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 ). Let and be disjoint subsets of whose union is . Given th
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 number such that P (r) = 0. Show that the n numbers cn r, cn
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 percentag
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,
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
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 play a game in which they take turns
removing stones f
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 given inequality is equivalent to
r2 + z 2 + 8 6r,
or
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 Find the least possible area of a convex set in the plane
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: Let C1 and C2 be the curves y = x2 + 1 1 x + 24 and x
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 that there exists a function g : R R such that f (x, y) =
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 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 t
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?
[
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 n even, this
cfw_ 1 , n , cfw_ 2 , n 1 , . . . ;
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 , . . . , Pn
such that n 2 and:
The
directed
line
segments
P0 P1
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
The 64th William Lowell Putnam Mathematical Competition Saturday, December 6, 2003
A1 Let be a xed positive integer. How many ways are there to write as a sum of positive integers, , with an arbitrary positive integer and ? For example, with there are fou
Solutions to the 63rd William Lowell Putnam Mathematical Competition
Saturday, December 7, 2002
Kiran Kedlaya and Lenny Ng
A1 By differentiating Pn (x)/(xk 1)n+1 , we nd that
Pn+1 (x) = (xk 1)Pn (x) (n + 1)kxk1 Pn (x); substituting x = 1 yields Pn+1 (1) =
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 other words, if but that is simply , then must also go to
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 that the two squares can be packed in the rectangle (wit
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
with equality for sired value of .
. Hence this value is
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 Evaluate
where bounded.
for all real . Prove that
B3 For eac
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 commutativity of
Again, we may assume
. This implies tha
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
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 integers. We conclude that
Q g 8 4 Q g 4 y' G 8 4 y' A
g
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 greater than 1. Suppose that for each integer there is some s
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
a2 , b2n+1 = an (an1 + an+1 ). Then
n
2b2n+1 + b2n = 2an
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 j numbers for which j=0 xj = A? A-2 Prove that there e
Solutions to the 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng
A1 The possible values comprise the interval (0, A2 ). To see that the values must lie in this interval, note that
The 62nd William Lowell Putnam Mathematical Competition Saturday, December 1, 2001
A-2 You have coins . For each , is biased so that, when tossed, it has probability of fallings heads. If the coins are tossed, what is the probability that the number of he