Problem 1
Find the maximum value of $xy^3 + yz^3 + zx^3 - x^3y - y^3z - z^3x$ where $0 \le x \le 1$, $0 \le y \le 1$, $0 \le z \le 1$.
Problem 2
How many sequences of 1’s and 3’s sum to 16? (Examples of such sequences are ${1, 3, 3, 3, 3, 3}$ and ${1, 3, 1, 3, 1, 3, 1, 3}$.)
Problem 3
Find the area of the region of points $(x, y)$ in the $xy$-plane such that $x^4 + y^4 \le x^2 - x^2 y^2 + y^2$.
Problem 4
Let $ABC$ be a triangle, let $M$ be the midpoint of $BC$, and let $X$ be a point on $AM$. Let $BX$ meet $AC$ at $N$, and let $CX$ meet $AB$ at $P$. If $\angle MAC = \angle BCP$, prove that $\angle BNC = \angle CPA$.
Problem 5
Let $a_1, a_2, \dots$ be a sequence of nonnegative real numbers and let $\pi, \rho$ be permutations of the positive integers $\mathbb{N}$ (thus $\pi, \rho : \mathbb{N} \to \mathbb{N}$ are one-to-one and onto maps).
Suppose that $\displaystyle{\sum_{n=1}^{\infty} a_n = 1}$ and $\varepsilon$ is a real number such that $$ \sum_{n=1}^{\infty} |a_n - a_{\pi n}| + \sum_{n=1}^{\infty} |a_n - a_{\rho n}| < \varepsilon. $$
Prove that there exists a finite subset $X$ of $\mathbb{N}$ such that $|X \cap \pi X|, |X \cap \rho X| > (1 - \varepsilon)|X|$ (here $|X|$ indicates the number of elements in $X$; also the inequalities $<, >$ are strict).
Problem 6
Find all pairs of positive (nonzero) integers $a, b$ such that $ab - 1$ divides $a^4 - 3a^2 + 1$.
Problem 7
Let $f_1(x) = x$ and $f_{n+1}(x) = x^{f_n(x)}$ for $n$ a positive integer. Thus $f_2(x) = x^x$ and $f_3(x) = x^{(x^x)}$. Now define $g(x) = \displaystyle{\lim_{n \to \infty} \dfrac{1}{f_n(x)}}$ for $x > 1$. Is $g$ continuous on the open interval $(1, \infty)$? Justify your answer.