Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\] [i]2019 CCA Math Bonanza Team Round #4[/i]

[b]Problem 2[/b] : $k$ number of unit squares selected from a $99 \times 99$ square grid are coloured using five colours Red, Blue, Yellow, Green and Black such that each colour appears the same number of times and on each row and on each column there are no differently coloured unit squares. Find the maximum possible value of $k$.

Find all polynomials $P(x, y)$ with real coefficients which for all real numbers $x$ and $y$ satisfy $P(x + y, x - y) = 2P(x, y)$.

Answer the following questions: (1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$. Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$. (2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant　for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$

Given $$a = bc$$ $$b = ca$$ $$c = a + b$$ $$c > a$$ Evaluate $a+b+c$. [i]2022 CCA Math Bonanza Lightning Round 1.1[/i]

It is known that for some $x{}$ and $y{}$ the sums $\sin x+ \cos y$ and $\sin y + \cos x$ are positive rational numbers. Prove that there exist natural numbers $m{}$ and $n{}$ such that $m\sin x+n\cos x$ is a natural number. [i]Proposed by N. Agakhanov[/i]

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Each square of a $10\times 10$ board contains a chip. One may choose a diagonal containing an even number of chips and remove any chip from it. Find the maximal number of chips that can be removed from the board by these operations.

Find all integers satisfying the equation $ 2^x\cdot(4\minus{}x)\equal{}2x\plus{}4$.

Some frogs, some red and some others green, are going to move in an $11 \times 11$ grid, according to the following rules. If a frog is located, say, on the square marked with # in the following diagram, then [list] [*]If it is red, it can jump to any square marked with an x. [*]if it is green, it can jump to any square marked with an o.[/list] \[\begin{tabular}{| p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | l} \hline &&&&&&\\ \hline &&x&&o&&\\ \hline &o&&&&x&\\ \hline &&&\small{\#}&&&\\ \hline &x&&&&o&\\ \hline &&o&&x&&\\ \hline &&&&&&\\ \hline \end{tabular} \] We say 2 frogs (of any color) can meet at a square if both can get to the same square in one or more jumps, not neccesarily with the same amount of jumps. [list=a] [*]Prove if 6 frogs are placed, then there exist at least 2 that can meet at a square. [*]For which values of $k$ is it possible to place one green and one red frog such that they can meet at exactly $k$ squares?[/list]

Let $a,b,c$ be the sides and $\alpha,\beta,\gamma$ be the corresponding angles of a triangle. Prove the equality $$\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.$$

Let $ I$ be the incentre and $ O$ the circumcentre of a given acute triangle $ ABC$. The incircle is tangent to $ BC$ at $ D$. Assume that $ \angle B < \angle C$ and the segments $ AO$ and $ HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $ OD$ and $ AH$ be $ E$. If the midpoint of $ CI$ is $ F$, prove that $ E,F,I,O$ are concyclic.

Let $ABCD$ be a convex quadrilateral. The circumcenter and the incenter of triangle $ABC$ coincide with the incenter and the circumcenter of triangle $ADC$ respectively. It is known that $AB = 1$. Find the remaining sidelengths and the angles of $ABCD$.

Define the sequence $a_1,a_2,a_3,\ldots$ by $a_n=\sum_{k=1}^n\sin(k)$, where $k$ represents radian measure. Find the index of the $100$th term for which $a_n<0$.

Let $ABC$ be an acute triangle. Let $M_A, M_B$ and $M_C$ be the midpoints of sides $BC,CA$, respectively $AB$. Let $M'_A , M'_B$ and $M'_C$ be the the midpoints of the arcs $BC, CA$ and $AB$ respectively of the circumscriberd circle of triangle $ABC$. Let $P_A$ be the intersection of the straight line $M_BM_C$ and the perpendicular to $M'_BM'_C$ through $A$. Define $P_B$ and $P_C$ similarly. Show that the straight line $M_AP_A, M_BP_B$ and $M_CP_C$ intersect at one point.

We define a sequence $a_n$ so that $a_0=1$ and \[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \] for all postive integers $n$. Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.

Initially given $31$ tuplets $$(1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1)$$ were written on the blackboard. At every move we choose two written $31$ tuplets as $(a_1,a_2,a_3,\dots, a_{31})$ and $(b_1,b_2,b_3,\dots,b_{31})$, then write the $31$ tuplet $(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})$ to the blackboard too. Find the least possible value of the moves such that one can write the $31$ tuplets $$(0,1,1,\dots,1),(1,0,1,\dots,1),\dots, (1,1,1,\dots,0)$$ to the blackboard by using those moves.

A fair coin is to be tossed $10$ times. Let $i/j$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j$.

The number of regular triangles that three apexes are among eight vertex of a cube is $\text{(A)}4\qquad\text{(B)}8\qquad\text{(C)}12\qquad\text{(D)}24$

The real numbers $x, y, z$, distinct in pairs satisfy $$\begin{cases} x^2=2 + y \\ y^2=2 + z \\ z^2=2 + x.\end{cases}$$ Find the possible values of $x^2 + y^2 + z^2$.

Suppose that $x$ and $y$ are real numbers that satisfy the system of equations $2^x-2^y=1$ $4^x-4^y=\frac{5}{3}$ Determine $x-y$

Let $C$ and $D$ be points inside angle $\angle AOB$ such that $5\angle COD = 4\angle AOC$ and $3\angle COD = 2\angle DOB$. If $\angle AOB = 105^{\circ}$, find $\angle COD$

A cube is given with base $ ABCD $, where $ AB = a $ cm. Calculate the distance of the line $ BC $ from the line passing through the point $ A $ and the center $ S $ of the face opposite the base.

Determine all the integers solutions $(x,y)$ of the following equation $$\frac{x^2-4}{2x-1}+\frac{y^2-4}{2y-1}=x+y$$

In the triangle $ABC$, let $\ell$ be the bisector of the external angle at $C$. The line through the midpoint $O$ of $AB$ parallel to $\ell$ meets $AC$ at $E$. Determine $|CE|$, if $|AC|=7$ and $|CB|=4$.