The Fifty-Sixth William Lowell Putnam Mathematical Competition Saturday, December 2, 1995
A1 Let be a set of real numbers which is closed under multiplication (that is, if and are in , then so is ). L
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
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
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 + y 2 + z 2 + 8)2 36(x2 + y 2 ).
A2 Alice and Bob
Solutions to the 67th William Lowell Putnam Mathematical Competition
Saturday, December 2, 2006
Kiran Kedlaya and Lenny Ng
A1 We change to cylindrical coordinates, i.e., we put r =
x2 + y 2 . Then 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 = y 2 + y + 24 are tangent to each
other.
A2 Fin
Solutions to the 68th William Lowell Putnam Mathematical Competition Saturday, December 1, 2007
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng 601)/12.
A1 The only such are 2/3, 3/2, (13
First solution
The 69th William Lowell Putnam Mathematical Competition Saturday, December 6, 2008
A1 Let f : R2 R be a function such that f (x, y) + f (y, z) + f (z, x) = 0 for all real numbers x, y, and z. Prove 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) int
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
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
Solutions to the 71st William Lowell Putnam Mathematical Competition
Saturday, December 4, 2010
Kiran Kedlaya and Lenny Ng
A1 The largest such k is n+1 =
2
value is achieved by the partition
n
2
. For
The 72nd William Lowell Putnam Mathematical Competition
Saturday, December 3, 2011
A1 Dene a growing spiral in the plane to be a sequence of
points with integer coordinates P0 = (0, 0), P1 , . . . , P
Solutions to the 72nd William Lowell Putnam Mathematical Competition
Saturday, December 3, 2011
Kiran Kedlaya and Lenny Ng
A1 We claim that the set of points with 0 x 2011 and
0 y 2011 that cannot be
Exercises III
Joshua Egger
MATH-191
Exercise 3.3: Prove by hand that the homology of the sphere Hk (S n ) is zero whenever n 6= k, 0.
More can be said:
Claim:
(
Z2
Hk (S ) =
0
n
n = 0, n
n>0
Proof. We
DE RHAM COHOMOLOGY
JOSHUA P. EGGER
Abstract. De Rahm Cohomology is a powerful tool which allows one to extract purely topological information about a manifold, essentially by doing algebra on its cota
Exercises I
Joshua Egger
MATH-191
Exercise 1.2 - Bridges of K
onigsberg A walk is a sequence of vertices which are pairwise connected
by edges, and we are allowed to possibly repeat vertices. Show tha
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% an
The 65th William Lowell Putnam Mathematical Competition Saturday, December 4, 2004
B1 Let P (x) = cn xn + cn-1 xn-1 + + c0 be a polynomial with integer coefficients. Suppose that r is a rational numbe
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 Fifty-Sixth William Lowell Putnam Mathematical Competition Saturday, December 2, 1995
Kiran Kedlaya
Choose a vector orthogonal to but not to . Since as , the same is true of ; . In ot
The Fifty-Seventh William Lowell Putnam Mathematical Competition Saturday, December 7, 1996
A1 Find the least number such that for any two squares of combined area 1, a rectangle of area exists such t
Solutions to the Fifty-Eighth William Lowell Putnam Mathematical Competition Saturday, December 7, 1996
Manjul Bhargava and Kiran Kedlaya
To find this maximum, we let . Then we are to maximize
with
wi
The Fifty-Eighth William Lowell Putnam Mathematical Competition Saturday, December 6, 1997
(Here
denotes the minimum of
and .)
B2 Let be a twice-differentiable real-valued function satisfying
A3 Evalu
Solutions to the Fifty-Eighth William Lowell Putnam Mathematical Competition Saturday, December 6, 1997
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
Thus the desired integral is simply
and using the c
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
Solutions to the 59th William Lowell Putnam Mathematical Competition Saturday, December 5, 1998
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
I O F
and that the first and second expressions are both in
The 60th William Lowell Putnam Mathematical Competition Saturday, December 4, 1999
.
A-4 Sum the series
Show that, for all n,
is an integer multiple of .
B-6 Let be a nite set of integers, each greate
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
a
The 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000
A-1 Let A be a positive real number. What are the possible values of j=0 x2 , given that x0 , x1 , . . . are positive