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Dear All,Our next meeting is on Friday, November 6th
at GC 283B, from Noon till 2:00 p.m.
We will be doing (A2 and B2) problems from Putnam
competitions: the rest of those I already sent to you.
2008/A2:Alan and Barbara play a game in which they take turns
filling entries of an initially empty $2008 \times 2008$ array.
Alan plays first. At each turn, a player chooses a real
number and places it in a vacant entry. The game ends
when all the entries are filled. Alan wins if the determi-
nant of the resulting matrix is nonzero; Barbara wins if
it is zero. Which player has a winning strategy?
2008/A3:Start with a finite sequence $a_1 , a_2 , . . . , a_n$ of positive
integers. If possible, choose two indices $j < k$ such
that $a_j$ does not divide $a_k$ , and replace $a_j$ and $a_k$ by
$\gcd(a_j , a_k )$ and $\text{lcm}\,(a_j , a_k )$ , respectively. Prove that if
this process is repeated, it must eventually stop and the
final sequence does not depend on the choices made.
(Note: gcd means greatest common divisor and lcm
means least common multiple.)
2008/B2: Let $F_0 (x) = \ln x$ . For $n \geq 0$ and $x > 0$ , let
$F_{n+1} (x) = \int_0^x F_n(t) dt$
Evaluate
$\lim_{x \to \infty} n! \frac{F(1)}{ \ln n}.$
2008/B3: What is the largest possible radius of a circle contained
in a $4$ -dimensional hypercube of side length $1$ ?
2007/A1: Find all values of $\alpha$ for which the curves
$y = \alpha x^2 + \alpha x + 1/24$
and
$x = \alpha y ^2 + \alpha y +1/ 24$
are tangent to each other.
2007/A2:Find the least possible area of a convex set in the plane
that intersects both branches of the hyperbola $xy = 1$
and both branches of the hyperbola $xy = -1$ . (A set $S$
in the plane is called convex if for any two points in $S$
the line segment connecting them is contained in $S$ .)
2007/A3:Let $k$ be a positive integer. Suppose that the integers
$1, 2, 3, \ldots , 3k + 1$
are written down in random order. What is the probability that at
no time during this process, the sum of the integers that have
been written up to that time is a positive integer divisible by $3$ ? Your answer should be in closed form, but may include factorials.
2007/B2:Suppose that $f : [0, 1] \rightarrow \RR$ has a continuous
derivative, and that
$\int_0^1 f(x) dx = 0.$
Prove that for every $\alpha \in (0,1)$ , $| \int_0^{\alpha} f(x)\, dx | \leq \text{max}_{[0, 1]} | f'(x) \,dx |.$

Friday, October 16, 2009: Math Club Meeting Today and Putnam Problems

Dear All,
Our next meeting is on Friday, October 16th,
at GC 283B, from Noon till 2:00 p.m.

We will be doing (A1 and B1) problems from Putnam
competitions.
Here are some recent B1 Putnam Problems:
2001. Let $n$ be an even positive integer.
Write the numbers $1, 2, . . . , n_2$ in the squares of an $n \times n$ grid so that the $k$ -th row, from left to right, is $(k - 1)n + 1, (k - 1)n + 2, . . . , (k - 1)n + n$ .Color the squares of the grid so that half of the squares
in each row and in each column are red and the other half
are black (a checkerboard coloring is one possibility). Provethat for each coloring, the sum of the numbers on the red
squares is equal to the sum
of the numbers on the black
squares.
2002.Shanille O’Keal shoots free throws on a basketball court.
She hits the first and misses the second, and thereafter
the probability that she hits the next shot is equal to theproportion of shots she has hit so far. What is the probability
she hits exactly $50$ of her first $100$ shots?
2003.Do there exist polynomials $a(x), b(x), c(y), d(y)$ such that $1 + xy + x^2 y^2 = a(x)c(y) + b(x)d(y)$ holds identically?
2004.Let $P (x) = c_n X^n +c_{n-1} X^{n-1} +\cdots +c_0$ be a polynomial
with integer coefficients. Suppose
that r is a rational number
such that $P (r) = 0$ . Show that the $n$ numbers $c_n r$ ,
$c_n r^2 + c_{n-1} r$ ,
$c_n r^3 + c_{n-1} r^2 + c_{n-2} r$ ,
$. . .$ ,
$c_n r^n + c_{n-1} r^{n-1} + \cdots + c_1 r$
are integers.

See you at the Club!C. Morales,
M. Yotov

Friday, October 09, 2009: Math Club Meeting Today and Other News

Dear All,

Our next meeting is on Friday, October 9th, at GC 283B, from Noon till 2:00 p.m.

We will be doing (A1 and B1) problems from Putnam
competitions.

Please send me your name if you are willing to attend the Competition on December 5th.

Here are some recent A1 Putnam Problems:

2001.Consider a set $S$ and a binary operation $\ast$ , i.e., for each $a, b \in S$ , $a \ast b \in S$ . Assume $(a \ast b) \ast a = b$ for all $a, b \in S$ . Prove that $a \ast (b \ast a) = b$ for all $a, b \in S$ .

2002.Let $k$ be a fixed positive integer. The $n$ -th derivative of $1/(x^k - 1)$ has the form $\frac{P_n(x)}{(x^k - 1)^n}$ where $P_n(x)$ is a polynomial. Find $P_n(1)$ .

2003.Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, $n = a_1 + a_2 + \cdots + a_k$ , with $k$ an arbitrary positive integer and $a_1 \leq a_2 \leq \cdots \leq a_k \leq a_1 + 1$ For example, with $n = 4$ there are four ways: $4$ , $2+2$ , $1+1+2$ , $1+1+1+1$ .

2004.Basketball star Shanille O’Keal’s team statistician keeps track of the number, $S(N )$ , of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N )$ was less than $80\%$ of $N$ , but by the end of the season, $S(N )$ was more than $80\%$ of $N$ . Was there necessarily a moment in between when $S(N )$ was exactly $80\%$ of $N$ ?

Saturday, May 30th, 2009: Algebraic Geometry Seminar continues at a new day and time

Where: DM 409A (Math Lab)

When: Friday June 5th, 2009 at 1:30pm

Dear All,

We are returning to the previous schedule of the Seminar
Friday, 1:30 p.m., DM 409B.

After discussing the need for projective coordinates in the plane,
we are turning next time to the need for complex coordinates for
better understanding the link Algebra-Geometry in Algebraic
Geometry. The discussion will be based again on curves of low
degree: conics, cubics, quartics.

See you all there,

T. Draghici,
G. Grantcharov,
M. Yotov

Tuesday, May 26th, 2009: Introduction to Algebraic Geometry Seminar continues at a new day and time

Where: DM 409A (Math Lab)
When: Thursday May 28th, 2009 at 4:30pm

Dear All,

This meeting is devoted to an introduction to Algebraic Geometry.
We begin with a discussion of plane curves of small degree over
the real and complex numbers. The material should be accessible
to anyone with rudimentary knowledge of Calculus, Linear Algebra,
Polynomials of two variables.
We hope that the time of the meeting is convenient for all of you.

See you there,

T. Draghici

G. Garntcharov
M. Yotov

Thursday, May 14th, 2009: Announcement - Research Seminar is Moved Earlier - 12pm

Dear All,

The Research Seminar on Geometry this Friday
starts at 12:00 noon in DM 409B.
The speaker is Prof. Draghici.
The Student Seminar will begin as scheduled at
1:00 p.m.

Best regards,
M.Yotov

Monday, May 11th, 2009: Announcement; Student Seminar Second Meeting on Friday, May 15th at 1:00pm

Dear All,

The second meeting of the Seminar will take place again
in DM 409 B, at 1:00 p.m. on Friday.

Last time we discussed examples from the Math courses
most of the students attend, and discussed their categorical
nature. This time we are starting with the definition of
categories, and are discussing their basic features. The idea
is to learn the modern language of dealing with math notions,
and to get closer to one of the goals of the Seminar: Algebraic
Geometry.

Some of potential attendees of the Seminar informed me they
are busy on Fridays. We will try to reschedule the meetings for
another day (right before the Number Theory class for instance).

Hope to see you at the Seminar,

T. Draghici,
G. Grantcharov,
M. Yotov

Monday, May 4th, 2009: Announcement; Student Seminar Resumes on Friday, May 8th at 1:00pm

First meeting will be this week in DM 409B (the computer lab in the mathematics department). This meeting is mostly organizational: the topics of the Seminar for the Summer will be discussed as well as a better time for the meetings if need be. The Seminar is intended to give the participants some knowledge from the modern mathematics including topics from current investigations in Algebra and Geometry.

Featured Presenter (M. Yotov):

The first couple of lectures will introduce some basic language from Category Theory (a Mathematical Area) and give motivating examples for the use of and need for this language.

When: Friday, May 8th, at 1:00 p.m.

Where: DM 409B (computer lab in the mathematics department)

For more information about the Student Seminar, see below.

Monday, May 4th, 2009: Announcement; Research Seminar in Geometry on Friday, May 8th at 3:00pm

After the Student Seminar, there is a Research Seminar on Geometry: the same room[DM 409B], at 3:00 p.m.

Monday, May 4th, 2009: More Information: Student Seminar - Algebraic Geometry and Commutative Algebra

Dear All,

This is an advertisement about resuming the work of the Student Seminar: Algebraic Geometry and Commutative Algebra after more than a year long break (due to the change in the schedule of classes at FIU and impossibility to find time for the meetings of the Seminar which would be convenient for at least three participants!).

The Seminar is intended to give the participants some knowledge from the modern mathematics including topics from current investigations in Algebra and Geometry. Originally, the emphasis was on the "commutative" objects in Mathematics. After two-three years of work, we realized that such a constrain is artificial, so we decide to consider "non-commutative" objects as well. Thus, the change in the name of the Seminar: Algebra and Geometry.

We are glad to report that among the precious participants in the Seminar, there are McNair fellows, as well as successful graduate students in prestige Universities.

Traditionally, the Seminar relies upon the active participation of the students. This includes presentation of topics in Math the students work on (these may be different from the main ones of the Seminar), of proofs the students have independently obtained or learned in connection with their work and interests. We encourage all students share their thoughts and take part in the work of the Seminar. In particular, students from the McNair Program in Mathematics are expected to do so at the Seminar.

Some suggested topics to be considered are:

Polynomial rings, Hilbert's Basis Theorem and the Nullstellensatz;
Plane algebraic curves: conic sections, elliptic curves;
Projective plane and the Bezout's Theorem;
Addition on an elliptic curve; link to complex variable;
Quaternions; link to Geometry; generalizations;
Clifford Algebras, applications;
Other topics may be added.

Our first meeting is scheduled for this coming

Friday, May 8th, at 1:00 p.m. in DM 409B.

This meeting is mostly organizational: the topics of the Seminar for the Summer will be discussed as well as a better time for the meetings if need be.

Featured Presentation (M. Yotov):

The first couple of lectures will introduce some basic language from Category Theory (a Mathematical Area) and give motivating examples for the use of and need for this language.

Prerequisites: the concepts to be introduced are so general that no specific knowledge in Math is needed in order to grasp them. The examples would be faster and easier to understand if the participants know a bit of set theory, a bit of Linear Algebra, and a bit of Algebraic Structures. In case some of the latter are not in their math background, the participants may find the considerations on the Seminar as a valuable introduction to the mentioned areas.

We hope to see you there!

Note: After the Student Seminar, there is a Research Seminar on Geometry: the same room, 3:00 p.m.

T. Draghici
G. Grantcharov
M.Yotov

Algebraic Geometry and Commutative Algebra Seminar for Summer 2009 - 04/29/2009

Hello all,

Professor Mirroslav Yotov would like to re-open the Student Seminar with Commutative Algebra and Algebraic Geometry. There will be weekly meetings on the subject and also notes in pdf format for each lecture.

For those interested, please email a314sces@gmail.com with the
Subject: "Commutative Algebra and Algebraic Geometry"
and I'll be sure to keep you posted on the hour of the meetings. Also tell others! The more people, the more likely the Seminar will happen.

More information on the topic of the seminar: Here is an excellent summary of the Algebraic Geometry course in the Independent University of Moscow:
Algebraic geometry studies geometric loci defined by polynomial equations, for example the complex plane curve f(x,y)=0. It plays an important role at both elementary and sophisticated levels in many areas of mathematics and theoretical physics, and provides the most visual and elegant tools to express all aspects of the interaction between different branches of mathematical knowledge. The course gives the flavor of the subject by presenting examples and applications of the ideas of algebraic geometry, as well as a first discussion of its technical apparatus.

As for Commutative Algebra:
Commutative and homological algebra studies algebraic structures, say, modules over commutative rings, in terms of their generators and relations. It provides the most powerful algebraic tools for applications in algebraic and differential geometry, number theory, algebraic topology, etc. This course gives a quick introduction to these techniques. It is designed as the starting point for those who intend to study algebraic geometry and related topics.

Tuesday, March 24th, 2009: Announcement:

The Math Club will be holding its final general meeting of Spring Semester 2009 on Thursday, April 2, at 3:15 pm in The Math Department. There are many announcements and events during the month of April. Also, the structure of the club and its executive board for the following year, 2009-2010, will be discussed. As such, all those who are interested in holding a position should be in attendance, or should contact myself and state their intentions.

Hope to see you there,

Olivia Shand

President

Wednesday, February 11th, 2009: Math Club Meeting:

Hello All,

The Math Club will be holding a meeting tomorrow, Thursday, February 12, 2009, at 3:15 pm in the Math Department, DM 4th floor. The topic for discussion will be Analysis. Please bring any problems you are interested in discussing.

Mark your calendars! Julian Edward, Chair of The Math Department, will be giving a talk on Arrow's Theorem on Thursday, February 19, 2009, at 3:15 pm. We will be securing a room to hold the talk in, and will announce the location after it is secured.

Thanks,
Olivia Shand
President

Thursday, November 20th, 2008: New Problem:

Evaluate

Wednesday, November 19th, 2008: Meeting Friday, November 21th, 2008:

We are having our regular meeting on Friday,November 21st, at PCA 171, 12:30 p.m.

We continue doing problems as a preparation for thePutnam Competition which is coming on December 5th.

The featured presentation this time is about the powerseries expansion of tan(x). Leonard Foret will explainhis idea how to do the problem.

Wednesday, November 12th, 2008: Meeting Friday, November 14th, 2008:

We are having our regular meeting on Friday, November 14th, at PCA 171, 12:30 p.m.

We continue with a medley of series, sequences, integrals, number theory problems.

Thursday, November 6th, 2008: Meeting Tomorrow, Friday, November 7th, 2008:

We are having our regular meeting on Friday, November 7th, at PCA 171, 12:30 p.m.

Topics: a medley of series, sequences, integrals, and number theory problems.

Sunday, November 2nd, 2008: Problem of the Day:

0. Prove that

Where the number

is the number of permutations,

, of 1, 2, ... n such that

Some exercises:

1. Evaluate:

2. Evaluate:

3. Evaluate:

Thursday, October 30th, 2008: Meeting this Week on Friday, October 31th:

We are having our regular meeting on Friday, October 31st, at PCA 171, 12:30 p.m.

The featured presentation this time:

- Tony Wills will present a problem on numerical series.

- Problems on the topic of numerical and functional sequences and series.

Tuesday, October 21st, 2008: Meeting this Week on Friday, October 24th:

We are having our regular meeting Friday, October 24th, at PCA 171, 12:30 p.m., featuring the following two presentations:

1) A crash course on Pell's equations an often useful topic for the nearing Putnam Competition. It typically part of an advanced Number Theory course which but has been omitted at FIU due to time constraints.

2) A presentation prepared by Luis Rodriguez on an interesting, non-trivial logic problem.

We will also do problems from previous Putnam competitions if time permits.

Tuesday, October 14th, 2008: Problem of the Day!

1. Problem of the Day: Players 1,2,3, ..., n are seated around a table, and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing on penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.

Thursday, October 10th, 2008: Math Club Meeting Tomorrow!

We are having our regular meeting on Friday(tomorrow), October 10th, at PCA 171, 12:30 p.m.

New Problems for this week on the Weekly Problems page!

1. Let

be an $n \times n$ matrix all of whose entries are

and
whose rows are mutually orthogonal. Suppose

has an $a \times b$ submatrix
whose entries are all 1. Show that $ab \leq n$ .

Friday, October 3rd, 2008: Math Club Meeting Today!

We're having our regular meeting on Friday October 3rd in room PCA 171 at 12:30pm.
(really weird shaped building next to ECS/computer science building)

There are some extra problems posted in last weeks suggested problems that weren't solved (or attempted), those are part of this week's problems as well.

Also, anyone who wants to participate in the putnam this Fall 2008 must notify (someone) before wednesday of next week. You can post a comment in one of the forums, append the comment to a page, or send an email, or ask in person, but it needs to be done asap.

Tuseday, September 30th, 2008: Problems for Sept 29-October 3rd:

Three problems (including the one below) are now posted in the Weekly Problems page.

2. Sum the series:

Wednesday, September 23th, 2008: Problems for Sept 22-26th:

(More posted in the Weekly Problems page.)

1. Let

for some integer

. Say a permutation

of S has a local maximum at

if any of the following hold for k:

(i)

for k=1;

(ii)

and

for

(iii)

for k=n.

(For example, if n=5 and

takes values 1,2,3,4,5 of 2,1,4,5,3, then

has a local maximum of 2 at k=1, and a local maximum of 5 at k=4.)

Question:
What is the average number of local maxima of a permutation of S, averaging over all permutations of S ( averaging meaning, dividing by (n!) ) ?

Wednesday, September 23th, 2008: Idea for the website

I figure this format might draw more interest to the site, similar to other problem-oriented websites. I'll post a map to PCA 171 if anyone is having trouble finding the room.

Thursday, September 18th, 2008: Room Update!

We have a room! This week's meeting (tomorrow September 19th), will be in room PCA 171 at 12:30pm+. (PCA is the orange architecture building closer to ECS and 8th street in UP)

Friday, September 12th, 2008: New Problems for Week Sept 15-19th, 2008:

The meeting next week (September 19th, 2008 at 12:30pm+, DM423) will discuss the problems posted today in the Weekly Problems page.

Tuesday, September 9th, 2008: New Problems and Friday's Math Club Meeting:

Putnam Problems A2 and A3 from the 1985 exam will be discussed during this Friday's meeting at 12:30pm+ (in the lobby in front of DM 409A). The problems will be posted in the Weekly problems later tonight.

Friday, September 5th, 2008: Math Club & Math Seminar Meeting:

Math Club meeting lasted between 12:30pm-4:00pm. Math Club meetings are starting earlier, at around 12:30pm every Friday. If there are no rooms available, the meeting will be in the mathematics department lobby of DM, 4th floor.

Today there were two new problems introduced, and apple pi(thanks Olivia!). The two problems: one sent to Prof. Draghici, another suggested by Rafael (A2 in one of the Putnam problems). The details will be posted later on in the minutes page.

Tuesday, August 5th, 2008: Math Club & Math Seminar Meeting:

Math Seminar meeting this week Friday August 8th, 2008 in GC 276 at 1pm. Updated calender in Math Club Minutes , .
Math Club meeting this week will be: Friday August 8th, 2008; same room; after the seminar; starting approx 2-3pm.

(Note: For those who can't make it, there will be lecture notes posted on the website in the Math Club Minutes. Any suggestions for the website and future of the math club posted in the forum I'll compile a separate page.)

Wednesday, July 30th, 2008: Tomorrow Math Seminar Meeting:

Math Seminar Meeting on Thursday July 31st, 2008. Updated calender in Math Club Minutes , the exact time is TBA.

Thursday, July 24th, 2008: Solutions to July 14-18th Problem Set:

Solutions to 1, 3, 4, posted as a pdf in the Weekly Problems page.

Monday, July 21st, 2008: Updated Problems

Now the problems can be viewed and discussed without having to download a pdf file, check it out in the Weekly Problems page.

This is a trick I found using the website http://www.sitmo.com/latex/ which generates a permanent link for any equation you'd like to post.

To add an equation through EasyEdit, create the permanent link through the website above, copy the url, then upload as a photo in the EasyEdit Toolbar.

Posting the problems in this manner also circumvents wetpaint's 40 attachment limitation. This trick can also be used to discuss problems in the threads I believe.

Monday, July 21st, 2008: New Suggested Problems

by Prof. Draghici, can be found here. (or download the pdf)

About the FIU Math Club:

The Math Club at the Department of Mathematics was organized as a Fall semester weekly event aimed at preparing students with a passion for solving challenging math problems for the prestigious W. L. Putnam mathematics competition. Gradually, the Club became a full year weekly event, and is now a unique forum where students meet, solve challenging math problems, and discuss interesting math facts related to the subjects they are immersed in.

Following tradition, the Fall semester at the Club is devoted to the preparation of the participants for the Putnam competition. We use a wide range of problem solving books including the full collection of problems given in previous competitions.

The focus of the Spring and Summer semesters expands to include problems not related to the Putnam competition. These problems may be suggested by the students or the organizers of the Club. Interesting topics in math are also discussed. In Spring we invite faculty members to give talks on subjects chosen by the students. Regularly, participants themselves give talks sharing with their colleagues, fellow students and instructors what they have recently learned or proved. Past discussions have covered subjects including game theory, algebra, analytic number theory, commutative algebra and geometry. The discussions at the Club provide a nice opportunity for students to find interesting topics for their individual research projects. There have been three participants in the Club who recently became McNair fellows.

The Math Club activities are in close collaboration with the work of the Student Seminar on Commutative Algebra and Algebraic Geometry. The Club is also close in spirit to the work of the Math Circles organized by the Department. In particular, interested High School students are invited to come to the Club.

Traditionally, the Math Club meets on Friday afternoons. Participants are strongly encouraged to propose problems or topics for discussion. Most meetings are problem solving sessions meant as training for the Putnam Mathematical Competition. This is a prestigious math contest for undergraduate college students in the US and Canada. A good performance in this contest represents a strong point in the CV of a student applying to graduate school. FIU had a number of notable results in the Putnam contest in recent years, both as a team and via the individual scores of students. If you would like your name added to that list, the first step is to start attending the Math Club. The room and time for meetings in Summer 2008 will be announced by mid May. If you would like to be added to the mailing list please send an e-mail to Prof. Yotov or Prof. Draghici.

a314sces	Latest page update: made by a314sces , Nov 5 2009, 9:43 AM EST (about this update About This Update new post - a314sces 328 words added 31 images added view changes - complete history)
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