MathNet 2003 — Part 03

We wish to color the integers $1,2,3, \ldots, 10$ in red, green, and blue, so that no two numbers $a$ and $b$, with $a-b$ odd, have the same color. (We do not require that all three colors be used.) In how many ways can this be done?

In triangle $ABC$, $\angle ABC = 50^{\circ}$ and $\angle ACB = 70^{\circ}$. Let $D$ be the midpoint of side $BC$. A circle is tangent to $BC$ at $B$ and is also tangent to segment $AD$; this circle intersects $AB$ again at $P$. Another circle is tangent to $BC$ at $C$ and is also tangent to segment $AD$; this circle intersects $AC$ again at $Q$. Find $\angle APQ$ (in degrees).

For a pair of integers $a$ and $b$, with $0 \lt a \lt b \lt 1000$, the set $S \subseteq \{1, 2, \dots, 2003\}$ is called a *skipping set* for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2| \notin \{a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.

Convex quadrilateral $M A T H$ is given with $H M / M T = 3 / 4$, and $\angle A T M = \angle M A T = \angle A H M = 60^{\circ}$. $N$ is the midpoint of $M A$, and $O$ is a point on $T H$ such that lines $M T$, $A H$, $N O$ are concurrent. Find the ratio $H O / O T$.

$r$ and $s$ are integers such that
$$
3 r \geq 2 s - 3 \text{ and } 4 s \geq r + 12.
$$
What is the smallest possible value of $r / s$?

For $x$ a real number, let $f(x)=0$ if $x\lt 1$ and $f(x)=2x-2$ if $x \geq 1$. How many solutions are there to the equation
$$
f(f(f(f(x))))=x ?
$$

Five mathematicians find a bag of 100 gold coins in a room. They agree to split up the coins according to the following plan:
- The oldest person in the room proposes a division of the coins among those present. (No coin may be split.) Then all present, including the proposer, vote on the proposal.
- If at least $50\%$ of those present vote in favor of the proposal, the coins are distributed accordingly and everyone goes home. (In particular, a proposal wins on a tie vote.)
- If fewer than $50\%$ of those present vote in favor of the proposal, the proposer must leave the room, receiving no coins. Then the process is repeated: the oldest person remaining proposes a division, and so on.
- There is no communication or discussion of any kind allowed, other than what is needed for the proposer to state his or her proposal, and the voters to cast their vote.
Assume that each person is equally intelligent and each behaves optimally to maximize his or her share. How much will each person get?

A particle moves along the $x$-axis in such a way that its velocity at position $x$ is given by the formula $v(x) = 2 + \sin x$. What is its acceleration at $x = \frac{\pi}{6}$?

The circle $\omega$ is drawn through the vertices $A$ and $B$ of the triangle $ABC$. If $\omega$ intersects $AC$ at point $M$ and $BC$ at point $P$, and the segment $MP$ contains the center of the circle inscribed in $ABC$. Given that $AB = c$, $BC = a$ and $CA = b$, find $MP$.

Define the Fibonacci numbers by $F_{0}=0$, $F_{1}=1$, $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. For how many $n$, $0 \leq n \leq 100$, is $F_{n}$ a multiple of $13$?

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that
$$
f(m+n)f(m-n) = f(m^2)
$$
for all $m, n \in \mathbb{N}$.

You have $2003$ switches, numbered from $1$ to $2003$, arranged in a circle. Initially, each switch is either ON or OFF, and all configurations of switches are equally likely. You perform the following operation: for each switch $S$, if the two switches next to $S$ were initially in the same position, then you set $S$ to ON; otherwise, you set $S$ to OFF. What is the probability that all switches will now be ON?

A partition of a number $n$ is a sequence of positive integers, arranged in descending order, whose sum is $n$. For example, $n=4$ has 5 partitions: $1+1+1+1=2+1+1=2+2=3+1=4$. Given two different partitions of the same number, $n=a_{1}+a_{2}+\cdots+a_{k}=b_{1}+b_{2}+\cdots+b_{l}$, where $k \leq l$, the first partition is said to dominate the second if all of the following inequalities hold:
$$
\begin{aligned}
a_{1} & \geq b_{1} ; \\
a_{1}+a_{2} & \geq b_{1}+b_{2} ; \\
a_{1}+a_{2}+a_{3} & \geq b_{1}+b_{2}+b_{3} ; \\
& \vdots \\
a_{1}+a_{2}+\cdots+a_{k} & \geq b_{1}+b_{2}+\cdots+b_{k} .
\end{aligned}
$$
Find as many partitions of the number $n=20$ as possible such that none of the partitions dominates any other. Your score will be the number of partitions you find. If you make a mistake and one of your partitions does dominate another, your score is the largest $m$ such that the first $m$ partitions you list constitute a valid answer.

A broken calculator has the $+$ and $\times$ keys switched. For how many ordered pairs $(a, b)$ of integers will it correctly calculate $a+b$ using the labelled $+$ key?

Let $r, s, t$ be the solutions to the equation $x^{3}+a x^{2}+b x+c=0$. What is the value of $(r s)^{2}+(s t)^{2}+(r t)^{2}$ in terms of $a, b$, and $c$?

Take a clay sphere of radius $13$, and drill a circular hole of radius $5$ through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius?

Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a $60\%$ chance of winning each point, what is the probability that he will win the game?

A teacher must divide 221 apples evenly among 403 students. What is the minimal number of pieces into which she must cut the apples? (A whole uncut apple counts as one piece.)

What is the smallest number of regular hexagons of side length $1$ needed to completely cover a disc of radius $1$?

Find all ordered triples of primes $(p, q, r)$ such that
$$
p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1.
$$