### 44th Putnam 1983

 A1.  How many positive integers divide at least one of 1040 and 2030? A2.  A clock's minute hand has length 4 and its hour hand length 3. What is the distance between the tips at the moment when it is increasing most rapidly? A3.  Let f(n) = 1 + 2n + 3n2 + ... + (p - 1)np-2, where p is an odd prime. Prove that if f(m) = f(n) (mod p), then m = n (mod p). A4.  Prove that for m = 5 (mod 6), mC2 - mC5 + mC8 - mC11 + ... - mC(m-6) + mC(m-3) ≠ 0. A5.  Does there exist a positive real number α such that [αn] - n is even for all integers n > 0? A6.  Let T be the triangle with vertices (0, 0), (a, 0), and (0, a). Find lima→∞ a4exp(-a3) ∫T exp(x3+y3) dx dy. B1.  Let C be a cube side 4, center O. Let S be the sphere center O radius 2. Let A be one of the vertices of the cube. Let R be the set of points in C but not S, which are closer to A than to any other vertex of C. Find the volume of R. B2.  Let f(n) be the number of ways of representing n as a sum of powers of 2 with no power being used more than 3 times. For example, f(7) = 4 (the representations are 4 + 2 + 1, 4 + 1 + 1 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1). Can we find a real polynomial p(x) such that f(n) = [p(n)] ? B3.  y1, y2, y3 are solutions of y''' + a(x) y'' + b(x) y' + c(x) y = 0 such that y12 + y22 + y32 = 1 for all x. Find constants α, β such that y1'(x)2 + y2'(x)2 + y3'(x)2 is a solution of y' + α a(x) y + βc(x) = 0. B4.  Let f(n) = n + [√n]. Define the sequence ai by a0 = m, an+1 = f(an). Prove that it contains at least one square. B5.  Define ||x|| as the distance from x to the nearest integer. Find limn→∞ 1/n ∫1n ||n/x|| dx. You may assume that ∏1∞ 2n/(2n-1) 2n/(2n+1) = π/2. B6.  Let α be a complex (2n + 1)th root of unity. Prove that there always exist polynomials p(x), q(x) with integer coefficients, such that p(α)2 + q(α)2 = -1.

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 91 (1984) 489. 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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