2019 William Lowell Putnam Mathematical Competition
Problems
A1: Determine all possible values of the expression
A3 + B 3 + C 3 − 3ABC
where A, B, and C are nonnegative integers.
A2: In the triangle ∆ABC, let G be the centroid, and let I be the center of the inscribed circle.
Let α and β be the angles at the vertices A and B, respectively. Suppose that the segment
IG is parallel to AB and that β = 2 tan−1 (1/3). Find α.
A3: Given real numbers b0 , b1 , . . . , b2019 with b2019 6= 0, let z1 , z2 , . . . , z2019 be the roots in the
complex plane of the polynomial
2019
X
P (z) = bk z k .
k=0
Let µ = (|z1 | + · · · + |z2019 |)/2019 be the average of the distances from z1 , z2 , . . . , z2019
to the origin. Determine the largest constant M such that µ ≥ M for all choices of
b0 , b1 , . . . , b2019 that satisfy
1 ≤ b0 < b1 < b2 < · · · < b2019 ≤ 2019 .
A4: Let f be a continuous real-valued function on R3 . Suppose that for every sphere S of radius
1, the integral of f (x, y, z) over the surface of S equals 0. Must f (x, y, z) be identically 0?
A5: Let p be an odd prime number, and let Fp denote the field of integers modulo p. Let Fp [x]
be the ring of polynomials over Fp , and let q(x) ∈ Fp [x] be given by
p−1
X
q(x) = ak xk ,
k=1
where
ak = k (p−1)/2 mod p .
Find the greatest nonnegative integer n such that (x − 1)n divides q(x) in Fp [x].
A6: Let g be a real-valued function that is continuous on the closed interval [0, 1] and twice
differentiable on the open interval (0, 1). Suppose that for some real number r > 1,
g(x)
lim = 0.
x→0+ xr
Prove that either
lim g 0 (x) = 0 or lim sup xr |g 00 (x)| = ∞ .
x→0+ x→0+
, B1: Denote by Z2 the set of all points (x, y) in the plane with integer coordinates. For each
integer n ≥ 0, let Pn be the subset of Z2 consisting of the point (0, 0) together with all
points (x, y) such that x2 + y 2 = 2k for some integer k ≤ n. Determine, as a function of n,
the number of four-point subsets of Pn whose elements are the vertices of a square.
B2: For all n ≥ 1, let
n−1 (2k−1)π
X sin 2n
an = (k−1)π
.
cos2 kπ
2
k=1 cos 2n 2n
Determine
an
lim .
n→∞ n3
B3: Let Q be an n-by-n real orthogonal matrix, and let u ∈ Rn be a unit column vector (that
is, uT u = 1). Let P = I − 2uuT , where I is the n-by-n identity matrix. Show that if 1 is
not an eigenvalue of Q , then 1 is an eigenvalue of P Q .
B4: Let F be the set of functions f (x, y) that are twice continuously differentiable for x ≥ 1,
y ≥ 1 and that satisfy the following two equations (where subscripts denote partial deriva-
tives):
xfx + yfy = xy ln(xy) ,
x2 fxx + y 2 fyy = xy .
For each f ∈ F, let
m(f ) = min f (s + 1, s + 1) − f (s + 1, s) − f (s, s + 1) + f (s, s) .
s≥1
Determine m(f ), and show that it is independent of the choice of f .
B5: Let Fm be the mth Fibonacci number, defined by F1 = F2 = 1 and Fm = Fm−1 + Fm−2
for all m ≥ 3. Let p(x) be the polynomial of degree 1008 such that p(2n + 1) = F2n+1 for
n = 0, 1, 2, . . . , 1008. Find integers j and k such that p(2019) = Fj − Fk .
B6: Let Zn be the integer lattice in Rn . Two points in Zn are called neighbors if they differ
by exactly 1 in one coordinate and are equal in all other coordinates. For which integers
n ≥ 1 does there exist a set of points S ⊂ Zn satisfying the following two conditions?
(1) If p is in S, then none of the neighbors of p is in S.
(2) If p ∈ Zn is not in S, then exactly one of the neighbors of p is in S.
Problems
A1: Determine all possible values of the expression
A3 + B 3 + C 3 − 3ABC
where A, B, and C are nonnegative integers.
A2: In the triangle ∆ABC, let G be the centroid, and let I be the center of the inscribed circle.
Let α and β be the angles at the vertices A and B, respectively. Suppose that the segment
IG is parallel to AB and that β = 2 tan−1 (1/3). Find α.
A3: Given real numbers b0 , b1 , . . . , b2019 with b2019 6= 0, let z1 , z2 , . . . , z2019 be the roots in the
complex plane of the polynomial
2019
X
P (z) = bk z k .
k=0
Let µ = (|z1 | + · · · + |z2019 |)/2019 be the average of the distances from z1 , z2 , . . . , z2019
to the origin. Determine the largest constant M such that µ ≥ M for all choices of
b0 , b1 , . . . , b2019 that satisfy
1 ≤ b0 < b1 < b2 < · · · < b2019 ≤ 2019 .
A4: Let f be a continuous real-valued function on R3 . Suppose that for every sphere S of radius
1, the integral of f (x, y, z) over the surface of S equals 0. Must f (x, y, z) be identically 0?
A5: Let p be an odd prime number, and let Fp denote the field of integers modulo p. Let Fp [x]
be the ring of polynomials over Fp , and let q(x) ∈ Fp [x] be given by
p−1
X
q(x) = ak xk ,
k=1
where
ak = k (p−1)/2 mod p .
Find the greatest nonnegative integer n such that (x − 1)n divides q(x) in Fp [x].
A6: Let g be a real-valued function that is continuous on the closed interval [0, 1] and twice
differentiable on the open interval (0, 1). Suppose that for some real number r > 1,
g(x)
lim = 0.
x→0+ xr
Prove that either
lim g 0 (x) = 0 or lim sup xr |g 00 (x)| = ∞ .
x→0+ x→0+
, B1: Denote by Z2 the set of all points (x, y) in the plane with integer coordinates. For each
integer n ≥ 0, let Pn be the subset of Z2 consisting of the point (0, 0) together with all
points (x, y) such that x2 + y 2 = 2k for some integer k ≤ n. Determine, as a function of n,
the number of four-point subsets of Pn whose elements are the vertices of a square.
B2: For all n ≥ 1, let
n−1 (2k−1)π
X sin 2n
an = (k−1)π
.
cos2 kπ
2
k=1 cos 2n 2n
Determine
an
lim .
n→∞ n3
B3: Let Q be an n-by-n real orthogonal matrix, and let u ∈ Rn be a unit column vector (that
is, uT u = 1). Let P = I − 2uuT , where I is the n-by-n identity matrix. Show that if 1 is
not an eigenvalue of Q , then 1 is an eigenvalue of P Q .
B4: Let F be the set of functions f (x, y) that are twice continuously differentiable for x ≥ 1,
y ≥ 1 and that satisfy the following two equations (where subscripts denote partial deriva-
tives):
xfx + yfy = xy ln(xy) ,
x2 fxx + y 2 fyy = xy .
For each f ∈ F, let
m(f ) = min f (s + 1, s + 1) − f (s + 1, s) − f (s, s + 1) + f (s, s) .
s≥1
Determine m(f ), and show that it is independent of the choice of f .
B5: Let Fm be the mth Fibonacci number, defined by F1 = F2 = 1 and Fm = Fm−1 + Fm−2
for all m ≥ 3. Let p(x) be the polynomial of degree 1008 such that p(2n + 1) = F2n+1 for
n = 0, 1, 2, . . . , 1008. Find integers j and k such that p(2019) = Fj − Fk .
B6: Let Zn be the integer lattice in Rn . Two points in Zn are called neighbors if they differ
by exactly 1 in one coordinate and are equal in all other coordinates. For which integers
n ≥ 1 does there exist a set of points S ⊂ Zn satisfying the following two conditions?
(1) If p is in S, then none of the neighbors of p is in S.
(2) If p ∈ Zn is not in S, then exactly one of the neighbors of p is in S.
