I will try to keep a record here of topics discussed and upcoming problems and topics of discussion.
Date Chapter.Section/Topics or Page/Problems
T Aug 23 Organizational Meeting
We discussed, among other things, Problem A1 from the 2021
(last year's) Putnam exam.
Comments and notes in a mathematica notebook
Same document in pdf
We also discussed a proof by contradiction that
sqrt(2) + sqrt(3) + sqrt(5) is irrational.
It was conjectured that sqrt(p_1) + sqrt(p_2) + ... + sqrt(p_k)
is irrational where p_1, p_2,..., p_k are distinct primes.
(We don't have a proof though we attempted an induction on k.)
For next time, perhaps someone can give a STEP 1/ STEP 2 proof
for some other resultant points (n,n) in the spirit of
Problem A1-21 (2021). Perhaps the first interesting one is
(2,2).
Also, here are some suggestions:
1. Prove there are infinitely many prime numbers. (contradiction)
2. Prove the regions determined in the plane by finitely many
lines can be colored with two colors (so that no two adjacent
regions have the same color). (Example of proof by induction.)
3. Problem 2 of Putnam and Beyond.
4. Problem 11 of Putnam and Beyond.
5. The other conjectures from my document above on Problem A1 (2021).
6. Problem A2 (the limit problem) from the Putnam exam from last year.
T 30 Solution of Putnam problem A2 (2021) ---Brandon Manville
Comments on second posted solution of problem A2 (2021)
Big-O notation, Taylor's theorem, Moore-Osgood Theorem ---John McCuan
Principle of mathematical induction (we didn't get to this)
T 6 Some notes from the meeting including William's amazing calculation
Some good problems you should solve
T 13 Some notes from this meeting
Principle of inclusion/exclusion applied to areas---Joseph
Series on the Putnam Exams (compiled by Sushanth)
T 20 Problem B1 (Putnam exam 2010)---Taiki Aiba (also a series problem)
Problem A3 (Putnam exam 2020)---John McCuan (Sushanth's first problem)
T 27 We did some global trigonometric estimates via integration.
Drake solved Problem A3 (2020) and used one of those trig estimates.
Both topics can be reviewed (and more) in the document I've posted concerning this problem above under last week's meeting.
Note: I've also posted Taiki's solution from last week (above).
We did some work on formulas for sums of powers motivated by a problem posed by (I think) Ethan.
In particular, it came out that the sum of the first k cubes is the square of the sum of the first k integers.
Based on that the following problem was posed: If the q power of the sum of the first k integers
is equal to the sum of the first k integers to the power p, then
p = 3 and q = 2. (Drake says he can prove this.)
Then "Ethan" suggested we look at Problem B5 of the 2020 Putnam exam...and quickly left.
Here are some thoughts on the problem (which I do not know how to solve).
T Oct 4 I talked a little about B5 (2020)
Drake went though his proof mentioned above that p = 3 and q = 2.
T 11 Some notes from this meeting
Oct 17 Fall Break
T 18 Fall Break
Problem A3 (2018)
T 25 Some notes from this meeting
Sat 29 Practice Exam 10AM-1PM Skiles room 006 (in the basement)
T Nov 1 Some notes from this meeting
Paper of Dustin Jones related to A2 (2004)
T 8 Some notes from this meeting
Mathematica notebook with an animation of the parabolas
T 15 Some notes from this meeting
T 22 Some notes from this meeting
W 23 Student Recess
Th 24 Thanksgiving Holiday
F 25 Thanksgiving Holiday
...
Th Dec 1
Sat 3 Putnam Exam Skiles 006 Sessions 10AM-1PM and 3PM-6PM
M 5
T 6 Last meeting (Final Instructional Class Day)
B5 (2020) Solution I got it!
Th 8