One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel $50$ miles per hour while heading to our final vacation destination..." Hannah teases her husband, "You $\textit{would}$ drive that $\textit{slowly}\text{!}$" Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we travel at a constant rate of $50$ miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same route. If our faster return rate is an integer number of miles per hour, and our average speed for the $\textit{whole round trip}$ is $\textit{also}$ an integer number of miles per hour, what must be our speed during the return trip?" Michael pipes up, "How about $4950$ miles per hour?!" Wendy smiles, "For the sake of your $\textit{other}$ children, please don't let $\textit{Michael}$ drive." Jerry adds, "How about we assume that we never $\textit{ever}$ drive more than $100$ miles per hour. Michael and Wendy, let Josh and Alexis try this one." Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate. But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find?

Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.

Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

If for every real $x$, $ax^2 + bx+c \geq 0$, where $a,b,c$ are reals such that $a<b$, what is the smallest value of $\dfrac{a+b+c}{b-a}$? $ \textbf{(A)}\ \dfrac{5}{\sqrt 3} \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \dfrac{\sqrt 5}2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \dfrac{\sqrt 7}2 $

Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$? [asy] size(10cm); draw((0,0)--(26,0)--(26,10)--(0,10)--cycle); draw((1,0)--(25,10)); draw(circle((0,5),5)); draw(circle((26,5),5)); dot((1,0)); dot((25,10)); label("$E$",(1,0),SE); label("$F$",(25,10),NW); label("$A$", (0,0), SW); label("$B$", (0,10), NW); label("$C$", (26,10), NE); label("$D$", (26,0), SE); dot((0,0)); dot((0,10)); dot((26,0)); dot((26,10)); [/asy] [i]Proposed by Nathan Xiong[/i]

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?

It is known that there are $3$ buildings in the same shape which are located in an equilateral triangle. Each building has a $2015$ floor with each floor having one window. In all three buildings, every $1$st floor is uninhabited, while each floor of others have exactly one occupant. All windows will be colored with one of red, green or blue. The residents of each floor of a building can see the color of the window in the other buildings of the the same floor and one floor just below it, but they cannot see the colors of the other windows of the two buildings. Besides that, sresidents cannot see the color of the window from any floor in the building itself. For example, resident of the $10$th floor can see the colors of the $9$th and $10$th floor windows for the other buildings (a total of $4$ windows) and he can't see the color of the other window. We want to color the windows so that each resident can see at lest $1$ window of each color. How many ways are there to color those windows?

In trapezoid $ABCD$, we have $\overline{AD}\parallel\overline{BC}$, $BC=3$, and $CD=4$. In addition, $\cos\angle ADC=\tfrac{1}{3}$ and $\angle ABC=2\angle ADC$. Compute $AC$.

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

Given five segments. It is possible to construct a triangle of every subset of three of them. Prove that at least one of those triangles is acute-angled.

Find all values of $n$ for which all solutions of the equation $x^3-3x+n=0$ are integers.

A river is bounded by the lines $x=0$ and $x=25$, with a current of 2 units/s in the positive y-direction. At $t=0$, a mallard is at $(0, 0)$, and a wigeon is at $(25, 0)$. They start swimming with a constant speed such that they meet at $(x,22)$. The mallard has a speed of 4 units/s relative to the water, and the wigeon has a speed of 3 units/s relative to the water. Find the value of $x$. [i]2022 CCA Math Bonanza Individual Round #11[/i]

Two intelligent people playing a game on the $1403 \times 1403$ table with $1403^2$ cells. The first one in each turn chooses a cell that didn't select before and draws a vertical line segment from the top to the bottom of the cell. The second person in each turn chooses a cell that didn't select before and draws a horizontal line segment from the left to the right of the cell. After $1403^2$ steps the game will be over. The first person gets points equal to the longest verticals line segment and analogously the second person gets point equal to the longest horizonal line segment. At the end the person who gets the more point will win the game. What will be the result of the game?

[b]a)[/b] Prove that, for any integer $ k, $ the equation $ x^3-24x+k=0 $ has at most an integer solution. [b]b)[/b] Show that the equation $ x^3+24x-2016=0 $ has exactly one integer solution.

(i) For positive integers $a<b$, let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$. Calculate $[M(a,b)]$ (ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$.

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$. $$(x+y)f(x-y) = f(x^2-y^2).$$

If $ A,B,C,D$ are four points in space, such that \[ \angle ABC\equal{}\angle BCD\equal{}\angle CDA\equal{}\angle DAB\equal{}\pi/2, \] prove that $ A,B,C,D$ lie in a plane.

Let $ABCDEF$ be a convex hexagon that does not have two parallel sides, such that $\angle AF B = \angle F DE, \angle DF E = \angle BDC$ and $\angle BFC = \angle ADF.$ Prove that the lines $ AB, FC$ and $DE$ are concurrent if and only if the lines $ AF, BE$ and $CD$ are concurrent.

A semicircle has diameter $AB$ with $AB = 100$. Points $C$ and $D$ lie on the semicircle such that $AC = 28$ and $BD = 60$. Find $CD$.

There are $2020$ points on the coordinate plane {$A_i = (x_i, y_i):i = 1, ..., 2020$}, satisfying $$0=x_1<x_2<...<x_{2020}$$ $$0=y_{2020}<y_{2019}<...<y_1$$ Let $O=(0, 0)$ be the origin, $OA_1A_2...A_{2020}$ forms a polygon $C$. Now, you want to blacken the polygon $C$. Every time you can choose a point $(x,y)$ with $x, y > 0$, and blacken the area {$(x', y'): 0\leq x' \leq x, 0\leq y' \leq y$}. However, you have to pay $xy$ dollars for doing so. Prove that you could blacken the whole polygon $C$ by using $4|C|$ dollars. Here, $|C|$ stands for the area of the polygon $C$. [i]Proposed by me[/i]

Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

Let $ABC$ be a triangle with $AB < AC$ and $\omega$ be a circle through $A$ tangent to both the $B$-excircle and the $C$-excircle. Let $\omega$ intersect lines $AB, AC$ at $X,Y$ respectively and $X,Y$ lie outside of segments $AB, AC$. Let $O$ be the center of $\omega$ and let $OI_C, OI_B$ intersect line $BC$ at $J,K$ respectively. Suppose $KJ = 4$, $KO = 16$ and $OJ = 13$. Find $\frac{[KI_BI_C]}{[JI_BI_C]}$. [i]Proposed by Grant Yu[/i]

Evaluate \[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]

Let the sides of two rectangles be $ \{a,b\}$ and $ \{c,d\},$ respectively, with $ a < c \leq d < b$ and $ ab < cd.$ Prove that the first rectangle can be placed within the second one if and only if \[ \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.\]

Brice is eating bowls of rice. He takes a random amount of time $t_1 \in (0,1)$ minutes to consume his first bowl, and every bowl thereafter takes $t_n = t_{n-1} + r_n$ minutes, where $t_{n-1}$ is the time it took him to eat his previous bowl and $r_n \in (0,1)$ is chosen uniformly and randomly. The probability that it takes Brice at least 12 minutes to eat 5 bowls of rice can be expressed as simplified fraction $\tfrac{m}{n}$. Compute $m+n$.