43rd Putnam 1982

A1.  Let S be the set of points (x, y) in the plane such that |x| ≤ y ≤ |x| + 3, and y ≤ 4. Find the position of the centroid of S.
A2.  Let Bn(x) = 1x + 2x + ... + nx and let f(n) = Bn(logn2) / (n log2n)2. Does f(2) + f(3) + f(4) + ... converge?
A3.  Evaluate ∫0 (tan-1(πx) - tan-1x) / x dx.
A4.  Given that the equations y' = -z3, z' = y3 with initial conditions y(0) = 1, z(0) = 0 have the unique solution y = f(x), z = g(x) for all real x, prove f(x) and g(x) are both periodic with the same period.
A5.  a, b, c, d are positive integers satisfying a + c ≤ 1982 and a/b + c/d < 1. Prove that 1 - a/b - c/d > 1/19833.
A6.  ai are real numbers and ∑1 ai = 1. Also |a1| > |a2| > |a3| > ... . f(i) is a bijection of the positive integers onto itself, and |f(i) - i | |ai| → 0 as i → ∞. Prove or disprove that ∑1 af(i) = 1.
B1.  ABC is an arbitrary triangle, and M is the midpoint of BC. How many pieces are needed to dissect AMB into triangles which can be reassembled to give AMC?
B2.  Let a(r) be the number of lattice points inside the circle center the origin, radius r. Let k = 1 + e-1 + e-4 + ... + exp(-n2) + ... . Express ∫U a(√(x2 + y2) ) exp( -(x2+y2) ) dx dy as a polynomial in k, where U represents the entire plane.
B3.  Let pn be the probability that two numbers selected independently and randomly from {1, 2, 3, ... , n} have a sum which is a square. Find limn→∞ pn √n.
B4.  A set S of k distinct integers ni is such that ∏ ni divides ∏ (ni + m) for all integers m. Must 1 or -1 belong to S? If all members of S are positive, is S necessarily just {1, 2, ... , k}?
B5.  Given x > ee, define the sequence f(n) as follows: f(0) = e, f(n+1) = (ln x)/(ln f(n)). Prove that the sequence converges. Let the limit be g(x). Prove that g is continuous.
B6.  Let A(a, b, c) be the area of a triangle with sides a, b, c. Let f(a, b, c) = √A(a, b, c). Prove that for any two triangles with sides a, b, c and a', b', c' we have f(a, b, c) + f(a',b',c') ≤ f(a + a', b + b', c + c'). When do we have equality?

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and official solutions were published in American Mathematical Monthly 90 (1983) 547. They are also available (with the solutions expanded) in: Gerald L Alexanderson et al, The William Lowell Putnam Mathematical Competition, 1965-1984. Out of print, but available in some university libraries.

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© John Scholes
8 Oct 1999