### 16th Putnam 1956

 A1.  α ≠ 1 is a positive real. Find limx→∞ ( (αx - 1)/(αx - x) )1/x. A2.  Given any positive integer n, show that we can find a positive integer m such that mn uses all ten digits when written in the usual base 10. A3.  Find the trajectory of a particle which moves from rest in a vertical plane under (constant) gravity and a force kv perpendicular to its velocity v. A4.  Let p(x) be a real polynomial of degree n with leading coefficient 1 and all roots real. Let R be the reals and f : [a, b] → R be an n times differentiable function with at least n + 1 distinct zeros. Show that p(D) f(x) has at least one zero on [a, b], where D denotes d/dx. A5.  Show that there are just (n-k+1)Ck subsets of {1, 2, ... , n} with k elements and not containing both i and i+1 for any i. A6.  Let R be the reals. Find f : R → R which preserves all rational distances but not all distances. Show that if f : R2 → R2 preserves all rational distances then it preserves all distances. A7.  Show that for any given positive integer n, the number of odd nCm with 0 ≤ m ≤ n is a power of 2. B1.  The differential equation a(x, y) dx + b(x, y) dy = 0 is homogeneous and exact (meaning that a(x, y) and b(x, y) are homogeneous polynomials of the same degree and that ∂a/∂y = ∂b/∂x). Show that the solution y = y(x) satisfies x a(x, y) + y b(x, y) = c, for some constant c. B2.  Let P be the set of all subsets of the plane. f : P → P satisfies f(X ∪ Y) ⊇ f( f(X) ) ∪ f(Y) ∪ Y for all X, Y ∈ P (*). Show that (1) f(X) ⊇ X, (2) f( f(X) ) = f(X) , (3) if X ⊇ Y, then f(X) ⊇ f(Y), for all X, Y ∈ P. Show conversely that if f : P → P satisfies (1), (2), (3), then f satisfies (*). B3.  ABCD is an arbitrary tetrahedron. The inscribed sphere touches ABC at S, ABD at R, ACD at Q and BCD at P. Show that the four sets of angles {ASB, BSC, CSA}, {ARB, BRD, DRA}, {AQC, CQD, DQA}, {BPC, CPD, DPB} are the same. B4.  Show that for any triangle ABC, we have sin A cos C + A cos B > 0. B5.  Show that a graph with 2n points and n2 + 1 edges necessarily contains a 3-cycle, but that we can find a graph with 2n points and n2 edges without a 3-cycle. B6.  The sequence an is defined by a1 = 2, an+1 = an2 - an + 1. Show that any pair of values in the sequence are relatively prime and that ∑ 1/an = 1. B7.  p(z) and q(z) are complex polynomials with the same set of roots (but possibly different multiplicities). p(z) + 1 and q(z) + 1 also have the same set of roots. Show that p(z) = q(z).

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and solutions are available in: A M Gleason, R E Greenwood & L M Kelly, The William Lowell Putnam Mathematical Competition, Problems and Solutions, 1938-1964, MAA 1980. Out of print, but available in some university libraries.

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