MathNet 2021 — Part 05

Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M$, or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order.

Let $f$ be a monic cubic polynomial satisfying $f(x)+f(-x)=0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x))=y$. Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\{1,5,9\}$. Compute the sum of all possible values of $f(10)$.

Let $S$ be a set of positive integers satisfying the following two conditions:
- For each positive integer $n$, at least one of $n, 2n, \ldots, 100n$ is in $S$.
- If $a_{1}, a_{2}, b_{1}, b_{2}$ are positive integers such that $\operatorname{gcd}\left(a_{1} a_{2}, b_{1} b_{2}\right)=1$ and $a_{1} b_{1}, a_{2} b_{2} \in S$, then $a_{2} b_{1}, a_{1} b_{2} \in S$.
Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\left\lfloor 10^{5} r\right\rfloor$.
Note: $S$ has natural density $r$ if $\frac{1}{n}|S \cap\{1, \ldots, n\}|$ approaches $r$ as $n$ approaches $\infty$.

Three faces $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ of a unit cube share a common vertex. Suppose the projections of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ onto a fixed plane $\mathcal{P}$ have areas $x, y, z$, respectively. If $x: y: z=6: 10: 15$, then $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\cdots+a_{n}=2021$ and $a_{1} a_{2} \cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \mid M$. Proposed by: Sheldon Kieren Tan

For each positive integer $1 \leq m \leq 10$, Krit chooses an integer $0 \leq a_{m} \lt m$ uniformly at random. Let $p$ be the probability that there exists an integer $n$ for which $n \equiv a_{m} \pmod{m}$ for all $m$. If $p$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

Let $n$ be a positive integer. Given that $n^{n}$ has 861 positive divisors, find $n$.

Compute the number of labelings $f:\{0,1\}^{3} \rightarrow \{0,1, \ldots, 7\}$ of the vertices of the unit cube such that
$$
\left|f\left(v_{i}\right)-f\left(v_{j}\right)\right| \geq d\left(v_{i}, v_{j}\right)^{2}
$$
for all vertices $v_{i}, v_{j}$ of the unit cube, where $d\left(v_{i}, v_{j}\right)$ denotes the Euclidean distance between $v_{i}$ and $v_{j}$.

Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 10 can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Find the number of ways in which the nine numbers
$$
1, 12, 123, 1234, \ldots, 123456789
$$
can be arranged in a row so that adjacent numbers are relatively prime.

A tournament among $2021$ ranked teams is played over $2020$ rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probability that the second best ranked team is eliminated in the last round. Compute $\lfloor 2021 p \rfloor$.

In triangle $ABC$, $\angle A = 2 \angle C$. Suppose that $AC = 6$, $BC = 8$, and $AB = \sqrt{a} - b$, where $a$ and $b$ are positive integers. Compute $100a + b$.

In a $k \times k$ chessboard, a set $S$ of 25 cells that are in a $5 \times 5$ square is chosen uniformly at random. The probability that there are more black squares than white squares in $S$ is $48\%$. Find $k$.

Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius $1$ and $2$, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles in this fashion.

If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce $11$ times before returning to $(2,0)$ for the first time. If $\cot \theta = a - \sqrt{b}$ for positive integers $a$ and $b$, compute $100a + b$.

Tetrahedron $A B C D$ has side lengths $A B=6$, $B D=6 \sqrt{2}$, $B C=10$, $A C=8$, $C D=10$, and $A D=6$. The distance from vertex $A$ to face $B C D$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is square-free, and $\operatorname{gcd}(a, c)=1$. Find $100 a+10 b+c$.

Let $f: \mathbb{Z}^2 \rightarrow \mathbb{Z}$ be a function such that, for all positive integers $a$ and $b$,
$$
f(a, b)= \begin{cases}b & \text{ if } a\gt b \\ f(2 a, b) & \text{ if } a \leq b \text{ and } f(2 a, b)\lt a \\ f(2 a, b)-a & \text{ otherwise }\end{cases}
$$

Compute $f\left(1000,3^{2021}\right)$.

A counter begins at $0$. Then, every second, the counter either increases by $1$ or resets back to $0$ with equal probability. The expected value of the counter after ten seconds can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100m+n$.

Let $ABCDEFGH$ be an equilateral octagon with $\angle A \cong \angle C \cong \angle E \cong \angle G$ and $\angle B \cong \angle D \cong \angle F \cong \angle H$. If the area of $ABCDEFGH$ is three times the area of $ACEG$, then $\sin B$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100m+n$.

Triangle $A B C$ has side lengths $A B=19$, $B C=20$, and $C A=21$. Points $X$ and $Y$ are selected on sides $A B$ and $A C$, respectively, such that $A Y = X Y$ and $X Y$ is tangent to the incircle of $\triangle A B C$. If the length of segment $A X$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a + b$.

Let $x, y, z$ be real numbers satisfying
$$
\frac{1}{x} + y + z = x + \frac{1}{y} + z = x + y + \frac{1}{z} = 3.
$$
The sum of all possible values of $x + y + z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n) = 1$. Find $100m + n$.