16th Putnam 1956


A1. α ≠ 1 is a positive real. Find lim_x→∞ ( (α^x - 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 a_n is defined by a₁ = 2, a_n+1 = a_n² - a_n + 1. Show that any pair of values in the sequence are relatively prime and that ∑ 1/a_n = 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.