Solutions to the 70th William Lowell Putnam Mathematical Competition
Saturday, December 5, 2009
Kiran Kedlaya and Lenny Ng
A1 Yes, it does follow. Let P be any point in the plane. Let
ABCD be any square with center P . Let E, F, G, H
be the midpoints of t
A nswers to Textbook Questions
and Problems
CHAPTER
1
The Science of Macroeconomics
Questions for Review
1. Microeconomics is the study of how individual firms and households make decisions,
and how they interact with one another. Microeconomic models of
Figure 0.1
ABSTRACT ALGEBRA
DAVID S. DUMMIT AND RICHARD M. FOOTE
Solutions provided by Scott Larson.
Contents
0. Preliminaries
0.1. Basics
0.2. Properties of the Integers
0.3. Z/nZ: The Integers Modulo n
Part I Group Theory
1. Introduction to Groups
1.1.
1 The Rules of the Game
1.1: Denitions
Game theory is concerned with the actions of decision makers who are conscious that their
actions aect each other. When the only two publishers in a city choose prices for their
newspapers, aware that their sales are
2. Theory
Our model is in the spirit of Barro and Gordon (1983), who considered the behavior of a central
bank with a mandate to optimize a function of inflation and unemployment. It differs in that we
posit two agencies with the potential to intervene in
Undergraduate IO (Econ 431): Final exam (2007/12/17)
Abbreviated Solution
Page 1 of 2
(1) [5 points] See the textbook (Section 8.6) for a discussion of Section 2 of the Sherman Act (1890).
The most relevant part is attempting to monopolize.
(2) [10 points
Econ 431: Midterm exam (2007/10/03)
Abbreviated Solution
Page 1 of 4
(1) [10 pts.] The diaper industry in Albania consists of 5 rms producing identical diapers. Similarly,
the diaper industry in Bolivia consists of 6 rms. It has been recently observed tha
Econ 431: Midterm exam (2007/11/04)
Abbreviated Solution
Page 1 of 4
(1) Two rms have technologies for producing identical paper clips. Assume that all paper clips are
sold in boxes containing 100 paper clips. Firm A can produce each box at unit cost of c
Industrial Organization (Econ 431): Final exam (2008/06/20)
Abbreviated Solution
Page 1 of 4
(1) [5 points] Please refer to class discussion and/or Section 8.2 of the textbook. In your answer,
you should briey describe (a) horizontal merger, (b) vertical
Industrial Organization (Econ 431): Midterm exam (2008/05/20)
Abbreviated Solution
Page 1 of 4
(1) [10 points According to the four-largest rm concentration index, industry A is more concentrated
than industry B since
A
B
I4 = 40 + 15 + 15 + 15 = 85 > 78
Industrial Organization (Econ 431): Final exam (2009/06/22)
Abbreviated Solution
Page 1 of 1
(1) [15 points] See the solution for Problem Set # 10(f), or the Fall 2008 Final exam.
(2) [10 points] The subgame-perfect equilibrium strategies are
3
2
if x2 >