SIXTIETH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 4, 1999 Examination A

A1. Find polynomials f(x), g(x) and h(x), if they exist, such that, for all x,

|f(x)| − |g(x)| + h(x) = \{ \begin{matrix}−1 & if x < −1 \\ 3x + 2 & if −1 ≤ x ≤ 0 \\ −2x + 2 & if x > 0. \end{matrix}

A2. Let p(x) be a polynomial that is non-negative for all x. Prove that, for some k, there are polynomials f_{1}(x), …, f_{k}(x) such that

p(x) = ∑_{j=1}^{k} (f_{j}(x))^{2} .

A3. Consider the power series expansion

\frac{1}{1 − 2x − x^{2}} = ∑_{n=0}^{∞} a_{n}x^{n}.

Prove that, for each integer n ≥ 0, there is an integer m such that

a_{n}^{2} + a_{n+1}^{2} = a_{m}.

A4. Sum the series

∑_{m=1}^{∞} ∑_{n=1}^{∞} \frac{m^{2}n}{3^{m}(n3^{m} + m3^{n})}.

A5. Prove that there is a constant C such that, if p(x) is a polynomial of degree 1999, then

|p(0)| ≤ C ∫_{−1}^{1} |p(x)| dx.

A6. The sequence (a_{n})_{n≥1} is defined by a_{1} = 1, a_{2} = 2, a_{3} = 24, and, for n ≥ 4,

a_{n} = \frac{6a_{n−1}^{2}a_{n−3} − 8a_{n−1}a_{n−2}^{2}}{a_{n−2}a_{n−3}} .

Show that, for all n, a_{n} is an integer multiple of n.

SIXTIETH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 4, 1999 Examination B

B1. Right triangle ABC has right angle at C and ^{}B̸AC = θ; the point D is chosen on AB so that |AC| = |AD| = 1; the point E is chosen on BC so that ^{}C̸DE = θ. The perpendicular to BC at E meets AB at F. Evaluate lim_{θ→0} |EF |. [Here |P Q| denotes the length of the line segment P Q.]

A  B C D E F θ  θ

B2. Let P (x) be a polynomial of degree n such that P (x) + Q(x)P′′(x), where Q(x) is a quadratic polynomial and P′′(x) is the second derivative of P (x). Show that if P (x) has at least two distinct roots then it must have n distinct roots. [The roots may be either real or complex.]

B3. Let A = {(x, y) : 0 ≤ x, y < 1}. For (x, y) ∈ A, let

S(x, y) = ∑_{\frac{1}{2} ≤ \frac{m}{n} ≤2} x^{m}y^{n},

where the sum ranges over all pairs (m, n) of positive integers satisfying the indicated inequalities. Evaluate

lim_{{(x,y)→(1,1) \atop (x,y)∈A}}(1 − xy^{2})(1 − x^{2}y)S(x, y).

B4. Let f be a real function with a continuous third derivative such that f(x), f′(x), f′′(x), f′′′(x) are positive for all x. Suppose that f′′′(x) ≤ f(x) for all x. Show that f′(x) < 2f(x) for all x.

B5. For an integer n ≥ 3, let θ = 2π/n. Evaluate the determinant of the n × n matrix I + A, where I is the n × n identity matrix and A = (a_{jk}) has entries a_{jk} = cos(jθ + kθ) for all j, k.

B6. Let S be a finite set of integers, each greater than 1. Suppose that for each integer n there is some s ∈ S such that gcd(s, n) = 1 or gcd(s, n) = s. Show that there exist s, t ∈ S such that gcd(s, t) is prime. [Here gcd(a, b) denotes the greatest common divisor of a and b.]