2014 UNC) Math Contest II Problems

Twenty-second Annual UNC Math Contest Final Round January 25, 2014 Three hours; no electronic devices. Show your work and justify your answers. Clearer presentations will earn higher rank. We hope you enjoy thinking about these problems, but you are not expected to do them all.

You may write answers in terms of the Fibonacci numbers $F_n$.

The Fibonacci numbers are $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5, F_6 = 8, \ldots$

They are defined by the equations $F_1 = F_2 = 1$ and, for $n \ge 3, F_n = F_{n-1} + F_{n-2}.$

Problem 1

The Duchess had a child on May 1st every two years until she had five children. This year the youngest is $1$ and the ages of the children are $1, 3, 5, 7$, and $9$. Alice notices that the sum of the ages is a perfect square: $1 + 3 + 5 + 7 + 9 = 25$. How old will the youngest be the next time the sum of the ages of the five children is a perfect square, and what is that perfect square?

Solution

Problem 2

Define the Cheshire Cat function $\fbox{:)}$ by

\begin{align*}  \fbox{:)}(x) &= -x \quad \text {if } x \text{ is even and} \\ \fbox{:)}(x) &= x \quad \text{ if  }x \text{ is odd}  \end{align*}

Find the sum $\fbox{:)}(1) + \fbox{:)}(2) + \fbox{:)}(3) + \fbox{:)}(4) + . . .+ \fbox{:)}(289)$

Solution

Problem 3

Find $x$ and $y$ if $\frac{1}{1+\frac{1}{x}}=2$ and $\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{y}}}}=2$

Solution

Problem 4

On the first slate, the Queen’s jurors write the number $1$. On the second slate they write the numbers $2$ and $3$. On the third slate the jurors write $4, 5$, and $6$, and so on, writing $n$ integers on the $n$th slate.

(a) What is the largest number they write on the $20$th slate?

(b) What is the sum of the numbers written on the $20$th slate?

(c) What is the sum of the numbers written on the $n$th slate?

Solution

Problem 5

(a) The White Rabbit has a square garden with sides of length one meter. He builds a square cucumber frame in the center by connecting each corner of the garden to the midpoint of a far side of the garden, going clockwise, as shown in the diagram. What is the area of the region that is enclosed in the inner square frame?

(b) Suppose that the White Rabbit builds his square cucumber frame by connecting each corner of the garden to a point a distance $x$ from the next corner, going clockwise, as shown in the diagram. Now what is the area of the region that is enclosed in the inner square frame?

Solution

Problem 6

(a) Alice falls down a rabbit hole and finds herself in a circular room with five doors of five different sizes evenly spaced around the circumference. Alice tries keys in some or all of the doors. She must leave no pair of adjacent doors untried. How many different sets of doors left untried does Alice have to choose from? For example, Alice might try doors $1$, $2$, and $4$ and leave doors $3$ and $5$ untried. There are no adjacent doors in the set of untried doors. Note: doors $1$ and $5$ are adjacent.

(b) Suppose the circular room in which Alice finds herself has nine doors of nine different sizes evenly spaced around the circumference. Again, she is to try keys in some or all of the doors and must leave no pair of adjacent doors untried. Now how many different sets of doors left untried does Alice have to choose from?

Solution

Problem 7

The Caterpillar owns five different matched pairs of socks. He keeps the ten socks jumbled in random order inside a silk sack. Dressing in the dark, he selects socks, choosing randomly without replacement. If the two socks he puts on his first pair of feet are a mismatched pair and the two socks he puts on his second pair of feet are a mismatched pair, then what is the probability that the pair he selects for his third set of feet is a mismatched pair?

Solution

Problem 8

In the Queen’s croquet, a game begins with the ball at the bottom wicket. All players hit the same ball. Each player hits the ball from the place the previous player has left it. When the ball is hit from the bottom wicket, it has a $50$% chance of going to the top wicket and a $50$% chance of staying at the bottom wicket. When hit from the top wicket, it has a $50$% chance of hitting the goal post and a 50% chance of returning to the bottom wicket.

(a) If Alice makes the first hit and alternates hits with the Queen, what is the probability that Alice is the first player to hit the goal post with the ball?

(b) Suppose Alice, the King, and the Queen take turns hitting the ball, with Alice playing first. Now what is the probability that Alice is the first player to hit the goal post with the ball?


Solution

Problem 9

In the Queen’s croquet, as described in Problem $8$, what is the probability that the ball hits the goal post the $n$th time the ball is hit?

Solution

Problem 10

The March Hare invites $11$ guests to a tea party. He randomly assigns to each guest either tea or cake, but no guest receives both. The guests know that the March Hare always does this, but they never know which guests will receive tea and which will receive cake. The guests decide to play a game. Each one tries to guess who of all $11$ guests will get cake and who will get tea. If one guest has more correct guesses than all the others, that guest wins. When several guests tie for the most correct guesses, then the Dormouse selects one to be the winner by selecting at random one of the guessers who has tied.

(a) All the guests make their guesses at random, perhaps by tossing a coin. What is the probability that Tweedledee, the last guest to arrive, is the winner?

(b) Tweedledum is the first guest to arrive. What is the probability that one or the other of Tweedledee and Tweedledum is the winner?

(c) Suppose that instead of guessing randomly, Tweedledee always makes the guess opposite to Tweedledum’s guess. If Tweedledum guesses that a guest will have tea, then Tweedledee will guess cake. If all the other guests have guessed randomly, what is the probability that one or the other of Tweedledee and Tweedledum is the winner? Your answer should be an explicit number, but partial credit may be given for reasonable formulae.

Solution

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