SOLUTIONS TO HOMEWORK 4
1. One hundred points are given inside a cube of side length one. Prove that there are four
of them that span a tetrahedron whose volume is at most 1/99.
(Hint. You might directly use the geometric fact that the vol
due on March 9, 2016
Sections of the textbook relevant to this assignment are: 10.110.4. (Materials covered by previous homeworks might also be used in this assignment.)
Recall that we say a bipartite graph G on A t B is a good
due on March 2, 2016
Sections of the textbook relevant to this assignment are: 8.3, 9.1, 9.2 and
10.1. (Materials covered by previous homeworks might also be used in this
Note. This homework has a few problems that
Problem set 2 solution
1. (Putnam, 1978-A1) Let A be any set of 20 distinct integers chosen from the arithmetic
progression 1, 4, 7, , 100. Prove that there must be two distinct integers in A whose
sum is 104. [Actually, 20 can be replaced by 19.] (Junpin
due on February 3, 2016
Sections of the textbook relevant to this assignment are: 2.4, and 4.1
4.3. (Materials covered by previous homeworks might also be used in this
1. One hundred points are given inside a cube o
INFORMATION FOR MIDTERM
The midterm will be based on Homeworks 15 and lecture notes up to Section 7.2 (including Section 7.2). Please see details below.
(1) The exam will consist of 56 problems.
(2) The difficulty level will be similar to
DEFINITIONS AND NOTATIONS
In this notes, we summarize definitions and notations that are either used frequenctly in
the course or hard to understand.
Definition. If x belongs to S, we say x is an element of S or x is in S. Write x S. If x is
not in S, we
due on January 13, 2016
Sections of the textbook that are relevant to this assignment are: 1.2, 1.3,
For all the homework problems (including this assignment and all other
assignments), unless otherwise specified
Math 150A: Modern Algebra
This assignment is due at the end of class Thursday 8/4. As mentioned in class, your solutions
must be written in LaTeX. You do not necessarily have to use LaTeX for any images you include.
Complete exercise 2.2.3 from
Math 135B Midterm 2 Spring 2016
NAME(print in CAPITAL letters, ﬁrst name ﬁrst): “5%.-ij
Instructions: There are four problems. Make sure that you have all 4 problems. You must show
all your work to receive full credit.