The Forty-Sixth Annual William Lowell Putnam Competition
Saturday, December 7, 1985
A–1 Determine, with proof, the number of ordered triples has exactly k nonzero coefficients. Find, with proof, a
(A1 , A2 , A3 ) of sets which have the property that set of integers m1 , m2 , m3 , m4 , m5 for which this mini-
mum k is achieved.
(i) A1 ∪ A2 ∪ A3 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and B–2 Define polynomials fn (x) for n ≥ 0 by f0 (x) = 1,
(ii) A1 ∩ A2 ∩ A3 = 0.
/ fn (0) = 0 for n ≥ 1, and
Express your answer in the form 2a 3b 5c 7d , where d
a, b, c, d are nonnegative integers. fn+1 (x) = (n + 1) fn (x + 1)
dx
A–2 Let T be an acute triangle. Inscribe a rectangle R in T for n ≥ 0. Find, with proof, the explicit factorization of
with one side along a side of T . Then inscribe a rectan- f100 (1) into powers of distinct primes.
gle S in the triangle formed by the side of R opposite the
side on the boundary of T , and the other two sides of T , B–3 Let
with one side along the side of R. For any polygon X,
a1,1 a1,2 a1,3 ...
let A(X) denote the area of X. Find the maximum value,
a2,1 a2,2 a2,3 ...
or show that no maximum exists, of A(R)+A(S)
A(T ) , where T a3,1 a3,2 a3,3 ...
ranges over all triangles and R, S over all rectangles as .. .. .. ..
above. . . . .
A–3 Let d be a real number. For each integer m ≥ 0, define be a doubly infinite array of positive integers, and sup-
a sequence {am ( j)}, j = 0, 1, 2, . . . by the condition pose each positive integer appears exactly eight times in
the array. Prove that am,n > mn for some pair of positive
am (0) = d/2m , integers (m, n).
am ( j + 1) = (am ( j))2 + 2am ( j), j ≥ 0. B–4 Let C be the unit circle x2 + y2 = 1. A point p is chosen
randomly on the circumference C and another point q
Evaluate limn→∞ an (n). is chosen randomly from the interior of C (these points
A–4 Define a sequence {ai } by a1 = 3 and ai+1 = 3ai for i ≥ are chosen independently and uniformly over their do-
1. Which integers between 00 and 99 inclusive occur as mains). Let R be the rectangle with sides parallel to the
the last two digits in the decimal expansion of infinitely x and y-axes with diagonal pq. What is the probability
many ai ? that no point of R lies outside of C?
−1 )
B–5 Evaluate 0∞ t −1/2 e−1985(t+t
R
A–5 Let Im = 02π cos(x) cos(2x) · · · cos(mx) dx. For which
R dt. You may assume that
R ∞ −x2 √
integers m, 1 ≤ m ≤ 10 is Im 6= 0? −∞ e dx = π.
A–6 If p(x) = a0 +a1 x+· · ·+am xm is a polynomial with real B–6 Let G be a finite set of real n × n matrices {Mi }, 1 ≤
coefficients ai , then set i ≤ r, which form a group under matrix multiplication.
Suppose that ∑ri=1 tr(Mi ) = 0, where tr(A) denotes the
Γ(p(x)) = a20 + a21 + · · · + a2m . trace of the matrix A. Prove that ∑ri=1 Mi is the n × n
zero matrix.
Let F(x) = 3x2 + 7x + 2. Find, with proof, a polynomial
g(x) with real coefficients such that
(i) g(0) = 1, and
(ii) Γ( f (x)n ) = Γ(g(x)n )
for every integer n ≥ 1.
B–1 Let k be the smallest positive integer for which there
exist distinct integers m1 , m2 , m3 , m4 , m5 such that the
polynomial
p(x) = (x − m1 )(x − m2 )(x − m3 )(x − m4 )(x − m5 )
