Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality \[abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.\] Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

A friction-less board has the shape of an equilateral triangle of side length $1$ meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex $A$ towards the side $BC$. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form $\sqrt{N}$, where $N$ is an integer. What is the value of $N$ ?

Let be an open interval $ I $ and a convex function $ f:I\longrightarrow\mathbb{R} . $ Prove that the lateral derivatives of $ f $ are left-continuous on $ \mathbb{R} $ and also right-continuous on $ \mathbb{R} . $ [i]Marin Tolosi[/i]

Let $ n$ be a positive integer. Consider an $ n\times n$ matrix with entries $ 1,2,...,n^2$ written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for $ n \equal{} 3$ we get $ \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$) We choose $ n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

A blackboard contains $2018$ instances of the digit $1$ separated by spaces. Georg and his mother play a game where they take turns filling in one of the spaces between the digits with either a $+$ or a $\times$. Georg begins, and the game ends when all spaces have been filled. Georg wins if the value of the expression is even, and his mother wins if it is odd. Which player may prepare a strategy which secures him/her victory?

There are $ n \geq 3$ job openings at a factory, ranked $1$ to $ n$ in order of increasing pay. There are $ n$ job applicants, ranked from $1$ to $ n$ in order of increasing ability. Applicant $ i$ is qualified for job $ j$ if and only if $ i \geq j.$ The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking job for which he or she is qualified AND which is lower in rank than any job already filled. (Under these rules, job $1$ is always filled, and hiring terminates thereafter.) Show that applicants $ n$ and $ n \minus{} 1$ have the same probability of being hired.

4.Given is $n \in \mathbb Z$ and positive reals a,b. Find possible maximal value of the sum: $x_1y_1 + x_2y_2 + ... + x_ny_n$ when $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ are in $<0;1>$ and satisfies: $x_1 + x_2 + ... + x_n \leq a$ and $y_1 + y_2 + ... + y_n \leq b$

Prove that for every positive integer $ k$ there exists a sequence of $ k$ consecutive positive integers none of which can be represented as the sum of two squares.

A ladder style tournament is held with $2016$ participants. The players begin seeded $1,2,\cdots 2016$. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After $2015$ rounds, one player remains who wins the tournament. If each player has probability of $\tfrac{1}{2}$ to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has $\tfrac{p}{q}$ for coprime positive integers $p$ and $q$. Find the remainder when $p$ is divided by $1000$. [i]Proposed by Nathan Ramesh

Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.

10 geese are numbered 1-10. One goose leaves the pack, and the remaining nine geese assemble in a symmetric V-shaped formation with four geese on each side. Given that the product of the geese on both halves of the "V" are the same, what is the sum of the possible values of the goose that left? [i]2022 CCA Math Bonanza Lightning Round 2.4[/i]

Let $\omega=e^{\frac{2\pi i}{2017}}$ and $\zeta = e^{\frac{2\pi i}{2019}}$. Let $S=\{(a,b)\in\mathbb{Z}\,|\,0\leq a \leq 2016, 0 \leq b \leq 2018, (a,b)\neq (0,0)\}$. Compute $$\prod_{(a,b)\in S}(\omega^a-\zeta^b).$$

Show that the equation $x^3 + 2y^2 + 4z = n$ has an integral solution $(x, y, z)$ for all integers $n.$

The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.

a.) Prove that for $n = 4$ or $n \geq 6$ each triangle $ABC$ can be decomposed in $n$ similar (not necessarily congruent) triangles. b.) Show: An equilateral triangle can neither be composed in 3 nor 5 triangles. c.) Is there a triangle $ABC$ which can be decomposed in 3 and 5 triangles, analogously to a.). Either give an example or prove that there is not such a triangle.

What is the greatest amount of rooks that can be placed on an $n\times n$ board, such that each rooks beats an even number of rooks? A rook is considered to beat another rook, if they lie on one vertical or one horizontal line and no rooks are between them. [I]Proposed by D. Karpov[/i]

Let us call the [i]distance [/i] between the numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ the maximum $i$ for which $a_i \ne b_i$. All five-digit numbers are written out one after another in some order. What is the minimum possible sum of distances between adjacent numbers?

$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$. Find $\lim_{n \to \infty} \sqrt[n]{a_n}$.

Find the shortest distance between the graphs of $y=x^2+5$ and $x=y^2+5$. [i]Proposed by Nathan Ramesh

On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took 25 percent longer than usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?

Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.

Let $R$ be a commutative ring with 1. Prove that $R[x]$ has infinitely many maximal ideals.

There is a point $M$ inside a circle, at a distance $OM = d$ of the center $O$. Two chords $AB$ and $CD$ are traced through $M$ that form a right angle . Join $A$ with $C$ and $B$ with $D$. Determine the cosine of the angle that must form the chord $AB$ with $OM$ so that the sum of the areas of the triangles $AMC$ and $BMD$ be minimal.