29th Putnam 1968

A1.  Prove that ∫01 x4(1 - x)4/(1 + x2) dx = 22/7 - π.
A2.  Given integers a, b, c, d such that ad - bc ≠ 0, integers m, n and a real ε > 0, show that we can find rationals x, y, such that 0 < |ax + by - m| < ε and 0 < |cx + dy - n| < ε.
A3.  S is a finite set. P is the set of all subsets of S. Show that we can label the elements of P as Ai, such that A1 = ∅ and for each n ≥ 1, either An-1 ⊂ An and |An - An-1| = 1, or An-1 ⊃ An and |An-1 - An| = 1.
A4.  Let S2 be the 2-sphere { (x, y, z) : x2 + y2 + z2 = 1}. Show that for any n points on S2, the sum of the squares of the n(n - 1)/2 distances between them (measured in space, not in S2) is at most n2.
A5.  Find the smallest possible α such that if p(x) ≡ ax2 + bx + c satisfies |p(x)| ≤ 1 on [0, 1], then |p'(0)| ≤ α.
A6.  Find all finite polynomials whose coefficients are all ±1 and whose roots are all real.
B1.  The random variables X, Y can each take a finite number of integer values. They are not necessarily independent. Express prob( min(X, Y) = k) in terms of p1 = prob( X = k), p2 = prob(Y = k) and p3 = prob( max(X, Y) = k).
B2.  (G, *) is a finite group with n elements. K is a subset of G with more than n/2 elements. Prove that for every g ∈ G, we can find h, k ∈ K such that g = h * k.
B3.  Given that a 60o angle cannot be trisected with ruler and compass, prove that a 120o/n angle cannot be trisected with ruler and compass for n = 1, 2, 3, ... .
B4.  R is the reals. f : R → R is continuous and L = ∫-∞ f(x) dx exists. Show that ∫-∞ f(x - 1/x) dx = L.
B5.  Let F be the field with p elements. Let S be the set of 2 x 2 matrices over F with trace 1 and determinant 0. Find |S|.
B6.  A compact set of real numbers is closed and bounded. Show that we cannot find compact sets A1, A2, A3, ... such that (1) all elements of An are rational and (2) given any compact set K whose members are all rationals, K ⊆ some An.

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 76 (1969) 911-5. 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 in some university libraries.

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© John Scholes
16 Jan 2002