31st Putnam 1970

A1.  ebx cos cx is expanded in a Taylor series ∑ an xn. b and c are positive reals. Show that either all an are non-zero, or infinitely many an 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 limn→∞ (an+2 - an) = 0. Prove that limn→∞ (an+1 - an)/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 limn→∞ 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.  kn(x) = -n on (-∞, -n], x on [-n, n], and n on [n, ∞). Prove that the (real valued) function f(x) is continuous iff all kn( 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.

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© John Scholes
25 Aug 2001