26th Putnam 1965

A1.  The triangle ABC has an obtuse angle at B, and ∠A < ∠C. The external angle bisector at A meets the line BC at D, and the external angle bisector at B meets the line AC at E. Also, BA = AD = BE. Find ∠A.
A2.  Let k = [(n - 1)/2]. Prove that ∑0k ( (n - 2r)/n )2 (nCr)2 = 1/n (2n-2)C(n-1) (where nCr is the binomial coefficient).
A3.  {ar} is an infinite sequence of real numbers. Let bn = 1/n ∑1n exp(i ar). Prove that b1, b2, b3, b4, ... converges to k iff b1, b4, b9, b16, ... converges to k.
A4.  S and T and finite sets. U is a collection of ordered pairs (s, t) with s ∈ S and t ∈ T. There is no element s ∈ S such that all possible pairs (s, t) ∈ U. Every element t ∈ T appears in at least one element of U. Prove that we can find distinct s1, s2 ∈ S and distinct t1, t2 ∈ T such that (s1, t1), (s2, t2) ∈ U, but (s1, t2), (s2, t1) ∉ U.
A5.  How many possible bijections f on {1, 2, ... , n} are there such that for each i = 2, 3, ... , n we can find j < i with f(i) - f(j) = ±1?
A6.  α and β are positive real numbers such that 1/α + 1/β = 1. Prove that the line mx + ny = 1 with m, n positive reals is tangent to the curve xα + yα = 1 in the first quadrant (x, y ≥ 0) iff mβ + nβ = 1.
B1.  X is the unit n-cube, [0, 1]n. Let kn = ∫X cos2( π(x1 + x2 + ... + xn)/(2n) ) dx1 ... dxn. What is limn→∞ kn ?
B2.  Every two players play each other once. The outcome of each game is a win for one of the players. Player n wins an games and loses bn games. Prove that ∑ an2 = ∑ bn2.
B3.  Show that there are just three right angled triangles with integral side lengths a < b < c such that ab = 4(a + b + c).
B4.  Define fn :R → R by fn(x) = (nC0 + nC2 x + nC4 x2 + ... ) / (nC1 + nC3 x + nC5 x2 + ... ), where nCm is the binomial coefficient. Find a formula for fn+1(x) in terms of fn(x) and x, and determine limn→∞fn(x) for all real x.
B5.  Let S be a set with n > 3 elements. Prove that we can find a collection of [n2/4] 2-subsets of S such that for any three distinct elements A, B, C of the collection, A ∪ B ∪ C has at least 4 elements.
B6.  Four distinct points A1, A2, B1, B2 have the property that any circle through A1 and A2 has at least one point in common with any circle through B1 and B2. Show that the four points are collinear or lie on a circle.

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 73 (1966) 727-32. 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
9 Oct 1999