega_1$ at $A_1,B_1$. Prove that $\angle A_1O_1B_1=\angle A_2O_2B_2$ if and only if $C_1C_2||O_1O_2$.

We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$? [i]2020 CCA Math Bonanza Lightning Round #2.1[/i]

Arianna and Brianna play a game in which they alternate turns writing numbers on a paper. Before the game begins, a referee randomly selects an integer $N$ with $1 \leq N \leq 2019$, such that $i$ has probability $\frac{i}{1 + 2 + \dots + 2019}$ of being chosen. First, Arianna writes $1$ on the paper. On any move thereafter, the player whose turn it is writes $a+1$ or $2a$, where $a$ is any number on the paper, under the conditions that no number is ever written twice and any number written does not exceed $N$. No number is ever erased. The winner is the person who first writes the number $N$. Assuming both Arianna and Brianna play optimally, the probability that Brianna wins can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n.$ [i]Proposed by Edward Wan[/i]

The infinite sequence of integers $a_1, a_2, \cdots $ is defined recursively as follows: $a_1 = 3$, $a_2 = 7$, and $a_n$ equals the alternating sum $$a_1 - 2a_2 + 3a_3 - 4a_4 + \cdots (-1)^n \cdot (n-1)a_{n-1}$$ for all $n > 2$. Let $a_x$ be the smallest positive multiple of $1090$ appearing in this sequence. Find the remainder of $a_x$ when divided by $113$.

Let $ABC$ be an isosceles triangle with $BC = a$ and $AB = AC = b$. Segment $AC$ is the base of an isosceles triangle $ADC$ with $AD = DC = a$ such that points $D$ and $B$ share the opposite sides of AC. Let $CM$ and $CN$ be the bisectors in triangles $ABC$ and $ADC$ respectively. Determine the circumradius of triangle $CMN$. (M.Rozhkova)

Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle internal tangent to $(O)$ at $A_1$ and also tangent to segments $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B_1, C_1, B_c , B_a, C_a, C_b$ similarly. 1. Prove that $AA_1, BB_1, CC_1$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear. 2. Prove that the circle $(I)$ is inscribed in the hexagon with $6$ vertices $A_b,A_c , B_c , B_a, C_a, C_b$.

Esteban the alchemist have $8088$ copper pieces, $6066$ bronze pieces, $4044$ silver pieces and $2022$ gold pieces. He can take two pieces of different metals and use a magic hammer to turn them into two pieces of different metals that he take and different each other. Find the largest number of gold pieces that Esteban can obtain after using the magic hammer a finite number of times. $\textbf{Note:}$ [i]If Esteban takes a copper and bronze pieces, then he turn them into a silver and a gold pieces.[/i]

Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

Consider two circles $k_1,k_2$ touching at point $T$. A line touches $k_2$ at point $X$ and intersects $k_1$ at points $A,B$ where $B$ lies between $A$ and $X$.Let $S$ be the second intersection point of $k_1$ with $XT$. On the arc $\overarc{TS}$ not containing $A$ and $B$ , a point $C$ is choosen. Let $CY$ be the tangent line to $k_2$ with $Y\in{k_2}$ , such that the segment $CY$ doesn't intersect the segment $ST$ .If $I=XY\cap{SC}$ , prove that : $(a)$ the points $C,T,Y,I$ are concyclic. $(b)$ $I$ is the $A-excenter$ of $\triangle ABC$

If $x,y,z \in\mathbb{N}$ and $2x^2+3y^3=4z^4$, Prove that $6|x,y,z$

Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$. [i]Proposed by Ali Zamani [/i]

Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. What possible values can $a^3 + \frac{1}{a^3}$ and $a^5 + \frac{1}{a^5}$ take?

Two numbers have a sum of $32$. If one of the numbers is $ – 36$, what is the other number? $\textbf{(A)}\ 68 \qquad \textbf{(B)}\ -4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 72 \qquad \textbf{(E)}\ -68$

If $$x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1 \ \ with \ \ x,y,z\in \mathbb{R},$$ prove that at least one of $x,y,z$ is equal to zero.

Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$. (a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$. (b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$. [i]James Lin[/i]

Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.

In a regular pyramid with top point $ T$ and equilateral base $ ABC$, let $ P$, $ Q$, $ R$, $ S$ be the midpoints of $ \left[AB\right]$, $ \left[BC\right]$, $ \left[CT\right]$ and $ \left[TA\right]$, respectively. If $ \left|AB\right| \equal{} 6$ and the altitude of pyramid is equal to $ 2\sqrt {15}$, then area of $ PQRS$ will be $\textbf{(A)}\ 4\sqrt {15} \qquad\textbf{(B)}\ 8\sqrt {2} \qquad\textbf{(C)}\ 8\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {5} \qquad\textbf{(E)}\ 9\sqrt {2}$

Points $P$ and $Q$ are taken sides $AB$ and $AC$ of a triangle $ABC$ respectively such that $\hat{APC}=\hat{AQB}=45^{0}$. The line through $P$ perpendicular to $AB$ intersects $BQ$ at $S$, and the line through $Q$ perpendicular to $AC$ intersects $CP$ at $R$. Let $D$ be the foot of the altitude of triangle $ABC$ from $A$. Prove that $SR\parallel BC$ and $PS,AD,QR$ are concurrent.

Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$, $CA\cap t_B$, and $BC\cap t_A$ exist, and that they are collinear.

We have four white equilateral triangles of $3$ cm on each side and join them by their sides to obtain a triangular base pyramid. At each edge of the pyramid we mark two red dots that divide it into three equal parts. Number the red dots, so that when you scroll them in the order they were numbered, result a path with the smallest possible perimeter. How much does that path measure?

Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$

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.

In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$. Prove that $|AR|\cdot |BQ|=|P I|^2$

Solve the equation in $\mathbb{R}$ $$\left\{\left\{\frac{x^2-x}{2021}\right \}-\left\{\frac{x^2+x}{2022}\right \} \right \}=0.$$