Farmer John is grazing his cows at the origin. There is a river that runs east to west 50 feet north of the origin. The barn is 100 feet to the south and 80 feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farmer John leads them to the barn. He does this using the shortest path possible, and the total distance he travels is $d$ feet. Find the value of $d$.
A tree grows in a rather peculiar manner. Lateral cross-sections of the trunk, leaves, branches, twigs, and so forth are circles. The trunk is $1$ meter in diameter to a height of $1$ meter, at which point it splits into two sections, each with diameter $0.5$ meter. These sections are each one meter long, at which point they each split into two sections, each with diameter $0.25$ meter. This continues indefinitely: every section of tree is $1$ meter long and splits into two smaller sections, each with half the diameter of the previous.
What is the total volume of the tree?
An $n$-string is a string of digits formed by writing the numbers $1, 2, \ldots, n$ in some order (in base ten). For example, one possible 10-string is
35728910461
What is the smallest $n \gt 1$ such that there exists a palindromic $n$-string?
Suppose $f$ is a function that assigns to each real number $x$ a value $f(x)$, and suppose the equation
$$
f\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}\right)=f\left(x_{1}\right)+f\left(x_{2}\right)+f\left(x_{3}\right)+f\left(x_{4}\right)+f\left(x_{5}\right)-8
$$
holds for all real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$. What is $f(0)$ ?
Shown on your answer sheet is a $20 \times 20$ grid. Place as many queens as you can so that each of them attacks at most one other queen. (A queen is a chess piece that can move any number of squares horizontally, vertically, or diagonally.) It's not very hard to get 20 queens, so you get no points for that, but you get 5 points for each further queen beyond 20. You can mark the grid by placing a dot in each square that contains a queen.
$S$ is a set of complex numbers such that if $u, v \in S$, then $u v \in S$ and $u^{2}+v^{2} \in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most $1$ is finite. What is the largest possible value of $N$?
Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence $H H$) or flips tails followed by heads (the sequence $T H$). What is the probability that she will stop after flipping $H H$?
Let $P(x) = x^{3} - \frac{3}{2} x^{2} + x + \frac{1}{4}$. Let $P^{[1]}(x) = P(x)$, and for $n \geq 1$, let $P^{[n+1]}(x) = P^{[n]}(P(x))$. Evaluate $\int_{0}^{1} P^{[2004]}(x) \, d x$.
Yet another trapezoid $ABCD$ has $AD$ parallel to $BC$. $AC$ and $BD$ intersect at $P$. If $[ADP]/[BCP] = 1/2$, find $[ADP]/[ABCD]$. (Here the notation $[P_1 \cdots P_n]$ denotes the area of the polygon $P_1 \cdots P_n$.)
How many ways can you mark 8 squares of an $8 \times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)
In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus $RQR$. The dealer proceeds to move. With each move, the dealer randomly switches the center card with one of the two edge cards (so the configuration after the first move is either $RRQ$ or $QRR$). What is the probability that, after $2004$ moves, the center card is the queen?
Let $n$ be a positive integer. Consider sequences $a_0, a_1, \dots, a_n$ such that $a_i \in \{1, 2, \dots, n\}$ for each $i$ and $a_n = a_0$.
a. Call such a sequence *good* if for all $i = 1, 2, \dots, n$, $a_i - a_{i-1} \not\equiv i \pmod n$. Suppose that $n$ is odd. Find the number of good sequences.
b. Call such a sequence *great* if for all $i = 1, 2, \dots, n$, $a_i - a_{i-1} \not\equiv i, 2i \pmod n$. Suppose that $n$ is an odd prime. Find the number of great sequences.
We say a point is contained in a square if it is in its interior or on its boundary. Three unit squares are given in the plane such that there is a point contained in all three. Furthermore, three points $A, B, C$, are given, each contained in at least one of the squares. Find the maximum area of triangle $A B C$.
Evaluate the sum
$$
\frac{1}{2\lfloor\sqrt{1}\rfloor+1}+\frac{1}{2\lfloor\sqrt{2}\rfloor+1}+\frac{1}{2\lfloor\sqrt{3}\rfloor+1}+\cdots+\frac{1}{2\lfloor\sqrt{100}\rfloor+1}.
$$
For a string of $P$ 's and $Q$ 's, the value is defined to be the product of the positions of the $P$ 's. For example, the string $P P Q P Q Q$ has value $1 \cdot 2 \cdot 4=8$.
Also, a string is called antipalindromic if writing it backwards, then turning all the $P$ 's into $Q$ 's and vice versa, produces the original string. For example, $P P Q P Q Q$ is antipalindromic.
There are $2^{1002}$ antipalindromic strings of length $2004$. Find the sum of the reciprocals of their values.
A sequence of positive integers is defined by $a_{0}=1$ and $a_{n+1}=a_{n}^{2}+1$ for each $n \geq 0$. Find $\operatorname{gcd}\left(a_{999}, a_{2004}\right)$.