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
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 . Supp
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
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 th
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 squa
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) int
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 solu
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 achie
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,
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
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 w
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
m
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
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
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
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
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
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)
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
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
c
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
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 +
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).
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
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
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 h
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
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 pla
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 , . . . ,
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
t