1997 Putnam Problems

As of 12/15/97, I have solved 11 problems out of 12.
Note: I introduced some modifications to problem statements, due to limitations of HTML.

    A1. A rectangle HOMF has sides HO = 11 and OM = 5. A triangle ABC has H as the intersection of the altitudes, O the center of the circumscribed circle, M the midpoint of BC, and F the foot of the altitude from A. What is the length of BC?
    A2. Players 1,2,3,...,n are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.
    A3. Evaluete ò₀^¥ ( x - x³ /2 + x⁵ /2.4 - x⁷ /2.4.6 + - . . . ) ( 1 + x² /2² + x⁴ /2²4² + x⁶ /2²4²6² + . . . ) dx.
    A4. Let G be a group with identity e and f : G ® G a function such that
f(g₁) f(g₂) f(g₃) = f(h₁) f(h₂) f(h₃) whenever g₁ g₂ g₃ = e = h₁ h₂ h₃ . Prove that there exists an element a in G such that y(x) = a y(x) is a homomorphism (that is, y(x y) = y(x) y(y) for all x and y in G ).
    A5. Let N_n denote the number of ordered n-tuples of positive integers (a₁, a₂, . . . , a_n) such that 1/a₁ + 1/a₂ + . . . + 1/a_n = 1. Determine whether N₁₀ is even or odd.
    A6. For a positive integer n and any real number c, define x_k recursively by x₀ = 0, x₁ = 1, and for k ³ 0,
x_k+2 = ( c x_k+1 - (n - k) x_k ) / (k + 1) .
Fix n and then take c to be the largest value for which x_n+1 = 0. Find x_k in terms of n and k, 1 £ k £ n.

    B1. Let {x} denote the distance between the real number x and the nearest integer. For each positive integer n, evaluate S_n = å_m=1^{6 n-1} min ( {m/6n}, {m/3n} ). [ min (a,b) means the minimum of a and b.]
    B2. Let f be a twice-differentiable real-valued function satisfying f (x) + f ''(x) = - x g(x) f '(x), where g(x) ³ 0 for all real x. Prove that | f (x)| is bounded.
    B3. For each positive integer n write the sum å_m=1ⁿ (1/m) in the form p_n / q_n where p_n and q_n are relatively prime positive integers. Determine all n such that 5 does not divide q_n .
    B4. Let a_m,n denote the coefficient of xⁿ in the expansion of (1 + x + x²)^m. Prove that for all k ³ 0,
0 £ å_i=0^{[ 2 k / 3]} (-1)ⁱ a _{k - i, i} £ 1, where [x] denotes the largest integer £ x for each real number x.
    B5. Let f (1) = 2, f (2) = 2 ^{f (1)}, . . . , and f (n) = 2 ^{f (n-1)} for integers n > 1. Prove that for n ³ 2, f (n) - f (n-1) is a multiple of n.
    B6. Let ABC be a triangle, A', B', C' being the midpoints of BC, CA, AB respectively. Let lengths AB = 4, BC = 3, AC = 5. For each possible dissection D of triangle ABC into 4 parts, define the diameter d(D) be the least upper bound of the distances between pairs of points belonging to the same part. Then the line segments C'B', BB', A'B' subdivides triangle ABC into 4 triangles, and the dissection has a diameter = 2.5 . Find the least diameter of all dissections of the triangle into 4 parts.