31st Putnam 1970


A1. e^bx cos cx is expanded in a Taylor series ∑ a_n xⁿ. b and c are positive reals. Show that either all a_n are non-zero, or infinitely many a_n are zero.
A2. p(x, y) = a x² + b x y + c y² is a homogeneous real polynomial of degree 2 such that b² < 4ac, and q(x, y) is a homogeneous real polynomial of degree 3. Show that we can find k > 0 such that p(x, y) = q(x, y) has no roots in the disk x² + y² < k except (0, 0).
A3. A perfect square has length n if its last n digits (in base 10) are the same and non-zero. What is the longest possible length? What is the smallest square achieving this length?
A4. The real sequence a₁, a₂, a₃, ... has the property that lim_n→∞ (a_n+2 - a_n) = 0. Prove that lim_n→∞ (a_n+1 - a_n)/n = 0.
A5. Find the radius of the largest circle on an ellipsoid with semi-axes a > b > c.
A6. x is chosen at random from the interval [0, a] (with the uniform distribution). y is chosen similarly from [0, b], and z from [0, c]. The three numbers are chosen independently, and a ≥ b ≥ c. Find the expected value of min(x, y, z).
B1. Let f(n) = (n² + 1)(n² + 4)(n² + 9) ... (n² + (2n)²). Find lim_n→∞ f(n)^1/n/n⁴.
B2. A weather station measures the temperature T continuously. It is found that on any given day T = p(t), where p is a polynomial of degree <= 3, and t is the time. Show that we can find times t₁ < t₂, which are independent of p, such that the average temperature over the period 9am to 3pm is ( p(t₁) + p(t₂) / 2. Show that t₁ ≈ 10:16am, t₂ ≈ 1:44pm.
B3. S is a closed subset of the real plane. Its projection onto the x-axis is bounded. Show that its projection onto the y-axis is closed.
B4. A vehicle covers a mile (= 5280 ft) in a minute, starting and ending at rest and never exceeding 90 miles/hour. Show that its acceleration or deceleration exceeded 6.6 ft/sec².
B5. k_n(x) = -n on (-∞, -n], x on [-n, n], and n on [n, ∞). Prove that the (real valued) function f(x) is continuous iff all k_n( f(x) ) are continuous.
B6. The quadrilateral Q contains a circle which touches each side. It has side lengths a, b, c, d and area √(abcd). Prove it is cyclic.

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 78 (1971) 765-9. 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.