Problem 1
Determine the number of real solutions to the equation $\sqrt{2 - x^2} = \sqrt[3]{3 - x^3}$.
Problem 2
Evaluate $\displaystyle \int_0^{\pi/2} \frac{dx}{1 + \cos x + \sin x}$ for $-\pi/2 < a < \pi$.
Use your answer to show that $\displaystyle \int_0^{\pi/2} \frac{dx}{1 + \cos x + \sin x} = \ln 2$.
Problem 3
Let $ABC$ be a triangle and let $P$ be a point in its interior. Suppose $\angle BAP = 10^\circ$, $\angle ABP = 20^\circ$, $\angle PCA = 30^\circ$ and $\angle PAC = 40^\circ$. Find $\angle PBC$.
Problem 4
Let $P$ be an interior point of a triangle of area $T$. Through the point $P$, draw lines parallel to the three sides, partitioning the triangle into three triangles and three parallelograms. Let $a, b$ and $c$ be the areas of the three triangles. Prove that $\sqrt{T} = \sqrt{a} + \sqrt{b} + \sqrt{c}$.
Problem 5
Let $f(x, y) = \dfrac{x+y}{2}, g(x, y) = \sqrt{xy}, h(x, y) = \dfrac{2xy}{x+y}$, and let
$$ S = \lbrace (a, b) \in \mathbb{N} \times \mathbb{N} \mid a \ne b \text{ and } f(a,b), g(a,b), h(a,b) \in \mathbb{N} \rbrace, $$
where $\mathbb{N}$ denotes the positive integers. Find the minimum of $f$ over $S$.
Problem 6
Let $f(x) \in \mathbb{Z}[x]$ be a polynomial with integer coefficients such that $f(1) = -1$, $f(4) = 2$ and $f(8) = 34$. Suppose $n \in \mathbb{Z}$ is an integer such that $f(n) = n^2 - 4n - 18$. Determine all possible values for $n$.
Problem 7
Find all pairs $(m, n)$ of nonnegative integers for which $m^2 + 2 \cdot 3^n = m(2^{n+1} - 1)$.