

1. Find the side lengths of the triangle ABC with area S and ∠BAC = x such that the side BC is as short as possible.


2. Find all positive integers m, n such that n + (n+1) + (n+2) + ... + (n+m) = 1000.


3. Find a polynomial with integer coefficients which has √2 + √3 and √2 + 3^{1/3} as roots.


4. Points H_{1}, H_{2}, ... , H_{n} are arranged in the plane so that each distance H_{i}H_{j} ≤ 1. The point P is chosen to minimise max(PH_{i}). Find the largest possible value of max(PH_{i}) for n = 3. Find the best upper bound you can for n = 4.


5. a_{1}, a_{2}, ... , a_{n} are constants such that f(x) = 1 + a_{1} cos x + a_{2} cos 2x + ... + a_{n} cos nx ≥ 0 for all x. We seek estimates of a_{1}. If n = 2, find the smallest and largest possible values of a_{1}. Find corresponding estimates for other values of n.

