Problem Solving in Math (Math 43900)
Fall 2015
Instructor: David Galvin
Abstract
This document comprises an introduction to the Putnam competition, and a collec-
tion of themed solving strategies, problems and solutions presented during the semester
in preparation for the 2015 Putnam competition.
Contents
1 Introduction 4
1.1 The Putnam Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Problem Solving in Math seminar . . . . . . . . . . . . . . . . . . . . . 4
2 Week one (August 25) — A grab-bag 6
2.1 Week one problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Week one solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Week two (September 1) — Induction 14
3.1 Week two problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Week two solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Week three (September 8) — Pigeon-hole principle 24
4.1 Week three problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Week three solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Week four (September 15) — Binomial coefficients 31
5.1 Week four problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Week four solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Week five (September 22) — Modular arithmetic and the greatest common
divisor 45
6.1 Week five problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Week five solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7 Week six (September 29) — Inequalities 56
7.1 Week six problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2 Week six solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1
,8 Week seven (October 6) — Polynomials 68
8.1 Week seven problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.2 Week seven solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9 Week eight (October 13) — A grab-bag 74
9.1 Week eight problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.2 Week eight solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
10 Week nine (October 27) — Recurrences 83
10.1 Week nine problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.2 Week nine solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
11 Week ten (November 3) — Graphs 94
11.1 Week ten problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.2 Week ten solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
12 Week eleven (November 10) — Probability 103
12.1 Week eleven problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
12.2 Week eleven solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
13 Week twelve (November 17) — Games 113
13.1 Week twelve problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
13.2 Week twelve solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
14 Week thirteen (November 24) — Preparing for the Putnam Competition118
14.1 Week 13 problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
14.2 Week 13 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2
,.
3
, 1 Introduction
1.1 The Putnam Competition
The seventy-sixth annual William Lowell Putnam Mathematical Competition will take place
on Saturday, December 5, 2015. Every year, around 4000 US & Canadian undergraduates
participate in the competition.
The competition consists of two three-hour sessions (morning and afternoon), with each
session having six problems. The problems are hard, not because they are made up of lots
of parts, or involve extensive computation, or require very advanced mathematics to solve.
They are hard because they each require a moment of cleverness, intuition and ingenuity to
reach a solution. Typically, the median score out of 120 (10 possible points per question) is 1!
The Putnam Competition may be the most challenging and rewarding tests of mathematical
skill that you will ever encounter.
1.2 The Problem Solving in Math seminar
To help prepare for the Putnam Competition, the math department runs the 1 credit course
Math 43900. If you are signed up for Math 43900, then you are also signed up to participate
in the Putnam competition on December 5. This year, the details of the course are as follows:
• Instructor: David Galvin, 248 Hayes-Healy, .
• Meetings: Tuesdays, 3.30pm-4.20pm, De Bartolo 201, starting August 25 (just a short
organizational meeting), ending December 1 (the last Tuesday before the competition).
• Office hours: Email me for an appointment.
• Text: There is no required text. The following books (one available online through
the library, the others on reserve at the Math library) are worth looking at. First, two
classics that deal with the art of problem-solving:
– Problem-solving through problems by Larson (QA 43 .L37 1983)
– How to solve it by Pólya (QA 11 .P6 2004).
Next, a doorstop filled with problems and strategies for the Putnam itself:
– Putnam and beyond by Gelca and Andreescu (online).
Finally, three books that exhaustively catalog all Putnam Competitions up to 2000:
– The William Lowell Putnam Mathematical Competition 1985-2000: problems, so-
lutions, and commentary by Kedlaya, Poonen and Vakil (QA 43 .W5425 2002)
– The William Lowell Putnam Mathematical Competition problems and solutions:
1938-1964 by Gleason, Greenwood and Kelly (QA 43 .W54)
– The William Lowell Putnam Mathematical Competition problems and solutions:
1965-1984 by Alexanderson, Klosinski and Larson (QA 43 .W542 1985).
4
Fall 2015
Instructor: David Galvin
Abstract
This document comprises an introduction to the Putnam competition, and a collec-
tion of themed solving strategies, problems and solutions presented during the semester
in preparation for the 2015 Putnam competition.
Contents
1 Introduction 4
1.1 The Putnam Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Problem Solving in Math seminar . . . . . . . . . . . . . . . . . . . . . 4
2 Week one (August 25) — A grab-bag 6
2.1 Week one problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Week one solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Week two (September 1) — Induction 14
3.1 Week two problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Week two solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Week three (September 8) — Pigeon-hole principle 24
4.1 Week three problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Week three solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Week four (September 15) — Binomial coefficients 31
5.1 Week four problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Week four solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Week five (September 22) — Modular arithmetic and the greatest common
divisor 45
6.1 Week five problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Week five solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7 Week six (September 29) — Inequalities 56
7.1 Week six problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2 Week six solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1
,8 Week seven (October 6) — Polynomials 68
8.1 Week seven problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.2 Week seven solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9 Week eight (October 13) — A grab-bag 74
9.1 Week eight problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.2 Week eight solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
10 Week nine (October 27) — Recurrences 83
10.1 Week nine problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.2 Week nine solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
11 Week ten (November 3) — Graphs 94
11.1 Week ten problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.2 Week ten solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
12 Week eleven (November 10) — Probability 103
12.1 Week eleven problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
12.2 Week eleven solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
13 Week twelve (November 17) — Games 113
13.1 Week twelve problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
13.2 Week twelve solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
14 Week thirteen (November 24) — Preparing for the Putnam Competition118
14.1 Week 13 problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
14.2 Week 13 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2
,.
3
, 1 Introduction
1.1 The Putnam Competition
The seventy-sixth annual William Lowell Putnam Mathematical Competition will take place
on Saturday, December 5, 2015. Every year, around 4000 US & Canadian undergraduates
participate in the competition.
The competition consists of two three-hour sessions (morning and afternoon), with each
session having six problems. The problems are hard, not because they are made up of lots
of parts, or involve extensive computation, or require very advanced mathematics to solve.
They are hard because they each require a moment of cleverness, intuition and ingenuity to
reach a solution. Typically, the median score out of 120 (10 possible points per question) is 1!
The Putnam Competition may be the most challenging and rewarding tests of mathematical
skill that you will ever encounter.
1.2 The Problem Solving in Math seminar
To help prepare for the Putnam Competition, the math department runs the 1 credit course
Math 43900. If you are signed up for Math 43900, then you are also signed up to participate
in the Putnam competition on December 5. This year, the details of the course are as follows:
• Instructor: David Galvin, 248 Hayes-Healy, .
• Meetings: Tuesdays, 3.30pm-4.20pm, De Bartolo 201, starting August 25 (just a short
organizational meeting), ending December 1 (the last Tuesday before the competition).
• Office hours: Email me for an appointment.
• Text: There is no required text. The following books (one available online through
the library, the others on reserve at the Math library) are worth looking at. First, two
classics that deal with the art of problem-solving:
– Problem-solving through problems by Larson (QA 43 .L37 1983)
– How to solve it by Pólya (QA 11 .P6 2004).
Next, a doorstop filled with problems and strategies for the Putnam itself:
– Putnam and beyond by Gelca and Andreescu (online).
Finally, three books that exhaustively catalog all Putnam Competitions up to 2000:
– The William Lowell Putnam Mathematical Competition 1985-2000: problems, so-
lutions, and commentary by Kedlaya, Poonen and Vakil (QA 43 .W5425 2002)
– The William Lowell Putnam Mathematical Competition problems and solutions:
1938-1964 by Gleason, Greenwood and Kelly (QA 43 .W54)
– The William Lowell Putnam Mathematical Competition problems and solutions:
1965-1984 by Alexanderson, Klosinski and Larson (QA 43 .W542 1985).
4
