

A1. The parallelogram ABCD has AB = a, AD = 1, angle BAD = A, and the triangle ABD has all angles acute. Prove that circles radius 1 and center A, B, C, D cover the parallelogram iff
a ≤ cos A + √3 sin A.


A2. Prove that a tetrahedron with just one edge length greater than 1 has volume at most 1/8.


A3. Let k, m, n be natural numbers such that m + k + 1 is a prime greater than n + 1. Let c_{s} = s(s+1). Prove that:
(c_{m+1}  c_{k})(c_{m+2}  c_{k}) ... (c_{m+n}  c_{k})
is divisible by the product c_{1}c_{2} ... c_{n}.


B1. A_{0}B_{0}C_{0} and A_{1}B_{1}C_{1} are acuteangled triangles. Construct the triangle ABC with the largest possible area which is circumscribed about A_{0}B_{0}C_{0} (BC contains A_{0}, CA contains B_{0}, and AB contains C_{0}) and similar to A_{1}B_{1}C_{1}.


B2. a_{1}, ... , a_{8} are reals, not all zero. Let c_{n} = a_{1}^{n} + a_{2}^{n} + ... + a_{8}^{n} for n = 1, 2, 3, ... . Given that an infinite number of c_{n} are zero, find all n for which c_{n} is zero.


B3. In a sports contest a total of m medals were awarded over n days. On the first day one medal and 1/7 of the remaining medals were awarded. On the second day two medals and 1/7 of the remaining medals were awarded, and so on. On the last day, the remaining n medals were awarded. How many medals were awarded, and over how many days?

