Part 1
Foundations
Chapter 1
Sets and vector spaces
We assume some familiarity with basic notions from set theory and linear
algebra. For example the reader should be comfortable working with nite
dimensional vector spaces over a eld, their bases, and lin
Calulus
Rie Mathematis Tournament 2000
1. Find the slope of the tangent at the point of inetion of y = x3
3x2 + 6x + 2000.
2. Karen is attempting to limb a rope that is not seurely fastened. If she pulls herself
up x feet at one, then the rope slips x3 fe
1998 Harvard/MIT Math Tournament
CALCULUS Answer Sheet
Name:
School:
Grade:
1
6
2
7
3
8
4
9
5
10
TOTAL:
CALCULUS
Question One. [3 points]
Farmer Tim is lost in the densely-forested Cartesian plane. Starting
from the origin he walks a sinusoidal path in se
Harvard-MIT Mathematics Tournament
February 19, 2005
Team Round A
Disconnected Domino Rally [175]
On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) domino. A domino is said to be of type (a, b), with a b in
Harvard-MIT Mathematics Tournament
March 15, 2003
Individual Round: Calculus Subject Test
1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
1. What is its expected distance from the center of the circle?
2. A pa
Harvard-MIT Mathematics Tournament
February 19, 2005
Team Round B
Disconnected Domino Rally [150]
On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) domino. A domino is said to be of type (a, b), with a b in
HMMT 1998: Calculus Solutions
1. Problem: Farmer Tim is lost in the densely-forested Cartesian plane. Starting from the origin he walks
a sinusoidal path in search of home; that is, after t minutes he is at position (t, sin t).
Five minutes after he sets
Harvard-MIT Mathematics Tournament
March 15, 2003
Individual Round: Calculus Subject Test Solutions
1. A point is chosen randomly with uniform distribution in the interior of a circle of radius
1. What is its expected distance from the center of the circl
Power Question - Coloring Graphs
Perhaps you have heard of the Four Color Theorem (if not, dont panic!), which essentially says that any
map (e.g. a map of the United States) can be colored with four or fewer colors without giving neighboring
regions (e.g
Calculus Solutions
Harvard-MIT Math Tournament
February 27, 1999
Problem C1 [3 points]
Find all twice differentiable functions f (x) such that f (x) = 0, f (0) = 19, and f (1) = 99.
Solution: Since f (x) = 0 we must have f (x) = ax + b for some real numbe
Algebra: Assignment 8
Due on Wednesday, December 5, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited December 6, 2012
Contents
Exercise 4 . . . . . . . . . . . . . . . . . . . . . . .
Algebra: Assignment 7
Due on Wednesday, November 21, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 25, 2012
Contents
Exercise 2
Part (a)
Part (b)
Part (c)
.
.
.
.
3
3
3
3
E
Algebra Midterm 1
Fall 2012
Work out if each of the following statements is TRUE or FALSE.
Justify your answer carefully by supplying a PROOF or a COUNTEREXAMPLE.
1. Let Vecf pRq be the category of nite dimensional vector spaces
over R, and D : Vecf pRq V
Modules review
True or False?
1. Let V W X and V W Y be decompositions of a left
R-module V as direct sums of submodules. Then X Y .
2. Let V W X and V W Y be decompositions of a left
R-module V as direct sums of submodules. Then X Y .
3. If V is a simple
Sample homework
solutions 1
1.2.3 Let V be a nite dimensional vector space and W V be a subspace. Show that W is nite dimensional and any basis of W can be
extended to a basis of V . Deduce that dim W dim V with equality
if and only if W V .
If W is not n
Abstract Algebra: Assignment 1
Due on Friday, October 5, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exercise 1.2.3 . . . . . . . . . . . . . . . . . .
Abstract Algebra: Assignment 2
Due on Friday, October 12, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exercise 2.4.5 . . . . . . . . . . . . . . . . . .
Abstract Algebra: Assignment 3
Due on Friday, October 19, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exercise 3.4.2 . . . . . . . . . . . . . . . . . .
Algebra: Assignment 4
Due on Friday, October 26, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exercise 4.1.6 . . . . . . . . . . . . . . . . . . . . . .
Algebra: Assignment 5
Due on Firday, November 2, 2012
Brundan 1:00pm
A digital copy of this document can be found at http:/pages.uoregon.edu/raies
Dan Raies
Last edited November 21, 2012
Contents
Exercise 4.4.10
Part (a) . . .
Part (b) . . .
Part (c) . .
1998 Power Question Solutions
I. Graphs, total of 20 points
a. completely correct gets 1 point, total of 6 points
i. yes. vertices A,B,C,D, edges AB,AC,AD,BD
ii. no. A and B are connected twice
iii. yes. vertices A,B,C, edges ABC
iv. yes. vertices A,B,C,D