Found problems: 81
Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?
A student at Harvard named Kevin
Was counting his stones by $11$
He messed up $n$ times
And instead counted $9$s
And wound up at $2007$.
How many values of $n$ could make this limerick true?
Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a difference of 9.
16. Suppose trapezoid $JANE$ is inscribed in a circle of radius $25$ such that the center of the circle lies inside the trapezoid. If the two bases of $JANE$ have side lengths $14$ and $30$ and the average of the lengths of the two legs is $\sqrt{m}$, what is $m$?
17. What is the radius of the circle tangent to the $x$-axis, the line $y=\sqrt{3}x$, and the circle $(x-10\sqrt{3})^2+(y-10)^2=25$?
18. Find the smallest positive integer $n$ such that $3n^3-9n^2+5n-15$ is divisible by $121$ but not $2$.
Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$.
[i]Proposed by Andrew Wu[/i]
Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that
\[f(x+f(y+xy))=(y+1)f(x+1)-1\]for all $x,y\in\mathbb{R}^+$.
($\mathbb{R}^+$ denotes the set of positive real numbers.)