### 20th Putnam 1959

 A1.  Prove that we can find a real polynomial p(y) such that p(x - 1/x) = xn - 1/xn (where n is a positive integer) iff n is odd. A2.  Let ω3 = 1, ω ≠ 1. Show that z1, z2, -ωz1 - ω2z2 are the vertices of an equilateral triangle. A3.  Let C be the complex numbers. f : C → C satisfies f(z) + z f(1 - z) = 1 + z for all z. Find f. A4.  R is the reals. f, g : [0, 1] → R are arbitary functions. Show that we can find x, y such that |xy - f(x) - g(y)| ≥ 1/4. A5.  At a particular moment, A, T and B are in a vertical line, with A 50 feet above T, and T 100 feet above B. T flies in a horizontal line at a fixed speed. A flies at a fixed speed directly towards B, B flies at twice T's speed, also directly towards T. A and B reach T simultaneously. Find the distance traveled by each of A, B and T, and A's speed. A6.  Given any real numbers α1, α2, ... , αm, β, show that for m, n > 1 we can find m real n x n matrices A1, ... , Am such that det Ai = αi, and det(∑ Ai) = β. A7.  Let R be the reals. Let f : [a, b] → R have a continuous derivative, and suppose that if f(x) = 0, then f '(x) ≠ 0. Show that we can find g : [a, b] → R with a continuous derivative, such that f(x)g'(x) > f '(x)g(x) for all x ∈ [a, b]. B1.  Join each of m points on the positive x-axis to each of n points on the positive y-axis. Assume that no three of the resulting segments are concurrent (except at an endpoint). How many points of intersection are there (excluding endpoints)? B2.  Show that any positive real can be expressed in infinitely many ways as a sum ∑ 1/(10 an), where a1 < a2 < a3 < ... are positive integers. B3.  Find a continuous function f : [0, 1] → [0, 1] such that given any β ∈ [0, 1], we can find infinitely many α such that f(α) = β. B4.  A is the 5 x 5 array: ```11 17 25 19 16 24 10 13 15 3 12 5 14 2 18 23 4 1 8 22 6 20 7 21 9 ``` Pick 5 elements, one from each row and column, whose minimum is as large as possible (and prove it so). B5.  L1 is the line { (t + 1, 2t - 4, -3t + 5) : t real } and L2 is the line { (4t - 12, -t + 8, t + 17) : t real }. Find the smallest sphere touching L1 and L2. B6.  α and β are positive irrational numbers satisfying 1/α + 1/β = 1. Let an = [n α] and bn = [n β], for n = 1, 2, 3, ... . Show that the sequences an and bn are disjoint and that every positive integer belongs to one or the other. B7.  Given any finite ordered tuple of real numbers X, define a real number [X], so that for all xi, α: ```(1) [X] is unchanged if we permute the order of the numbers in the tuple X; (2) [(x1 + α, x2 + α, ... , xn + α)] = [(x1, x2, ... , xn)] + α; (3) [(-x1, -x2, ... , -xn)] = - [(x1, x2, ... , xn)]; (4) for y1 = y2 = ... = yn = [(x1, x2, ... , xn)], we have [(y1, y2, ... , yn, xn+1)] = [(x1, x2, ... , xn+1)]. ``` Show that [(x1, x2, ... , xn)] = (x1 + x2 + ... + xn)/n.

To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems.

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