View source for 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 <math>F_n</math>.

The Fibonacci numbers are <math>F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5, F_6 = 8, \ldots </math>

They are defined by the equations <math>F1 = F2 = 1</math> and, for <math>n > 2, Fn = Fn-1 + Fn-2.</math>

==Problem 1==

The Duchess had a child on May 1st every two years until she had five children. This year the
youngest is <math>1</math> and the ages of the children are <math>1, 3, 5, 7</math>, and <math>9</math>. Alice notices that the sum of the ages
is a perfect square: <math>1 + 3 + 5 + 7 + 9 = 25</math>. 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?

[[2014 UNC Math Contest II Problems/Problem 1|Solution]]

==Problem 2==

Define the Cheshire Cat function <math>\fbox{:)}</math> by

<cmath>\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*}</cmath>

Find the sum <math>\fbox{:)}(1) + \fbox{:)}(2) + \fbox{:)}(3) + \fbox{:)}(4) + . . .+ \fbox{:)}(289)</math>

[[2014 UNC Math Contest II Problems/Problem 2|Solution]]

==Problem 3==

Find <math>x</math> and <math>y</math> if <math>\frac{1}{1+\frac{1}{x}}=2</math> and <math>\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{y}}}}=2</math>

[[2014 UNC Math Contest II Problems/Problem 3|Solution]]

==Problem 4==

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

(a) What is the largest number they write on the <math>20</math>th slate?

(b) What is the sum of the numbers written on the <math>20</math>th slate?

(c) What is the sum of the numbers written on the <math>n</math>th slate?

[[2014 UNC Math Contest II Problems/Problem 4|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
<math>x</math> 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?

[[2014 UNC Math Contest II Problems/Problem 5|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 <math>1</math>, <math>2</math>, and <math>4</math> and leave doors <math>3</math> and
<math>5</math> untried. There are no adjacent doors in the set of untried doors. Note: doors <math>1</math> and <math>5</math> 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?

[[2014 UNC Math Contest II Problems/Problem 6|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?

[[2014 UNC Math Contest II Problems/Problem 7|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 <math>50</math>% chance of going to the top wicket
and a <math>50</math>% chance of staying at the bottom wicket. When hit from
the top wicket, it has a <math>50</math>% 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?


[[2014 UNC Math Contest II Problems/Problem 8|Solution]]

==Problem 9==

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

[[2014 UNC Math Contest II Problems/Problem 9|Solution]]

==Problem 10==

The March Hare invites <math>11</math> 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 <math>11</math> 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.

[[2014 UNC Math Contest II Problems/Problem 10|Solution]]