In a convex hexagon $ABCDEF$ the triangles $BDF, ACE$ are equilateral and congruent. Prove that the three lines connecting the midpoints of opposite sides are concurrent.

Let $ABC$ be a triangle and $A', B', C'$ the points in which its incircle touches the sides $BC, CA, AB$, respectively. We denote by $I$ the incenter and by $P$ its projection onto $AA' $. Let $M$ be the midpoint of the line segment $[A'B']$ and $N$ be the intersection point of the lines $MP$ and $AC$. Prove that $A'N $is parallel to $B'C'$

Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$. [i]2020 CCA Math Bonanza Lightning Round #4.3[/i]

The integers $ 1,2,\dots,20$ are written on the blackboard. Consider the following operation as one step: [i]choose two integers $ a$ and $ b$ such that $ a\minus{}b \ge 2$ and replace them with $ a\minus{}1$ and $ b\plus{}1$[/i]. Please, determine the maximum number of steps that can be done. [i]Yudi Satria, Jakarta[/i]

Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.

Given a graph $G$, consider the following two quantities, $\bullet$ Assign to each vertex a number in $\{0,1,2\}$ such that for every edge $e=uv$, the numbers assigned to $u$ and $v$ have sum at least $2$. Let $A(G)$ be the minimum possible sum of the numbers written to each vertex satisfying this condition. $\bullet$ Assign to each edge a number in $\{0,1,2\}$ such that for every vertex $v$, the sum of numbers on all edges containing $v$ is at most $2$. Let $B(G)$ be the maximum possible sum of the numbers written to each edge satisfying this condition. Prove that $A(G)=B(G)$ for every graph $G$. [Note: This question is not original] [Extra: Show that this statement is still true if we replace $2$ to $n$, if and only if $n$ is even (where we replace $\{0,1,2\}$ to $\{0,1,\cdots, n\}$)]

Let $ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Prove that $A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.

Let $A$ be a set of $20$ distinct positive integers which are all no greater than $397$. Prove that for any positive integer $n$ it is possible to pick four (not necessarily distinct) elements $x_1, x_2, x_3, x_4$ of $A$ satisfying $x_1 \neq x_2$ and $$(x_1-x_2)n\equiv x_3-x_4 \pmod{397}.$$

Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.

Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic. (A. Voidelevich)

Let $y_1$ and $y_2$ be two linearly independent solutions of the equation $$y''+P(x)y'+Q(x)=0.$$ Find the differential equation satisfied by the product $z=y_1 y_2$.

Find all $f: R\rightarrow R$ such that (i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite (ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$

Triangle $ABC$ has circumcircle $\omega$ where $B'$ is the point diametrically opposite $B$ and $C'$ is the point diametrically opposite $C$. Given $B'C'$ passes through the midpoint of $AB$, if $AC' = 3$ and $BC = 7$, find $AB'^2$..

Given a function $ f:\mathbb{N}\longrightarrow\mathbb{N} , $ find the necessary and sufficient condition that makes the sequence $$ \left(\left( 1+\frac{(-1)^{f(n)}}{n+1} \right)^{(-1)^{-f(n+1)}\cdot(n+2)}\right)_{n\ge 1} $$ to be monotone.

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

At the start of the game there are $ r$ red and $ g$ green pieces/stones on the table. Hojoo and Kestutis make moves in turn. Hojoo starts. The person due to make a move, chooses a colour and removes $ k$ pieces of this colour. The number $ k$ has to be a divisor of the current number of stones of the other colour. The person removing the last piece wins. Who can force the victory?

Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

In the plane are given 100 lines such that no 2 are parallel and no 3 meet in a point. The intersection points are marked. Then all the lines and k of the marked points are erased. Given the remained points of intersection for what max k one can reconstruct the lines? [i]Proposed by A. Golovanov[/i]

Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational.

For how many integers is the number $x^4-51x^2+50$ negative? $ \textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }14\qquad\textbf{(E) }16\qquad $

Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers. $(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number. $(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? $ \textbf{(A)}\ \frac{\sqrt{\pi}}{2} \qquad\textbf{(B)}\ \sqrt{\pi} \qquad\textbf{(C)}\ \pi \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \pi^{2}$

A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.