FORTY-FIRST ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 6, 1980 Examination A

A1. Let b and c be fixed real numbers and let the ten points (j, y_{j}), j = 1, 2, …, 10, lie on the parabola y = x^{2} + bx + c. For j = 1, 2, …, 9, let I_{j} be the point of intersection of the tangents to the given parabola at (j, y_{j}) and (j + 1, y_{j+1}). Determine the polynomial function y = g(x) of least degree whose graph passes through all nine points I_{j}.

A2. Let r and s be positive integers. Derive a formula for the number of ordered quadruples (a, b, c, d) of positive integers such that

3^{r} · 7^{s} = lcm[a, b, c] = lcm[a, b, d] = lcm[a, c, d] = lcm[b, c, d].

The answer should be a function of r and s. (Note that lcm[x, y, z] denotes the least common multiple of x, y, z.)

A3. Evaluate

∫_{0}^{π/2} \frac{dx}{1 + (tan x)^{\sqrt{2}}} .

A4.

(a) Prove that there exist integers a, b, c, not all zero and each of absolute value less than one million, such that

|a + b\sqrt{2} + c\sqrt{3}| < 10^{−11}.

(b) Let a, b, c be integers, not all zero and each of absolute value less than one million. Prove that

|a + b\sqrt{2} + c\sqrt{3}| > 10^{−21}.

A5. Let P (t) be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations

0 = ∫_{0}^{x} P (t) sin t dt = ∫_{0}^{x} P (t) cos t dt

has only finitely many real solutions x.

A6. Let C be the class of all real valued continuously differentiable functions f on the interval 0 ≤ x ≤ 1 with f(0) = 0 and f(1) = 1. Determine the largest real number u such that

u ≤ ∫_{0}^{1} |f′(x) − f(x)| dx

for all f in C.

FORTY-FIRST ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 6, 1980 Examination B

B1. For which real numbers c is (e^{x} + e^{−x})/2 ≤ e^{cx^{2}} for all real x.

B2. Let S be the solid in three-dimensional space consisting of all points (x, y, z) satisfying the following system of six simultaneous conditions:

x ≥ 0, y ≥ 0, z ≥ 0,
x + y + z ≤ 11,
2x + 4y + 3z ≤ 36,
2x + 3z ≤ 24.

(a) Determine the number of vertices v of S.

(b) Determine the number of edges e of S.

(c) Sketch in the bc-plane the set of points (b, c) such that (2, 5, 4) is one of the points (x, y, z) at which the linear function bx + cy + z assumes its maximum value on S.

B3. For which real numbers a does the sequence defined by the initial condition u_{0} = a and the recursion u_{n+1} = 2u_{n} − n^{2} have u_{n} > 0 for all n ≥ 0? (Express the answer in the simplest form.)

B4. Let A_{1}, A_{2}, …, A_{1066} be subsets of a finite set X such that |A_{i}| > \frac{1}{2}|X| for 1 ≤ i ≤ 1066. Prove that there exist ten elements x_{1}, …, x_{10} of X such that every A_{i} contains at least one of x_{1}, …, x_{10}. (Here |S| means the number of elements in the set S.)

B5. For each t ≥ 0, let S_{t} be the set of all nonnegative, increasing, convex, continuous, real-valued functions f(x) defined on the closed interval [0, 1] for which

f(1) − 2f(2/3) + f(1/3) ≥ t[f(2/3) − 2f(1/3) + f(0)].

Develop necessary and sufficient conditions on t for S_{t} to be closed under multiplication. (This closure means that, if the functions f(x) and g(x) are in S_{t}, so is their product f(x)g(x). A function f(x) is convex if and only if f(su + (1 − s)v) ≤ sf(u) + (1 − s)f(v) whenever 0 ≤ s ≤ 1.)

B6. An infinite array of rational numbers G(d, n) is defined for integers d and n with 1 ≤ d ≤ n as follows:

G(1, n) = \frac{1}{n}, G(d, n) = \frac{d}{n} ∑_{i=d}^{n} G(d − 1, i − 1) for d > 1.

For 1 < d ≤ p and p prime, prove that G(d, p) is expressible as a quotient s/t of integers s and t with t not an integral multiple of p. (For example, G(3, 5) = 7/4 with the denominator 4 not a multiple of 5.)