FORTY-FIFTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION^{†}

Saturday, December 1, 1984 Examination A

A-1. Let A be a solid a × b × c rectangular brick in three dimensions, where a, b, c > 0. Let B be the set of all points which are a distance at most one from some point of A (in particular, B contains A). Express the volume of B as a polynomial in a, b, and c.

A-2. Express ∑_{k=1}^{∞} \frac{6^{k}}{(3^{k+1} − 2^{k+1})(3^{k} − 2^{k})} as a rational number

A-3. Let n be a positive integer. Let a, b, x be real numbers, with a ≠ b, and let M_{n} denote the 2n × 2n matrix whose (i, j) entry m_{ij} is given by

m_{ij} = \{ \begin{matrix}x & if i = j, \\ a & if i ≠ j and i + j is even, \\ b & if i ≠ j and i + j is odd. \end{matrix}

Thus, for example, M_{2} = ( \begin{matrix}x & b & a & b \\ b & x & b & a \\ a & b & x & b \\ b & a & b & x \end{matrix} ). Express lim_{x→a} det M_{n}/(x − a)^{2n−2} as a

polynomial in a, b, and n, where det M_{n} denotes the determinant of M_{n}.

A-4. A convex pentagon P = ABCDE, with vertices labeled consecutively, is inscribed in a circle of radius 1. Find the maximum area of P subject to the condition that the chords AC and BD be perpendicular.

A-5. Let R be the region consisting of all triples (x, y, z) of nonnegative real numbers satisfying x + y + z ≤ 1. Let w = 1 − x − y − z. Express the value of the triple integral

∫∫∫_{R} x^{1}y^{9}z^{8}w^{4} dx dy dz

in the form a!b!c!d!/n!, where a, b, c, d, and n are positive integers.

A-6. Let n be a positive integer, and let f(n) denote the last nonzero digit in the decimal expansion of n!. For instance, f(5) = 2.

(a) Show that is a_{1}, a_{2}, …, a_{k} are distinct nonnegative integers, then f(5^{a_{1}} + 5^{a_{2}} + ⋯ + 5^{a_{k}}) depends only on the sum a_{1} + a_{2} + ⋯ + a_{k}.

(b) Assuming part (a), we can define

g(s) = f(5^{a_{1}} + 5^{a_{2}} + ⋯ + 5^{a_{k}}),

where s = a_{1} + a_{2} + ⋯ + a_{k}. Find the least positive integer p for which

g(s) = g(s + p), for all s ≥ 1,

or else show that no such p exists.

† Questions Commitee: Melvin Hochster (Chair), Bruce Reznick, Richard P. Stanley

FORTY-FIFTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 1, 1984 Examination B

B-1. Let n be a positive integer, and define

f(n) = 1! + 2! + ⋯ n!.

Find polynomials P (x) and Q(x) such that

f(n + 2) = P (n)f(n + 1) + Q(n)f(n),

for all n ≥ 1.

B-2. Find the minimum value of

(u − v)^{2} + (\sqrt{2 − u^{2}} − \frac{9}{v} )^{2}

for 0 < u < \sqrt{2} and v > 0.

B-3. Prove or disprove the following statement: If F is a finite set with two or more elements, then there exists a binary operation ∗ on F such that for all x, y, z in F,

(i) x ∗ z = y ∗ z implies x = y (right cancellation holds), and

(ii) x ∗ (y ∗ z) ≠ (x ∗ y) ∗ z (no case of associativity holds).

B-4. Find, with proof, all real-valued functions y = g(x) defined and continuous on [0, ∞), positive on (0, ∞), such that for all x > 0 the y-coordinate of the centroid of the region

R_{x} = {(s, t) : 0 ≤ s ≤ x, 0 ≤ t ≤ g(s)}

is the same as the average value of g on [0, x].

B-5. For each nonnegative integer k, let d(k) denote the number of 1’s in the binary expansion of k (for example, d(0) = 0 and d(5) = 2). Let m be a positive integer. Express

∑_{k=0}^{2^{m}−1}(−1)^{d(k)}k^{m}

in the form (−1)^{m}a^{f(m)}(g(m))!, where a is an integer and f and g are polynomials.

B-6. A sequence of convex polygons {P_{n}}, n ≥ 0, is defined inductively as follows. P_{0} is an equilateral triangle with sides of length 1. Once P_{n} has been determined, its sides are trisected; the vertices of P_{n+1} are the interior trisection points of the sides of P_{n}. Thus, P_{n+1} is obtained by cutting corners off P_{n}, and P_{n} has 3·2^{n} sides. (P_{1} is a regular hexagon with sides of length 1/3.)

Express lim_{n→∞} Area(P_{n}) in the form \sqrt{a}/b, where a and b are positive integers.