Math 310H Self-Quiz 2
1. Prove the following identity
n
k
n1
n1
+
k
k1
=
(a) using an argument based on the formula for
n
m
.
(b) using a combinatorial argument.
2. Prove the identity
n
k
n1
n2
n2
+
+
.
k1
k1
k
=
(a) using Pascals identity or an argument
Math 310H Problem Set 1
Due at 11:15am Wednesday, January 23, 2013
1. Let n be a nonnegative integer. Prove the following by Mathematical Induction:
(a) 1 + 3 + 5 + + (2n 1) = n2
(b)
n
0
+
n
1
+ +
n
n
= 2n
Solution. (a) For n = 0, the sum on the left is e
Math 310H Problem Set 2
Due at 11:15am Monday, January 28, 2013
Suggested Problems: Section 2.7 #810, 1214, 20, 21, 24, 28, 3032, 34, 39, 41, 4547, 51.
To Be Handed In:
1. A travel agent has to visit four cities, each of them ve times. In how many dieent
Math 310H Problem Set 3
Due at 11:15am Monday, February 4, 2013
Suggested Problems: Section 5.7 #311.
To Be Handed In:
1. Find the number of ways to distribute n pennies to k children such that the ith child gets at
least i pennies for i = 1, 2 . . . , k
Math 310H Problem Set 4
Due at 11:15am Monday, February 11, 2013
Suggested Problems: Section 5.7 #15, 16, 1820, 27, 28, 43.
To Be Handed In:
1. Give a combinatorial proof of the following binomial identity: for all positive integers n,
n
k=1
(Hint: Use
n
Math 310H Problem Set 5
Due at 11:15am Monday, February 18, 2013
Suggested Problems: Section 5.7 #36, 3841, 44, 45; Section 6.7 # 110. Self-Quiz 2,
To Be Handed In:
1. Find the number of integral solutions of the equation x1 + x2 + x3 + x4 = 17 such that
Math 310H Problem Set 6
Due at 11:15am Monday, March 18, 2013
Suggested Problems: Section 7.7 #1, 3, 8, 11, 1315, 22, 24.
To Be Handed In:
1. For the following expression, nd a simple formula which only involves one Fibonacci number.
Then prove it by indu
Math 310H Problem Set 7
Due at 11:15am Monday, March 25, 2013
Suggested Problems: Section 7.7 #31, 3336, 38.
To Be Handed In:
1. Let hn be the number of dierent ways to color the squares of an 1-by-n chessboard using red,
white, and blue so that no two ad
Math 310H Problem Set 8
Due at 11:15am Monday, April 1, 2013
Suggested Problems: Section 7.7 #42-48.
To Be Handed In:
1. Let an equal the number of ternary strings of length n made up of 0s, 1s, and 2s, such that
the substrings 00, 01, 10, and 11 never oc
Math 310H Self-Quiz 1
1. Given eight dierent English books, seven dierent French books, and ve dierent
German books:
(a) How many ways are there to select one book? 20 ways.
(b) How many ways are there to select three books, one of each language? 280 ways
Math 310H Self-Quiz 1
1. Given eight dierent English books, seven dierent French books, and ve dierent
German books:
(a) How many ways are there to select one book? 20 ways.
(b) How many ways are there to select three books, one of each language? 280 ways
Convex sets
I
affine and convex sets
I
some important examples
I
operations that preserve convexity
I
separating and supporting hyperplanes
I
generalized inequalities
I
dual cones and generalized inequalities
IOE 611: Nonlinear Programming, Fall 2016
2. C