

A1. Given n, how many ways can we write n as a sum of one or more positive integers a_{1} ≤ a_{2} ≤ ... ≤ a_{k} with a_{k}  a_{1} = 0 or 1.


A2. a_{1}, a_{2}, ... , a_{n}, b_{1}, ... , b_{n} are nonnegative reals. Show that (∏ a_{i})^{1/n} + (∏b_{i})^{1/n} ≤ (∏(a_{i}+b_{i}))^{1/n}.


A3. Find the minimum of sin x + cos x + tan x + cot x + sec x + cosec x for real x.


A4. a, b, c, A, B, C are reals with a, A nonzero such that ax^{2} + bx + c ≤ Ax^{2} + Bx + C for all real x. Show that b^{2}  4ac ≤ B^{2}  4AC.


A5. An npath is a lattice path starting at (0,0) made up of n upsteps (x,y) → (x+1,y+1) and n downsteps (x,y) → (x1,y1). A downramp of length m is an upstep followed by m downsteps ending on the line y = 0. Find a bijection between the (n1)paths and the npaths which have no downramps of even length.


A6. Is it possible to partition {0, 1, 2, 3, ... } into two parts such that n = x + y with x ≠ y has the same number of solutions in each part for each n?


B1. Do their exist polynomials a(x), b(x), c(y), d(y) such that 1 + xy + x^{2}y^{2} ≡ a(x)c(y) + b(x)d(y)?


B2. Given a sequence of n terms, a_{1}, a_{2}, ... , a_{n} the derived sequence is the sequence (a_{1}+a_{2})/2, (a_{2}+a_{3})/2, ... , (a_{n1}+a_{n})/2 of n1 terms. Thus the (n1)th derivative has a single term. Show that if the original sequence is 1, 1/2, 1/3, ... , 1/n and the (n1)th derivative is x, then x < 2/n.


B3. Show that ∏_{i=1}^{n} lcm(1, 2, 3, ... , [n/i]) = n!.


B4. az^{4} + bz^{3} + cz^{2} + dz + e has integer coefficients (with a ≠ 0) and roots r_{1}, r_{2}, r_{3}, r_{4} with r_{1}+r_{2} rational and r_{3}+r_{4} ≠ r_{1}+r_{2}. Show that r_{1}r_{2} is rational.


B5. ABC is an equilateral triangle with circumcenter O. P is a point inside the circumcircle. Show that there is a triangle with side lengths PA, PB, PC and that its area depends only on PO.


B6. Show that ∫_{0}^{1} ∫_{0}^{1} f(x) + f(y) dx dy ≥ ∫_{0}^{1} f(x) dx for any continuous realvalued function on [0,1].

