The Putnam and my big mistake

Found this recently on https://prase.cz/kalva/putnam/putn79.html For the other years, see https://prase.cz/kalva/putnam.html 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 x9 for all x. A3. an are defined by a1 = α, a2 = β, an+2 = anan+1/(2an - an+1). α, β are chosen so that an+1 ≠ 2an. For what α, β are infinitely many an 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 A1, A2, ... , An, and the points of B as B1, B2, ... , Bn so that no two of the n segments AiBi intersect? A5. Show that we can find two distinct real roots α, β of x3 - 10x2 + 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 αi ∈ [0, 1] show that we can find β ∈ [0, 1] such that ∑ 1/|β - αi| ≤ 8n ∑1n 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 limλ→0 ( ∫01 (βx + α(1 - x) )λ dx )1/λ. B3. F is a finite field with n elements. n is odd. x2 + bx + c is an irreducible polynomial over F. For how many elements d ∈ F is x2 + bx + c + d irreducible? B4. Find a non-trivial solution of the differential equation F(y) ≡ (3x2 + x - 1)y'' - (9x2 + 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. zi are complex numbers. Show that |Re[ (z12 + z22 + ... + zn2)1/2 ]| ≤ |Re z1| + |Re z2| + ... + |Re zn|. Out of a possible 120 I scored like 14. That first easy question which should have been a gimme I answered with all 2s and a 3 instead of the obvious all 3s and a 2! Of course, 3**659 * 2 529586439277951893965474092583036590810243522082200596517939604227123416013684342943500654729797215523804442022054078728685938631907140628741647186226659387625833568430667687420250306165230923151125081851800886369853847621855850099589166501531688975135868638057037196353126383349956287985876705820012772736921310134 and 2**988 * 3 7847963431540043854602722527294935135947789148233888726394656164821907112731465740916980313591522398395468795782859962098538661760441753249012141778343682717725513506951341053775201126754666363680142223256625836169615679639208081708373201132380661314383200545180572876704416380604949502078031495168 I think I got like 1 out of 10 points for that problem!