FIFTY-SECOND ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 7, 1991 Examination A

A-1. A 2 × 3 rectangle has vertices at (0, 0), (2, 0), (0, 3), and (2, 3). It rotates 90^{∘} clockwise about the point (2, 0). It then rotates 90^{∘} clockwise about the point (5, 0), then 90^{∘} clockwise about the point (7, 0), and finally, 90^{∘} clockwise about the point (10, 0). (The side originally on the x-axis is now back on the x-axis.) Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is (1, 1).

A-2. Let A and B be different n × n matrices with real entries. If A^{3} = B^{3} and A^{2} B = B^{2}A, can A^{2} + B^{2} be invertible?

A-3. Find all real polynomials p(x) of degree n ≥ 2 for which there exist real numbers r_{1} < r_{2} < ⋯ < r_{n} such that

(i) p(r_{i}) = 0, i = 1, 2, …, n,

and

(ii) p′ (\frac{r_{i} + r_{i+1}}{2} ) = 0 i = 1, 2, …, n − 1,

where p′(x) denotes the derivative of p(x).

A-4. Does there exist an infinite sequence of closed discs D_{1}, D_{2}, D_{3}, … in the plane, with centers c_{1}, c_{2}, c_{3}, …, respectively, such that

(i) the c_{i} have no limit point in the finite plane, (ii) the sum of the areas of the D_{i} is finite, and (iii) every line in the plane intersects at least one of the D_{i}?

A-5. Find the maximum value of

∫_{0}^{y} \sqrt{x^{4} + (y − y^{2})^{2}} dx

for 0 ≤ y ≤ 1.

A-6. Let A(n) denote the number of sums of positive integers a_{1} + a_{2} + ⋯ + a_{r} that add up to n with a_{1} > a_{2} + a_{3}, a_{2} > a_{3} + a_{4}, …, a_{r−2} > a_{r−1} + a_{r}, a_{r−1} > a_{r}. Let B(n) denote the number of b_{1} + b_{2} + ⋯ + b_{s} that add up to n, with

(i) b_{1} ≥ b_{2} ≥ … ≥ b_{s},

(ii) each b_{i} is in the sequence 1, 2, 4, …, g_{j}, … defined by g_{1} = 1, g_{2} = 2, and g_{j} =

g_{j−1} + g_{j−2} + 1, and

(iii) if b_{1} = g_{k} then every element in {1, 2, 4, …, g_{k}} appears at least once as a b_{i}. Prove that A(n) = B(n) for each n ≥ 1. (For example, A(7) = 5 because the relevant sums are 7, 6 + 1, 5 + 2, 4 + 3, 4 + 2 + 1,

and B(7) = 5 because the relevant sums are 4 + 2 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1.)

FIFTY-SECOND ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 7, 1991 Examination B

B-1. For each integer n ≥ 0, let S(n) = n − m^{2}, where m is the greatest integer with m^{2} ≤ n. Define a sequence (a_{k})_{k=0}^{∞} by a_{0} = A and a_{k+1} = a_{k} + S(a_{k}) for k ≥ 0. For what positive integers A is this sequence eventually constant?

B-2. Suppose f and g are non-constant, differentiable, real-valued functions on R. Furthermore, suppose that for each pair of real numbers x and y,

\begin{aligned}f(x + y) & = f(x)f(y) − g(x)g(y), \\ g(x + y) & = f(x)g(y) + g(x)f(y). \end{aligned}

If f′(0) = 0, prove that (f(x))^{2} + (g(x))^{2} = 1 for all x.

B-3. Does there exist a real number L such that, if m and n are integers greater than L, then an m × n rectangle may be expressed as a union of 4 × 6 and 5 × 7 rectangles, any two of which intersect at most along their boundaries?

B-4. Suppose p is an odd prime. Prove that

∑_{j=0}^{p} \binom{p}{j}\binom{p + j}{j} ≡ 2^{p} + 1 (mod p^{2}).

B-5. Let p be an odd prime and let Z_{p} denote (the field of) the integers modulo p. How many elements are in the set

{x^{2} : x ∈ Z_{p}} ∩ {y^{2} + 1 : y ∈ Z_{p}}?

B-6. Let a and b be positive numbers. Find the largest number c, in terms of a and b, such that

a^{x}b^{1−x} ≤ a\frac{sinh ux}{sinh u} + b\frac{sinh u(1 − x)}{sinh u}

for all u with 0 < |u| ≤ c and for all x, 0 < x < 1. (Note: sinh u = (e^{u} − e^{−u})/2.)