[b]p1.[/b] Let $ABCD$ be a square with side length $1$, $\Gamma_1$ be a circle centered at $B$ with radius 1, $\Gamma_2$ be a circle centered at $D$ with radius $1$, $E$ be a point on the segment $AB$ with $|AE| = x$ $(0 < x \le 1)$, and $\Gamma_3$ be a circle centered at $A$ with radius $|AE|$. $\Gamma_3$ intersects $\Gamma_1$ and $\Gamma_2$ inside the square at $G$ and $F$, respectively. Let region $I$ be the region bounded by the segment $GC$ and the minor arc $GC$ of $\Gamma_1$, and region II be the region bounded by the segment $FG$ and the minor arc $FG$ of $\Gamma_3$, as illustrated in the graph below. Let $r(x)$ be the ratio of the area of region I to the area of region II. (i) Find $r(1)$. Justify your answer. (ii) Find an explicit formula of $r(x)$ in terms of $x$ $(0 < x \le 1)$. Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/0/bd2379a1390a578d78dc7e9f4cde756d5f4723.png[/img] [b]p2.[/b] We call a [i]party [/i] any set of people $V$ . If $v_1 \in V$ knows $v_2 \in V$ in a party, we always assume that $v_2$ also knows $v_1$. For a person $v \in V$ in some party, the degree of v, denoted by $deg\,\,(v)$, is the number of people $v$ knows in the party. (i) Suppose that a party has four people with $V = \{v_1, v_2, v_3, v_4\}$, and that $deg\,\,(v_i) = i$ for $i = 1, 2, 3$ show that $deg\,\,(v_4) = 2$. (ii) Suppose that a party is attended by $n = 4k$ ($k \ge 1$) people with $V = \{v_1, v_2, ..., v_{4k}\}$, and that $deg\,\,(v_i) = i$ for $1 \le i \le n - 1$. Show that $deg\,\,(v_n) = \frac{n}{2}$ . [b]p3.[/b] Let $a, b$ be two real number parameters and consider the function $f(x) =\frac{b + \sin x}{a + \cos x}$. (i) Find an example of $(a, b)$ such that $f(x) \ge 2$ for all real numbers $x$. Justify your answer. (ii) If $a > 1$ and the range of the function $f(x)$ (when x varies over the set of all real numbers) is $[-1, 1]$, find the values of $a$ and $b$. Justify your answer. [b]p4.[/b] Let $f$ be the function that assigns to each positive multiple $x$ of $8$ the number of ways in which $x$ can be written as a difference of squares of positive odd integers. (For example, $f(8) = 1$, because $8 = 3^2 -1^2$, and $f(24) = 2$, because $24 = 5^2 - 1^2 = 7^2 - 5^2$.) (a) Determine with proof the value of $f(120)$. (b) Determine with proof the smallest value $x$ for which $f(x) = 8$. (c) Show that the range of this function is the set of all positive integers. [b]p5.[/b] Consider the binomial coefficients $C_{n,r} ={n \choose r}= \frac{n!}{r!(n - r)!}$, for $n \ge 2$. Prove that $C_{n,r}$ are even, for all $1 \le r \le n - 1$, if and only if $n = 2^m$, for some counting number $m$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.

A point moves on the sides of the triangle $ABC$, defined by the vertices $A(-1.8, 0)$, $B(3.2, 0)$, $C(0, 2.4)$ . Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum. [img]https://cdn.artofproblemsolving.com/attachments/2/5/9e5bb48cbeefaa5f4c069532bf5605b9c1f5ea.png[/img]

Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$

Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.

The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions: \begin{align*} \sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1 \end{align*} for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that \begin{align*} n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i} \end{align*}

Find $x\in\mathbb{R}$ for which \[\log_2(x^2 + 4) - \log_2x + x^2 - 4x + 2 = 0.\]

Let $ABC$ be an isosceles triangle with $CA=CB$ and $\angle ACB=20^{\circ}$. Let $D$ be a point on side $CA$ such that $\angle ADB=30^{\circ}$. Show that $AB=CD$.

The set $\{1, 2, 3, 4\}$ can be partitioned into two subsets $A = \{1, 4\}$ and $B = \{3, 2\}$ with no common elements and such that the sum of the elements of $A$ is equal to the sum of the elements of B. Such a partition is impossible for the set $\{1, 2, 3, 4, 5\}$ and also for the set $\{1, 2, 3, 4, 5, 6\}$. Determine all values of $n$ for which the set of the first $n$ natural numbers can be partitioned into two subsets with no common elements such that the sum of the elements of each subset is the same.

Calamitous Clod deceives the math beasts by changing a clock at Beast Academy. First, he removes both the minute and hour hands, then places each of them back in a random position, chosen uniformly along the circle. Professor Grok notices that the clock is not displaying a valid time. That is, the hour and minute hands are pointing in an orientation that a real clock would never display. One such example is the hour hand pointed at $6$ and the minute hand pointed at $3$. [center] [asy] import olympiad; size(4cm); defaultpen(fontsize(8pt)); draw(circle(origin, 4)); dot(origin); for(int i = 1; i <= 12; ++i){ label("$"+string(i)+"$", (3.6*sin(i * pi/6), 3.6*cos(i * pi/6))); } draw(origin -- (3.2, 0), EndArrow(5)); draw(origin -- (0, -2.2), EndArrow(5)); [/asy] [/center] The math beasts can fix this, though. They can turn both hands by the same number of degrees clockwise. On average, what is the minimal number of degrees they must turn the hands so that they display a valid time?

In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.

Suppose that $m$ and $n$ are integers with $1 \le m \le 49$ and $n \ge 0$ such that $m$ divides $n^{n+1} + 1$. What is the number of possible values of $m$?

Real numbers $x$ and $y$ with $x,y>1$ satisfy $\log_x(y^x)=\log_y(x^{4y})=10.$ What is the value of $xy$?

Let $m\in N$ and $E(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m$, where $x$ and $y$ are positive divisors of 72. a) Prove that there exist infinitely many natural numbers $m$ so, that 2005 divides $E(3,12,m)$ and $E(9,6,m)$. b) Find the smallest positive integer number $m_0$ so, that 2005 divides $E(3,12,m_0)$ and $E(9,6,m_0)$.

A three-digit number has, from left to right, the digits $ h$, $ t$, and $ u$, with $ h>u$. When the number with the digits reversed is subtracted from the original number, the units' digit in the difference of $ r$. The next two digits, from right to left, are: $ \textbf{(A)}\ \text{5 and 9} \qquad \textbf{(B)}\ \text{9 and 5} \qquad \textbf{(C)}\ \text{impossible to tell} \qquad \textbf{(D)}\ \text{5 and 4} \qquad \textbf{(E)}\ \text{4 and 5}$

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]