| A1. Let f(n) = ∑1n [r/2]. Show that f(m + n) - f(m - n) = mn for m > n > 0. |
|
| A2. A triangle has sides a, b, c. The radius of the inscribed circle is r and s = (a + b + c)/2. Show that 1/(s - a)2 + 1/(s - b)2 + 1/(s - c)2 ≥ 1/r2. |
|
| A3. Define the sequence {an} by a1 ∈ (0, 1), and an+1 = an(1 - an). Show that limn→∞n an = 1. |
|
| A4. Delete all the squares from the sequence 1, 2, 3, ... . Show that the nth number remaining is n + m, where m is the nearest integer to √n. |
|
| A5. Let S be the set of continuous real-valued functions on the reals. φ :S → S is a linear map such that if f, g ∈ S and f(x) = g(x) on an open interval (a, b), then φf = φg on (a, b). Prove that for some h ∈ S, (φf)(x) = h(x)f(x) for all f and x. |
|
| A6. Let an = √(1 + 2 √(1 + 3 √(1 + 4 √(1 + 5 √( ... + (n - 1) √(1 + n) ... ) ) ) ) ). Prove lim an = 3. |
|
| B1. A convex polygon does not extend outside a square side 1. Prove that the sum of the squares of its sides is at most 4. |
|
| B2. Prove that at least one integer in any set of ten consecutive integers is relatively prime to the others in the set. |
|
| B3. an is a sequence of positive reals such that ∑ 1/an converges. Let sn = ∑1n ai. Prove that ∑ n2an/sn2 converges. |
|
| B4. Given a set of (mn + 1) unequal positive integers, prove that we can either (1) find m + 1 integers biin the set such that bi does not divide bj for any unequal i, j, or (2) find n+1 integers ai in the set such that ai divides ai+1 for i = 1, 2, ... , n. |
|
| B5. Given n points in the plane, no three collinear, prove that we can label them Pi so that P1P2P3 ... Pn is a simple closed polygon (with no edge intersecting any other edge except at its endpoints). |
|
| B6. y = f(x) is a solution of y'' + exy = 0. Prove that f(x) is bounded. |
|
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 74 (1967) 772-7. 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.
Putnam home
© John Scholes
jscholes@kalva.demon.co.uk
9 Oct 1999