### 30th Putnam 1969

 A1.  R2 represents the usual plane (x, y) with -∞ < x, y < ∞. p: R2 → R is a polynomial with real coefficients. What are the possibilities for the image p(R2)? A2.  A is an n x n matrix with elements aij = |i - j|. Show that the determinant |A| = (-1)n-1 (n - 1) 2n-2. A3.  An n-gon (which is not self-intersecting) is triangulated using m interior vertices. In other words, there is a set of N triangles such that: (1) their union is the original n-gon; (2) the union of their vertices is the set consisting of the n vertices of the n-gon and the m interior vertices; (3) the intersection of any two distinct triangles in the set is either empty, a vertex of both triangles, or a side of both triangles. What is N? A4.  Prove that ∫01 xx dx = 1 - 1/22 + 1/33 - 1/44 + ... . A5.  u(t) is a continuous function. x(t), y(t) is the solution of x' = -2y + u(t), y' = -2x + u(t) satisfying the initial condition x(0) = x0, y(0) = y0. Show that if x0 ≠ y0, then we do not have x(t) = y(t) = 0 for any t, but that given any x0 = y0 and any T > 0, we can always find some u(t) such that x(T) = y(T) = 0. A6.  The sequence a1 + 2a2, a2 + 2a3, a3 + 2a4, ... converges. Prove that the sequence a1, a2, a3, ... also converges. B1.  The positive integer n is divisible by 24. Show that the sum of all the positive divisors of n - 1 (including 1 and n - 1) is also divisible by 24. B2.  G is a finite group with identity 1. Show that we cannot find two proper subgroups A and B (≠ {1} or G) such that A ∪ B = G. Can we find three proper subgroups A, B, C such that A ∪ B ∪ C = G? B3.  The sequence a1, a2, a3, ... satisfies a1a2 = 1, a2a3 = 2, a3a4 = 3, a4a5 = 4, ... . Also, limn→∞an/an+1 = 1. Prove that a1 = √(2/π). B4.  Γ is a plane curve of length 1. Show that we can find a closed rectangle area 1/4 which covers Γ. B5.  The sequence ai, i = 1, 2, 3, ... is strictly monotonic increasing and the sum of its inverses converges. Let f(x) = the largest i such that ai < x. Prove that f(x)/x → 0 as x → ∞. B6.  M is a 3 x 2 matrix, N is a 2 x 3 matrix. MN = ``` 8 2 -2 2 5 4 -2 4 5 ``` Show that NM = ``` 9 0 0 9 ```

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 77 (1970) 723-8. 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 available in some university libraries.

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