1Farmer John has a pasture where grass grows at a constant rate. 100 cows can be sustained for 10 days before the grass is gone. Alternatively, 100 cows can be sustained for 5 days, and then 120 cows for 3 more days. What is the maximum number of cows that can be sustained indefinitely?
2Sam places either a '+' or '-' in the blanks of the expression: 5 _ 8 _ 9 _ 7 _ 2 _ 3. How many ways can he place these operations such that the resulting integer is divisible by 3?
3Let $r_1, r_2, r_3$ be roots of a cubic polynomial $P(x)$. If $(P(2) + P(-2)) / P(0) = 200$, find the value of $m + n$ where $m/n = 1/(r_1r_2) + 1/(r_2r_3) + 1/(r_3r_1)$ in simplest form.
4In a rectangle $ABCD$ with $AB=3$ and $BC=1$, $O$ is the intersection of the diagonals. The circumcircle of triangle $ADO$ intersects line $AB$ at a second point $E$. What is the length of segment $BE$?
5How many pairs of real numbers $(x, y)$ with $0 < x, y < 1$ result in both $3x + 5y$ and $5x + 2y$ being integers?
6A circular spinner has 4 congruent sections. Each section is randomly colored one of four colors. Adjacent sections of the same color are fused into one. What is the expected number of sections in the final spinner?
7In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. A point $D$ lies on the extension of $BC$ past $C$. Line $l$ passes through $A$ perpendicular to $BC$. Lines through $B$ and $C$ perpendicular to $AD$ intersect $l$ at $H_1$ and $H_2$. If $H_1H_2 = 1001$, what is the length of $CD$?
8Consider the alternating sum $S_k = \sum_{n=1}^k (-1)^{n+1} \frac{n+1}{n!}$. For which of the following approximate values of $k$ does the sum first exceed $1 + 1/700^3$?
9In a $5 \times 6$ grid, two specific squares are colored red: $(2,4)$ and $(4,2)$. How many rectangles formed by gridlines contain at most one red square?
10A square $ABCD$ has side length 100. $M$ is the midpoint of $AB$. A circle with center $M$ and radius 50 intersects a circle with center $D$ and radius 100 at point $E$. If $CE$ intersects $AB$ at $F$, what is the value of $m+n$ for $AF = m/n$?
