Solutions to the 64th William Lowell Putnam Mathematical Competition
Saturday, December 6, 2003
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
A1 There are n such sums. More precisely, there is exactly
one such sum with k terms for each of k = 1, . . . , n

Solutions to the 66th William Lowell Putnam Mathematical Competition
Saturday, December 3, 2005
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
A1 We proceed by induction, with base case 1 = 20 30 . Suppose all integers less than n 1 can be represented. If
n

Solutions to the 65th William Lowell Putnam Mathematical Competition
Saturday, December 4, 2004
Kiran Kedlaya and Lenny Ng
A1 Yes. Suppose otherwise. Then there would be an N
such that S(N) < 80% and S(N + 1) > 80%; that is,
OKeals free throw percentage i

Solutions to the 67th William Lowell Putnam Mathematical Competition
Saturday, December 2, 2006
Kiran Kedlaya and Lenny Ng
A1 We
p change to cylindrical coordinates, i.e., we put r =
x2 + y2 . Then the given inequality is equivalent to
r2 + z2 + 8 6r,
or

Solutions to the 70th William Lowell Putnam Mathematical Competition
Saturday, December 5, 2009
Kiran Kedlaya and Lenny Ng
A1 Yes, it does follow. Let P be any point in the plane. Let
ABCD be any square with center P. Let E, F, G, H be
the midpoints of th

Solutions to the 69th William Lowell Putnam Mathematical Competition
Saturday, December 6, 2008
Kiran Kedlaya and Lenny Ng
A1 The function g(x) = f (x, 0) works.
Substituting
(x, y, z) = (0, 0, 0) into the given functional equation yields f (0, 0) = 0, wh

Solutions to the 68th William Lowell Putnam Mathematical Competition
Saturday, December 1, 2007
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
A1 The only such are 2/3, 3/2, (13 601)/12.
x2 +
First solution: Let C1 and C2 be the curves y =
1
1
x + 24
and x

Solutions to the 71st William Lowell Putnam Mathematical Competition
Saturday, December 4, 2010
Kiran Kedlaya and Lenny Ng
n
A1 The largest such k is b n+1
2 c = d 2 e. For n even, this value
is achieved by the partition
Since 10n n 2m m + 1, 10n is divis

Solutions to the 75th William Lowell Putnam Mathematical Competition
Saturday, December 6, 2014
Kiran Kedlaya and Lenny Ng
A1 The coefficient of xn in the Taylor series of (1 x +
x2 )ex for n = 0, 1, 2 is 1, 0, 12 , respectively. For n 3,
the coefficient

Solutions to the 73rd William Lowell Putnam Mathematical Competition
Saturday, December 1, 2012
Kiran Kedlaya and Lenny Ng
A1 Without loss of generality, assume d1 d2 d12 .
2 < d 2 + d 2 for some i 10, then d , d
If di+2
i i+1 , di+2
i
i+1
are the side le

Solutions to the 74th William Lowell Putnam Mathematical Competition
Saturday, December 7, 2013
Kiran Kedlaya and Lenny Ng
A1 Suppose otherwise. Then each vertex v is a vertex for
five faces, all of which have different labels, and so the
sum of the label

Solutions to the 60th William Lowell Putnam Mathematical Competition
Saturday, December 4, 1999
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
A1 Note that if r(x) and s(x) are any two functions, then
max(r, s) = (r + s + |r s|)/2.
an (an1 + an+1 ). Then
2b

Solutions to the 72nd William Lowell Putnam Mathematical Competition
Saturday, December 3, 2011
Kiran Kedlaya and Lenny Ng
bj
B
b j +2 B+2
B m
) ). Since
3/2(1 ( B+2
A1 We claim that the set of points with 0 x 2011 and
0 y 2011 that cannot be the last poi

Solutions to the 57th William Lowell Putnam Mathematical Competition
Saturday, December 7, 1996
Manjul Bhargava and Kiran Kedlaya
A1 If x and y are the sides of two squares with combined
area 1, then x2 + y2 = 1. Suppose without loss of generality that x

Solutions to the 56th William Lowell Putnam Mathematical Competition
Saturday, December 2, 1995
Kiran Kedlaya
A1 Suppose on the contrary that there exist t1 ,t2 T
with t1t2 U and u1 , u2 U with u1 u2 T . Then
(t1t2 )u1 u2 U while t1t2 (u1 u2 ) T , contrad

Solutions to the 58th William Lowell Putnam Mathematical Competition
Saturday, December 6, 1997
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
A1 The centroid G of the triangle is collinear with H and O
(Euler line), and the centroid lies two-thirds of the

Solutions to the 59th William Lowell Putnam Mathematical Competition
Saturday, December 5, 1998
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
A1 Consider the plane containing both the axis of the cone
and two opposite vertices of the cubes bottom face.
The

The 68th William Lowell Putnam Mathematical Competition
Saturday, December 1, 2007
A1 Find all values of for which the curves y = x2 +
1
1
x+ 24
and x = y2 +y+ 24
are tangent to each other.
f ( f (n) + 1) if and only if n = 1. [Editors note: one
must assu

The 74th William Lowell Putnam Mathematical Competition
Saturday, December 7, 2013
A1 Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular
icosah

The 59th William Lowell Putnam Mathematical Competition
Saturday, December 5, 1998
A1 A right circular cone has base of radius 1 and height 3.
A cube is inscribed in the cone so that one face of the
cube is contained in the base of the cone. What is the
s

The 66th William Lowell Putnam Mathematical Competition
Saturday, December 3, 2005
A1 Show that every positive integer is a sum of one or more
numbers of the form 2r 3s , where r and s are nonnegative
integers and no summand divides another. (For example,

The 64th William Lowell Putnam Mathematical Competition
Saturday, December 6, 2003
A1 Let n be a fixed positive integer. How many ways are
there to write n as a sum of positive integers, n = a1 +
a2 + + ak , with k an arbitrary positive integer and
a1 a2

The 56th William Lowell Putnam Mathematical Competition
Saturday, December 2, 1995
A1 Let S be a set of real numbers which is closed under
multiplication (that is, if a and b are in S, then so is ab).
Let T and U be disjoint subsets of S whose union is
S.

The 58th William Lowell Putnam Mathematical Competition
Saturday, December 6, 1997
A1 A rectangle, HOMF, has sides HO = 11 and OM = 5.
A triangle ABC has H as the intersection of the altitudes,
O the center of the circumscribed circle, M the midpoint
of B

The 71st William Lowell Putnam Mathematical Competition
Saturday, December 4, 2010
A1 Given a positive integer n, what is the largest k such that
the numbers 1, 2, . . . , n can be put into k boxes so that
the sum of the numbers in each box is the same? [

The 65th William Lowell Putnam Mathematical Competition
Saturday, December 4, 2004
A1 Basketball star Shanille OKeals team statistician
keeps track of the number, S(N), of successful free
throws she has made in her first N attempts of the season. Early in

The 75th William Lowell Putnam Mathematical Competition
Saturday, December 6, 2014
A1 Prove that every nonzero coefficient of the Taylor series
of
(1 x + x2 )ex
about x = 0 is a rational number whose numerator (in
lowest terms) is either 1 or a prime numb

The 67th William Lowell Putnam Mathematical Competition
Saturday, December 2, 2006
A1 Find the volume of the region of points (x, y, z) such that
(x2 + y2 + z2 + 8)2 36(x2 + y2 ).
A2 Alice and Bob play a game in which they take turns
removing stones from

The 72nd William Lowell Putnam Mathematical Competition
Saturday, December 3, 2011
A1 Define a growing spiral in the plane to be a sequence
of points with integer coordinates P0 = (0, 0), P1 , . . . , Pn
such that n 2 and:
The directed line segments P0 P

The 57th William Lowell Putnam Mathematical Competition
Saturday, December 7, 1996
A1 Find the least number A such that for any two squares of
combined area 1, a rectangle of area A exists such that
the two squares can be packed in the rectangle (without