### 48th Putnam 1987

 A1.  Four planar curves are defined as follows: C1 = {(x, y): x2 - y2 = x/(x2 + y2) }, C2 = {(x, y): 2xy + y/(x2 + y2) = 3}, C3 = {(x, y): x3 - 3xy2 + 3y = 1}, C4 = {(x, y): 3yx2 - 3x - y3 = 0}. Prove that C1 ∩ C2 = C3 ∩ C4. A2.  An infinite sequence of decimal digits is obtained by writing the positive integers in order: 1234567891011 ... . Define f(n) = m if the 10n th digit forms part of an m-digit number. For example, f(1) = 2, because the 10th digit is part of 10, and f(2) = 2, because the 100th digit is part of 55. Find f(1987). A3.  y = f(x) is a real-valued solution (for all real x) of the differential equation y'' - 2y' + y = 2ex which is positive for all x. Is f '(x) necessarily positive for all x? y = g(x) is another real valued solution, which satisfies g'(x) > 0 for all real x. Is g(x) necessarily positive for all x? A4.  p(x, y, z) is a polynomial with real coefficients such that: (1) p(tx, ty, tz) = t2f(y - x, z - x) for all real x, y, z, t (and some function f); (2) p(1, 0, 0) = 4, p(0 ,1, 0) = 5, and p(0, 0, 1) = 6; and (3) p(α, β, γ) = 0 for some complex numbers α, β, γ such that |β - α| = 10. Find |γ - α|. A5.  f: R2→R3 (where R is the real line) is defined by f(x, y) = (-y/(x2 + 4y2), x/(x2 + 4y2), 0). Can we find F: R3→R3, such that: (1)   if F = (F1, F2, F3), then Fi all have continuous partial derivatives for all (x, y, z) ≠ (0, 0, 0); (2)   ∇ x F = 0 for all (x, y, z) ≠ 0; (3)   F(x, y, 0) = f(x, y)? A6.  Define f(n) as the number of zeros in the base 3 representation of the positive integer n. For which positive real x does F(x) = xf(1)/13 + xf(2)/23 + ... + xf(n)/n3 + ... converge? B1.  Evaluate ∫24 ln1/2(9 - x)/( ln1/2(9 - x) + ln1/2(x + 3) )   dx. B2.  Let n, r, s be non-negative integers with n ≥ r + s, prove that ∑i=0s sCi / nC(r+i) = (n+1)/( (n+1-s) (n-s)Cr ), where mCn denotes the binomial coefficient. B3.  F is a field in which 1 + 1 ≠ 0. Define Pα = ( (α2 - 1)/(α2 + 1), 2α/(α2 + 1) ). Let A = { (β, γ) : β, γ ∈ F, and β2 + γ2 = 1}, and let B= { (1, 0) } ∪ { Pα: α ∈ F, and α2 ≠ -1}. Prove that A = B. B4.  Define the sequences xi and yi as follows. Let (x1, y1) = (0.8, 0.6) and let (xn+1, yn+1) = (xncos yn - ynsin yn, xnsin yn + yn cos yn) for n ≥ 1. Find limn→∞ xn and limn→∞ yn. B5.  A is a complex 2n x n matrix such that if z is a real 1 x 2n row vector then z A ≠ 0 unless z = 0. Prove that given any real 2n x 1 column vector x we can always find an n x 1 column vector z such that the real part of A z = x. B6.  F is a finite field with p2 elements, where p is an odd prime. S is a set of (p2 - 1)/2 distinct non-zero elements of F such that for each a ∈ F, just one of a and -a is in S. Prove that the number of elements in S ∩ {2a: a ∈ S} is even.

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 95 (1988) 719-727.

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