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Analysis SEP Summer 2015
Problem 1. (a)Suppose (an ) is a sequence of nonnegative real numbers such that an < , construct a
n=0
n=0
sequence (bn ) , such that bn 0, an 0, but still bn < ;
n=0
n=0
bn
(b)Suppose (an ) is a sequence of nonnegative real numbe

6
Combinatorics and Probability
We conclude the book with combinatorics. First, we train combinatorial skills in set
theory and geometry, with a glimpse at permutations. Then we turn to some specic
techniques: generating functions, counting arguments, the

3
Real Analysis
The chapter on real analysis groups material covering differential and integral calculus,
ordinary differential equations, and also a rigorous introduction to real analysis with -
proofs.
We found it natural, and also friendly, to begin wi

2
Algebra
It is now time to split mathematics into branches. First, algebra. A section on algebraic
identities hones computational skills. It is followed naturally by inequalities. In general,
any inequality can be reduced to the problem of nding the mini

1
Methods of Proof
In this introductory chapter we explain some methods of mathematical proof. They
are argument by contradiction, the principle of mathematical induction, the pigeonhole
principle, the use of an ordering on a set, and the principle of inv

Putnam and Beyond
R zvan Gelca
a
Titu Andreescu
Putnam and Beyond
R zvan Gelca
a
Texas Tech University
Department of Mathematics and Statistics
MA 229
Lubbock, TX 79409
USA
rgelca@gmail.com
Titu Andreescu
University of Texas at Dallas
School of Natural Sc

6.1 Combinatorial Arguments in Set Theory and Geometry
291
4. m = 4, n = 3, in which case E = 12, V = 6, F = 8; this is the regular octahedron.
5. m = 5, n = 3, in which case E = 30, V = 12, F = 20; this is the regular
icosahedron.
We have proved the well

4
Geometry and Trigonometry
Geometry is the oldest of the mathematical sciences. Its age-old theorems and the sharp
logic of its proofs make you think of the words of Andrew Wiles, Mathematics seems to
have a permanence that nothing else has.
This chapter

5
Number Theory
This chapter on number theory is truly elementary, although its problems are far from
easy. (In fact, here, as elsewhere in the book, we tried to follow Felix Kleins advice:
Dont ever be absolutely boring.)1 We avoided the intricacies of a