

A1. Find the set of positive integers with sum 1979 and maximum possible product.


A2. R is the reals. For what real k can we find a continuous function f : R → R such that f(f(x)) = k x^{9} for all x.


A3. a_{n} are defined by a_{1} = α, a_{2} = β, a_{n+2} = a_{n}a_{n+1}/(2a_{n}  a_{n+1}). α, β are chosen so that a_{n+1} ≠ 2a_{n}. For what α, β are infinitely many a_{n} integral?


A4. A and B are disjoint sets of n points in the plane. No three points of A ∪ B are collinear. Can we always label the points of A as A_{1}, A_{2}, ... , A_{n}, and the points of B as B_{1}, B_{2}, ... , B_{n} so that no two of the n segments A_{i}B_{i} intersect?


A5. Show that we can find two distinct real roots α, β of x^{3}  10x^{2} + 29x  25 such that we can find infinitely many positive integers n which can be written as n = [rα] = [sβ] for some integers r, s.


A6. Given n reals α_{i} ∈ [0, 1] show that we can find β ∈ [0, 1] such that ∑ 1/β  α_{i} ≤ 8n ∑_{1}^{n} 1/(2i  1).


B1. Can we find a line normal to the curves y = cosh x and y = sinh x?


B2. Given 0 < α < β, find lim_{λ→0} ( ∫_{0}^{1} (βx + α(1  x) )^{λ} dx )^{1/λ}.


B3. F is a finite field with n elements. n is odd. x^{2} + bx + c is an irreducible polynomial over F. For how many elements d ∈ F is x^{2} + bx + c + d irreducible?


B4. Find a nontrivial solution of the differential equation F(y) ≡ (3x^{2} + x  1)y''  (9x^{2} + 9x  2)y' + (18x + 3)y = 0. The solution of F(y) = 6(6x + 1) such that f(0) = 1, and ( f(1)  2)( f(1)  6) = 1 is y=f(x). Find a relation of the form ( f(2)  a)( f(2)  b) = c.


B5. A convex set S in the plane contains (0, 0) but no other lattice points. The intersections of S with each of the four quadrants have the same area. Show that the area of S is at most 4.


B6. z_{i} are complex numbers. Show that Re[ (z_{1}^{2} + z_{2}^{2} + ... + z_{n}^{2})^{1/2} ] ≤ Re z_{1} + Re z_{2} + ... + Re z_{n}.

