Andy's lawn has twice as much area as Beth's lawn and three times as much area as Carlos' lawn. Carlos' lawn mower cuts half as fast as Beth's mower and one third as fast as Andy's mower. If they all start to mow their lawns at the same time, who will finish first? $ \textbf{(A)}\ \text{Andy} \qquad \textbf{(B)}\ \text{Beth} \qquad \textbf{(C)}\ \text{Carlos} \qquad \textbf{(D)}\ \text{Andy and Carlos tie for first.}$ $\textbf{(E)}\ \text{All three tie.}$

Let $A = A(x,y)$ and $B = B(x,y)$ be two-variable polynomials with real coefficients. Suppose that $A(x,y)/B(x,y)$ is a polynomial in $x$ for infinitely many values of $y$, and a polynomial in $y$ for infinitely many values of $x$. Prove that $B$ divides $A$, meaning there exists a third polynomial $C$ with real coefficients such that $A = B \cdot C$. [i]Proposed by Victor Wang[/i]

Let the inscribed circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$ , side $CA$ at $N$ and side $AB$ at $P$ . Let $D$ be a point from $\left[ NP \right]$ such that $\frac{DP}{DN}=\frac{BD}{CD}$ . Show that $DM \perp PN$ .

Given an isosceles triangle $\triangle ABC$, $AB=AC$. A line passes through $M$, the midpoint of $BC$, and intersects segment $AB$ and ray $CA$ at $D$ and $E$, respectively. Let $F$ be a point of $ME$ such that $EF=DM$, and $K$ be a point on $MD$. Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$. $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$. Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$. Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$, then $BF$ is perpendicular to $BC$.

Let $n\ge 2$ be an integer. A sequence $\alpha = (a_1, a_2,..., a_n)$ of $n$ integers is called [i]Lima [/i] if $\gcd \{a_i - a_j \text{ such that } a_i> a_j \text{ and } 1\le i, j\le n\} = 1$, that is, if the greatest common divisor of all the differences $a_i - a_j$ with $a_i> a_j$ is $1$. One operation consists of choosing two elements $a_k$ and $a_{\ell}$ from a sequence, with $k\ne \ell $ , and replacing $a_{\ell}$ by $a'_{\ell} = 2a_k - a_{\ell}$ . Show that, given a collection of $2^n - 1$ Lima sequences, each one formed by $n$ integers, there are two of them, say $\beta$ and $\gamma$, such that it is possible to transform $\beta$ into $\gamma$ through a finite number of operations. Notes. The sequences $(1,2,2,7)$ and $(2,7,2,1)$ have the same elements but are different. If all the elements of a sequence are equal, then that sequence is not Lima.

Prove that if the quadratic $x^2 +ax+b$ is always positive (for all real $x$) then it can be written as the quotient of two polynomials whose coefficients are all positive.

Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median.

Find all pairs \((m,n)\) of positive integers such that there exists a polyhedron, with all faces being regular polygons, such that each vertex of the polyhedron is the vertex of exactly three faces, two of them having \(m\) sides, and the other having \(n\) sides.

When a positive integer $N$ is divided by $60$, the remainder is $49$. When $N$ is divided by $15$, the remainder is $ \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 3 \qquad \mathrm {(C) \ } 4 \qquad \mathrm{(D) \ } 5 \qquad \mathrm{(E) \ } 8$

There are $ 6n \plus{} 4$ mathematicians participating in a conference which includes $ 2n \plus{} 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time. 1. Determine whether or not there is the case $ n \equal{} 1$. 2. Determine whether or not there is the case $ n > 1$.

In triangle $\vartriangle ABC$, $\angle BAC = 135^o$. $M$ is the midpoint of $BC$, and $N \ne M$ is on $BC$ such that $AN = AM$. The line $AM$ meets the circumcircle of $\vartriangle ABC$ at $D$. Point $E$ is chosen on segment $AN$ such that $AE = MD$. Show that $ME = BC$.

Let $P$ be a set of $n\ge 3$ points in the plane, no three of which are on a line. How many possibilities are there to choose a set $T$ of $\binom{n-1}{2}$ triangles, whose vertices are all in $P$, such that each triangle in $T$ has a side that is not a side of any other triangle in $T$?

Alice, Bob, Cassie, Dave, and Ethan are going on a road trip and need to arrange themselves among a drivers seat, a passenger seat, and three distinguishable back row seats. Alice, Bob, and Cassie are not allowed to drive. Alice and Bob are also not allowed to sit in the front passenger seat. Find the number of possible seating arrangements. [i]2022 CCA Math Bonanza Individual Round #2[/i]

5. We call $(P_n)_{n\in \mathbb{N}}$ an arithmetic sequence with common difference $Q(x)$ if $\forall n: P_{n+1} = P_n + Q$ $\newline$ We have an arithmetic sequence with a common difference $Q(x)$ and the first term $P(x)$ such that $P,Q$ are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that: $\newline$ a) $P(x)$ is divisible by $Q(x)$ $\newline$ b) $\text{deg}(\frac{P(x)}{Q(x)}) = 1$

Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.

Let $x$, $y$, $z$ be non-zero real numbers, such that $x + y + z = 0$ and the numbers $$ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \quad \text{and} \quad \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1 $$ are equal. Determine their common value.

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly. Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$. Proposed by Anant Mudgal

$10$ integers are written in a row. A second row of $10$ integers is formed as follows: the integer written under each integer $A$ of the first row is equal to the total number of integers that stand to the right side of $A$ (in the first row) and are strictly greater than A. A third row is formed by the same way under the second one, and so on. (a) Prove that after several steps a “zero row” (i.e. a row consisting entirely of zeros) appears. (b) What is the maximal possible number of non-zero rows (i.e. rows in which at least one entry is not zero)? (S Tokarev)

Compute the area of an octagon inscribed in a circle, whose four sides have length $1$ and the other four sides have length $2$.

There are two families of convex polygons in the plane. Each family has a pair of disjoint polygons. Any polygon from one family intersects any polygon from the other family. Show that there is a line which intersects all the polygons.

For a natural number $n$ let $W (n)$ be the number of possibilities, to distribute weights with the masses $1, 2,..., n$ all of them between the two bowls of a beam balance so that they are in balance/ Show that $W (100)$ is really larger than $W (99)$.

Find the number of subsets $X$ of $\{1,2,\dots,10\}$ such that $X$ contains at least two elements and such that no two elements of $X$ differ by $1$.

Solve the system of simultaneous equations \[\sqrt x - \frac 1y - 2w + 3z = 1,\]\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

A rhombus with sidelength $1$ has an inscribed circle with radius $\frac{1}{3}.$ If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive $a,b,$ evaluate $a+b.$ [i]Proposed by Alex Li[/i]