Putnam Exam Seminar
Fall 2010
Assignment 2
Due September 13
Exercise 1. ROM N is a rectangle with |RO| = 11 and |OM | = 5. The triangle ABC has
circumcenter O and its altitudes intersect at R. M is the midpoint of BC and AN is the
altitude from A to BC .
Putnam Exam Seminar
Fall 2010
Assignment 1
Due September 8
Exercise 1. Dene a sequence cfw_an by a1 = 1 and an+1 = cos [arctan(an )] for n 1. Find
a formula for an and determine lim an .
n
n
Exercise 2. Let n 1. Prove that 22 1 has at least n distinct pr
Putnam Exam Seminar
Fall 2010
Handout
September 27
Exercise 1. Dene a sequence cfw_an recursively by setting a0 = 1, a1 = 2010 and
an+1 = Ban an1
for n 1. Determine the value of B so that
an+1
= 3 + 2 2.
lim
n an
Exercise 2. Determine the exact value of
Putnam Exam Seminar
Fall 2010
August 30
Exercise 1. Consider the following game. You are given a pile of at least two marbles and
allowed to divide that pile into two smaller (nonempty) piles of sizes a and b. You earn
exactly ab points for this move. You
Putnam Exam Seminar
Fall 2010
Quiz 11
December 1
Problem 1. A game involves jumping to the right on the real number line. If a and b are
real numbers and b > a, the cost of jumping from a to b is b3 ab2 . For what real numbers
c can one travel from 0 to 1
Putnam Exam Seminar
Fall 2010
Quiz 10
November 29
Problem 1. Determine, with proof, the number of ordered triples (A1 , A2 , A3 ) of sets which
have the property that
(i) A1 A2 A3 = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and
(ii) A1 A2 A3 = .
Express your ans
Putnam Exam Seminar
Fall 2010
Quiz 9
November 15
Problem 1. Consider a set S with a binary operation , that is, for each a, b S , a b S .
Assume that (a b) a = b for all a, b S . Prove that a (b a) = b for all a, b S . [Putnam
Exam, 2001, A1]
Problem 2. L
Putnam Exam Seminar
Fall 2010
Quiz 8
November 10
Problem 1. Find the maximum value of f (x) = x3 3x on the set of all real numbers x
satisfying x4 + 36 13x2 . [Putnam Exam, 1986, A1]
Problem 2. Determine the minimum value of
(r 1)2 +
s
1
r
2
+
t
1
s
2
+
4
Putnam Exam Seminar
Fall 2010
Quiz 7
November 1
Problem 1. Suppose p(x) is a polynomial of degree seven such that (x 1)4 is a factor of
p(x) + 1 and (x + 1)4 is a factor of p(x) 1. Find p(x).
Problem 2. Let k be a xed positive integer. The n-th derivative
Putnam Exam Seminar
Fall 2010
Quiz 6
October 25
Problem 1. Find all ordered pairs of real numbers (x, y ) which satisfy the system of
equations
x+ y = 3
3x + 2y = 14.
Problem 2. Curves A, B , C and D are dened in the plane as follows:
A=
(x, y ) : x2 y 2
Putnam Exam Seminar
Fall 2010
Quiz 5
October 18
Problem 1. Find the unique function u(t) so that
1
u(s) ds
u (t) = u(t) +
0
and u(0) = 1. [Putnam Exam, 1958, 3]
Problem 2. If
x3 x6
+
+ ,
3!
6!
x4 x 7
v = x+
+
+ ,
4!
7!
x2 x5 x8
+
+
+ ,
w=
2!
5!
8!
u = 1+
Putnam Exam Seminar
Fall 2010
Problem 1. Evaluate
Quiz 4
October 11
4
2
ln(9 x)
ln(9 x) +
ln(x + 3)
dx.
[Putnam Exam, 1987, B1]
Problem 2. Find the volume of the region of points (x, y, z ) such that
(x2 + y 2 + z 2 + 8)2 36(x2 + y 2 ).
[Putnam Exam, 2006
Putnam Exam Seminar
Fall 2010
Quiz 3
September 22
Problem 1. Given any thirteen real numbers, prove that there are two of them, say x and
y , so that
xy
2 3
.
<
0<
1 + xy
2+ 3
Problem 2. Prove that if any ve points are chosen on a sphere, then four of the
Putnam Exam Seminar
Fall 2010
Quiz 2
September 15
Problem 1. Suppose an arbitrary triangle has interior angles , and . Show that
sin
1
sin sin .
2
2
2
4
Problem 2. The area A and an angle of a triangle are given. Determine the lengths of
the sides a and b
Putnam Exam Seminar
Fall 2010
Quiz 1
September 1
Problem 1. Evaluate the innite product
n3 + 1
.
n3 1
n=2
Problem 2. A function f is dened for all positive integers and satises
f (1) = 2010
and
f (1) + f (2) + + f (n) = n2 f (n).
Compute the exact value o
Putnam Exam Seminar
Fall 2010
Assignment 10
Due November 22
Exercise 1. Let G be a group with identity e and : G G a function such that
(g1 )(g2 )(g3 ) = (h1 )(h2 )(h3 )
whenever g1 g2 g3 = e = h1 h2 h3 . Prove that there exists an element a G such that (
Putnam Exam Seminar
Fall 2010
Assignment 9
Due November 8
Exercise 1. Do there exist polynomials a(x), b(x), c(y ), d(y ) such that
1 + xy + x2 y 2 = a(x)c(y ) + b(x)d(y )
holds identically? [Putnam Exam, 2003, B1]
Exercise 2. Let n be a positive integer
Putnam Exam Seminar
Fall 2010
Assignment 8
Due November 1
Exercise 1. Find all pairs of real numbers (x, y ) satisfying the system of equations
1
1
+
= (x2 + 3y 2 )(3x2 + y 2 )
x 2y
1
1
= 2(y 4 x4 ).
x 2y
[Putnam Exam, 2001, B-2]
Exercise 2. Assume that x
Putnam Exam Seminar
Fall 2010
Assignment 7
Due October 25
Exercise 1. Functions f , g , h are dierentiable on some open interval around 0 and satisfy
the equations and initial conditions
1
f = 2f 2 gh + , f (0) = 1,
gh
4
g = f g2h +
, g (0) = 1,
fh
1
h =
Putnam Exam Seminar
Fall 2010
Assignment 6
Due October 18
Exercise 1. Find all real-valued continuously dierentiable functions f dened on the real
line such that for all x,
x
(f (t)2 + (f (t)2 dt.
(f (x)2 = 1990 +
0
[Putnam Exam, 1990, B1]
Exercise 2. Eva
Putnam Exam Seminar
Fall 2010
Assignment 4
Due October 4
Exercise 1. The sequence cfw_an is dened by a1 = 1, a2 = 2, a3 = 24, and, for n 4,
an =
6a2 1 an3 8an1 a2 2
n
n
.
an2 an3
Show that, for all n, an is an integer multiple of n. [Putnam Exam, 1999, A
Putnam Exam Seminar
Fall 2010
Assignment 3
Due September 20
Exercise 1. Six points are in general position in space (no three in a line, no four in a
plane). The fteen line segments joining them in pairs are drawn and each one is colored
either red or blu