### 58th Putnam 1997

 A1.  ROMN is a rectangle with vertices in that order and RO = 11, OM = 5. The triangle ABC has circumcenter O and its altitudes intersect at R. M is the midpoint of BC, and AN is the altitude from A to BC. What is the length of BC? A2.  Players 1, 2, ... , n are seated in a circle. Each has one penny. Starting with player 1 passing one penny to player 2, players alternately pass one or two pennies to the next player still seated. A player leaves as soon as he runs out of pennies. So player 1 leaves after move 1, and player 2 leaves after move 2. Find an infinite set of numbers n for which some player ends up with all n pennies. A3.  Let f(x) = (x - x3/2 + x5/(2.4) - x7/(2.4.6) + ... ), and g(x) = (1 + x2/22 + x4/(2242) + x6/(224262) + ... ). Find ∫0∞ f(x) g(x) dx. A4.  G is a group, not necessarily abelian. We write the operation as juxtapostion and the identity as 1. There is a function φ : G → G such that if abc = def = 1, then φ(a)φ(b)φ(c) = φ(d)φ(e)φ(f). Prove that there exists an element k ∈ G such that kφ(x) is a homomorphism. A5.  Is the number of ordered 10-tuples of positive integers (a1, a2, ... , a10) such that 1/a1 + 1/a2 + ... + 1/a10 = 1 even or odd? A6.  Let N be a fixed positive integer. For real α, define the sequence xk by: x0 = 0, x1 = 1, xk+2 = (αxk+1 - (N - k)xk)/(k + 1). Find the largest α such that xN+1 = 0 and the resulting xk. B1.  Find ∑16N-1 min({r/3N}, {r/3N}), where {α} = min (α - [α], [α] + 1 - α), the distance to the nearest integer. B2.  Let R be the reals. f : R → R is twice-differentiable and we can find g : R → R such that g(x) ≥ 0 and f(x) + f ''(x) = -xg(x)f '(x) for all x. Prove that f(x) is bounded. B3.  Let (1 + 1/2 + 1/3 + ... + 1/n) = pn/qn, where pn and qn are relatively prime positive integers. For which n is qn not divisible by 5? B4.  Let (1 + x + x2)m = ∑02m am,nxn. Prove that for all k ≥ 0, ∑0[2k/3] (-1)iak-i,i ∈ [0,1]. B5.  Define the sequence an by a1 = 2, an+1 = 2an. Prove that an = an-1 (mod n) for n ≥ 2. B6.  For a plane set S define d(S), the diameter of S, to be sup {PQ : P, Q ∈ S}. Let K be a triangle with sides 3, 4, 5 and its interior. If K = H1 ∪ H2 ∪ H3 ∪ H4, what is the smallest possible value of max(d(H1), d(H2), d(H3), d(H4))?

To avoid possible copyright problems, I have changed the wording, but not the substance of all the problems. The original text of the problems and the official solutions are in American Mathematical Monthly 105 (1998) 746.

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