Cezar Lupu

The 2016 William Lowell Putnam Competition Exam at the University of Pittsburgh

The 77th annual William Lowell Putnam Mathematical Competition took place on December 5th in 705 Thackeray Hall. Eight Pitt students had the mission of solving the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official Pitt Team members were Alex Mang, Andrew Tindall and Jack Hafer. Other participating students were: Matthew Gerstbrein, Terry Tan, Andrew Klang, Tianke Li and Haoming Yan. Below one can find this year’s Putnam problems. Congratulations to all participants!

SESSION A:

Problem A1. Find the smallest positive integer j such that for every polynomial p(x) with integer coefficients and for every integer k, the integer

\displaystyle p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}

(the j-th derivative of p(x) at k) is divisible by 2016.

Problem A2. Given a positive integer n, let M(n) be the largest integer m such that

\displaystyle \binom{m}{n-1}>\binom{m-1}{n}.

Evaluate

\displaystyle\lim_{n\to\infty}\frac{M(n)}{n}.

Problem A3. Suppose that f is a function from \mathbb{R} to \mathbb{R} such that

\displaystyle f(x)+f\left(1-\frac{1}{x}\right)=\arctan x

for real x\neq 0. (As usual, y=\arctan x means -\frac{\pi}{2}<y<\frac{\pi}{2} and \tan y=x).

Find

\displaystyle \int_0^1f(x)dx.

Problem A4. Consider a (2m-1)\times(2n-1) rectangular region, where m and n are integers such that m,n\ge 4. The region is to be tiled using tiles of the two types shown:

(The dotted lines divide the tiles into 1\times 1 squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.

What is the minimum number of tiles required to tile the region?

Problem A5. Suppose that G is a finite group generated by the two elements g and h, where the order of g is odd. Show that every element of G can be written in the form

\displaystyle g^{m_{1}}h^{n_{1}} g^{m_{2}}h^{n_{2}}\ldots g^{m_{r}}h^{n_{r}}

with 1\leq r\leq |G| and m_{1}, n_{1}, m_{2}, n_{2}, \ldots m_{r}, n_{r}\in \{1, -1\}. (Here |G| is the number of elements of G.)

Problem A6. Find the smallest constant C such that got every real polynomial P(x) of degree 3 that has a root in the interval [0, 1],

\displaystyle \int_0^1|P(x)|dx\leq C\cdot\max_{x\in [0,1]}|P(x)|.

SESSION B:

Problem B1. Let x_{0}, x_{1}, \ldots, x_{2}, \ldots be the sequence such that x_{0}=1 and for n\geq 0,

\displaystyle x_{n+1}=ln(e^{x_{n}}-x_{n}).

(as usual, the function ln is the natural logarithm). Show that the infinite series

\displaystyle x_{0}+x_{1}+x_{2}+\ldots

converges and find its sum.

Problem B2. Define a positive integer n to be squarish if either n is itself a perfect square of the distance from n to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is 45^2=2025 and 2025-2016=9 is a perfect square. (of the positive integers between 1 and 10, only 6 and 7 are not squarish.) For a positive integer N, let S(N) be the number of squarish between 1 and N inclusive. Find positive constants \alpha and \beta such that

\displaystyle\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,

or show that no such constants exist.

Problem B3. Suppose that S is a finite set of points in the plane such that the area of the triangle \Delta ABC is at most 1 whenever A, B and C are in S. Show that there exists a triangle of area 4 that (together with its interior) covers the set S.

Problem B4. Let A be a 2n\times 2n matrix, with entries choasen indepedently at random. Every entry is chosen to be 0 or 1, each with probability \frac{1}{2}. Find the expected value of \det(A-A^{t}) (as a function of n), where A^{t} is the transpose of A.

Problem B5. Find all functions f from the interval (1, \infty) to (1, \infty) with the following property:

if x, y\in (1, \infty) and x^2\leq y\leq x^3, then (f(x))^2\leq f(y)\leq (f(x)^3.

Problem B6. Evaluate

\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\cdot\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.

 


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NEAM-1st Northeastern Analysis Meeting, October 14-16, 2016, Brockport, NY, USA

The 1st Northeastern Analysis Meeting (NEAM 1) took place at the SUNY-College at Brockport, Brockport, NY between 14-16 October. More than 70 mathematicians affiliated with universities all around the world participated at the meeting.
The conference was organized by Gabriel Prajitura and Ruhan Zhao of SUNY-College at Brockport and it included special sessions on Fluid Dynamics and Dynamical Systems as well as 4 contributed sections. Although, the contributed sections involved more operator theory, it featured many other fields such as PDEs, number theory, combinatorics and probability. More details about the conference including the abstract of the talks are given below:

       Main Speakers                 Participants                Abstract of Talks                 Schedule

 


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The 2016 Putnam seminar, Fall 2016, University of Pittsburgh

Information about the competition and seminar (course description)

The William Lowell Putnam Mathematical Competition is the premiere competition for undergraduate students in North America. More than 500 universities compete in this contest organized by the Mathematical Association of America (MAA).  The competition takes place in the FIRST Saturday of December. This year’s Putnam competition will be held on Saturday, December 3 in Thackeray 703 or 705, from 10.00 am-1.00 pm and 3.00 pm-6.00 pm. The test is supervised by faculty members of each participating school. Every problem is graded on a scale of 0-10. The problems are usually listed in increasing order of difficulty, with A1 and B1 the easisest, and A6 and B6 the hardest. Top 5 scoring students on the Putnam exam are named Putnam Fellows. A student can take this exam maximum four times and the Putnam official team of the university consists of 3 members.
 The purpose of this class is to provide a comprehensive introduction into the world of problem solving in different branches of mathematics such as: real & complex analysis, linear algebra, abstract algebra, combinatorics, probability, geometry and trigonometry and number theory.
The teaching style will be a mixture of a lecture and a problem-solving session. By the end of this course, students should develop fundamental problem solving skills, and become accustomed to concentrating on a problem for an extended period of time. Indeed, this seminar concentrates on the raw creative problem-solving skills which can serve as an essential ingredient in almost every field of activity.  On the other hand, starting this Fall, the Putnam seminar has honors designation.
Last year, Pitt official Team ranked 24th in the nation and this marks the best performance since 2002. More details about this can be found here and here.

Course number, lecturers and webpage
Schedule and locations
  • Tuesday, 5.15-7.00 PM in Thackeray Hall, room 427

This is a lecture given by the instructor on a certain topic. The students will learn different concepts and techniques. Moreover, the lecturer will also present solutions of some problems and will assign other problems as homework for the students.

  • Thursday, 5.15-7.00 PM in Thackeray Hall, room 703 

This is more like a recitation rather than a lecture. The students will meet and discuss with the coordinators the problems assigned by the lecturer as homework.

Detailed program (lectures & recitations)

Week 1. What is Putnam competition? Organizational meeting

Abstract: This is an introduction to the seminar and the competition itself. There will be discussed training techniques for a competition of this caliber.

Lecturers: George Sparling & Cezar Lupu
Date: September 8


Week 2. (Elementary) Algebra

Abstract: This seminar will cover problems on topics such as algebraic identities and inequalities as well as polynomials in one variable.

Lecturer: Cezar Lupu
Date: September 13

Recitation instructor: Cezar Lupu
Date: September 15


Week 3. Real Analysis I

Abstract: This will cover problems on topics such as sequences of real numbers, convergence, limits and series of real numbers.

Lecturer: Cezar Lupu
Date: September 20

Recitation instructor: Cezar Lupu
Date: September 22


Week 4. Geometry and Trigonometry

Abstract: This will cover problems on topics such as vectors, conics, quadratics, and other curves in the plane as well as trigonometric formulae.

Lecturer: Derek Orr
Date: September 27

Recitation instructors: George Sparling & Cezar Lupu
Date: September 29


Week 5. Abstract Algebra

Abstract: This will cover problems on topics such as groups, rings, and finite fields.

Lecturer: George Sparling
Date: October 4

Recitation instructor: George Sparling
Date: October 6


Week 6. Real Analysis II

Abstract: This will cover problems on topics such as continuity and differentiability of functions of single variable.

Lecturer: Roxana Popescu
Date: October 11

Recitation instructor: George Sparling & Cezar Lupu
Date: October 13


Week 7. Linear Algebra I (video)

Abstract: This will cover problems on topics such as matrices and determinants.

Lecturer: Cezar Lupu
Date: October 18

Recitation instructor: George Sparling & Cezar Lupu
Date: October 20


Week 8. Number Theory I

Abstract: This will cover problems on topics such as arithmetic of numbers, prime numbers and diophantine equations.

Lecturer: Thomas Hales
Date: October 25

Recitation instructor: George Sparling
Date: October 27


Week 9. Number Theory II

Abstract: This will cover problems on topics such as arithmetic functions and quadratic residues.

Lecturer: Thomas Hales
Date: November 1

Recitation instructor: George Sparling
Date: November 3


Week 10. Real Analysis III (video)

Abstract: This will cover problems on topics such as Riemann sums and integrals.

Lecturer: Cezar Lupu
Date: November 8

Special lecture! Generating Functions and Applications

Lecturer: Vlad Matei (University of Wisconsin-Madison)
Date: November 10


Week 11. Linear Algebra II

Abstract: This will cover problems on topics such as vector spaces, linear transformations, characteristic polynomial, eigenvalues  and eigenvectors.

Lecturer: Cezar Lupu
Date: November 15

Recitation instructor: George Sparling & Cezar Lupu
Date: November 17


Week 12. Thanksgiving break!

Abstract: No seminar this week!

Date: November 22 & 24


Week 13. Real Analysis IV (video)

Abstract: This will cover problems on special topics in analysis and beyond.

Lecturer: Cezar Lupu
Date: November 29

Special lecture! Problems with similar ideas in Integral Calculus (video)

Lecturer: Cezar Lupu
Date: December 1


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Putnam’16 squad

Overview of the course CALCULUS III (MATH 0240), Summer 2016, University of Pittsburgh

This Summer I shall be teaching CALCULUS III (MATH 0240) in the mathematics department at University of Pittsburgh. Mainly, we shall cover vectors, vector functions and space functions, arc length and curvature, functions of several variables, double and triple integrals, line integrals, surface integrals and some other related topics.

About the course

This is the third sequence of three calculus courses for science and engineering students. The goal is to prepare you to make use of calculus as a practical problem-solving tool.

Prerequisite

Math 0230 or equivalent, with grade of C or better.

Lectures and Recitations

  • Lectures are from Monday to Thursday, 7-8.45 PM  in 704 Thackeray Hall.
  • Recitations are from Monday to Thursday, 6-6.50 PM  in 704 Thackeray Hall. The recitation instructor for this class is Fawwaz Battayneh. His office hours are Monday and Thursday, 12-1 PM in the MAC.
  • Moreover, you can also check the regular teaching page for this class: https://lupucezar.wordpress.com/teaching/ (Calculus 3-Summer 2016).
  • Quizzes and exams will appear on the teaching section above.

Office Hours

My office hours are from Monday to Thursday, 5-6 PM in the MAC. If you want to schedule an additional meeting with me in my office (Thackeray hall, room 415) do not hesitate to send me an e-mail to lupucezar@gmail.com.

References (Lecture notes, textbook, and practice exams)

  1. P. Hajlasz, Lecture Notes in Calculus III, Part 1, Part 2, Part 3.
  2. J. Stewart, Essential Calculus, Early Transcendentals, 2nd edition.
  3. A. Athanas, Calculus 3-practice exams.
  4. I. Sysoeva, Calculus 3-practice exams.
  5. E. Trofimov, Calculus 3-webpage.
  6. A. Yarosh, Calculus 3-webpage.

Grades

Your final grade will be determined as follows:

  • Homework: 20% (2 homeworks of 10% each)
  • Quizzes: 20% (4 quizzes of 5% each)
  • Two exams: 60% (2 exams of 30% each)
  • LON CAPA homework: 5% (extra credit)

Homework

I shall assign two homeworks which will appear on the teaching section. The first homework is hereEach one will be due before the exams. Additionally, one can do problems from the LON CAPA system. If you find it really annoying, you can also do this.

Final Grade Policy

A: 90-100%, A-: 85-90%, B+: 80-85%, B: 75-80%, B-: 70-75%, C+: 65-70%, C: 60-65%.

Exam Dates

The first exam will be on Thursday, July 14th, 7-9 PM in Thackeray 704.

The second exam will be on Wednesday, August 3rd, 7-9 PM in Thackeray 704.

Disability Resource Services

If you have any disability for which you are or may be requesting an accomodation, you are encouraged to contact both your instructor and the Office of Disability and Services, 216 William Pitt Union (412) 624-7890 as early as possible in the term.

Gauss normal distribution proof


spherical_coordinates_integral_formula_2

The integral formula in spherical coordinates system

fubini_theorem

Fubini’s theorem

green_theorem_2

Green’s theorem


The 2016 Mathematical Olympiad Summer Program, June 7-July 2, Carnegie Mellon University, Pittsburgh, PA

The 2016 Mathematical Olympiad Summer Program (MOP or MOSP) is organized for the second time by the Mathematical Association of America (MAA) at Carnegie Mellon University (CMU) in the city of Pittsburgh between June 7th and July 2nd.  The camp is run by the CMU faculty Po-Shen Loh (director) with the help of its associate director Razvan Gelca of Texas Tech University. They will be accompanied by other instructors (faculty, postdocs and Ph.D. students) from schools such as Massachusetts Institute of Technology (MIT), Stanford University, Harvard University, Columbia University, Carnegie Mellon University (CMU), North Carolina State University and University of Pittsburgh. This year’s MOP staff is here. The news about this year’s Mathematical Olympiad Summer Program appears on CMU’s front page here. This year’s guest speaker is Steve Shreve from CMU. Moreover, Noam Elkies from Harvard University will be the invited speaker for one of the evening’s seminars. Carnegie Mellon University also released a video about MOP:

Some details about the schedule are below:

  • Classrooms are Gates 4101, Gates 4102, Scaife 125, and Wean 8220. The room numbers are indicated on the top row of the MOP schedule, in the leftmost column.
  • Students will be separated into four groups.
    • Black: USAMO winners and IMO team
    • Blue: next top few from USAMO
    • Red: students in grades 9 and 10.
    • Many Red students will come with no prior MOP experience. Black level is quite impressive
  • The timetable will be:
    • 8:30am – 10:00am (Lecture 1)
    • 10:15am – 11:45am (Lecture 2)
    • 1:15pm – 2:45pm (Lecture 3), or 1:15pm – 5:45pm
    • 7:30pm: optional-attendance evening research seminar

Moreover, this year is for the first time when MOP welcomes international students from countries such as Australia, Canada, Hungary, India, Romania, and Singapore. The two amazing Romanian students are Ioana Teodorescu and Stefan Tudose from the International Computer High School of Bucharest (ICHB).  Another important news is the fact that four Romanians will serve as academic instructors. Besides the associate director, Razvan Gelca and myself, Bogdan Ion and Irina-Roxana Popescu of University of Pittsburgh are in the staff as well. This is for the second time in the history of MOP when this happens. The last time when MOP had four Romanian instructors was back in 2002. This was covered by Bogdan Suceava in the Romanian newspaper Evenimentul Zilei here. Also, the entire summary of courses is here. I  shall deliver 15 lectures/ problem sessions & 1 seminar as follows:


 

1. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: black
Time and date: 8.30-10 AM, Thursday June 9th
Location: Wean Hall, Room 8820

SEMINAR. Arcsine power series expansion and Euler’s \zeta(2).

Groups: black. green, red 1 & 2
Time and date: 7.30-8.20 PM, Sunday June 12
Location: Stever Dorm

2. Real analysis techniques in solving elementary problems.

Group: black
Time and Date:  8.30-10 AM, Tuesday, June 14th
Location: Wean Hall, Room 8820

3. Real analysis techniques in solving elementary problems.

Group: blue
Time and Date:  1.15-2.45 PM, Tuesday, June 14th
Location: Scaife Hall, Room 125

4. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: red 2
Time and Date: 8.30-10 AM. Wednesday, June 15th
Location: Gates Hall, Room 4102

5. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: red 1
Time and Date: 10.15-11.45 AM, Wednesday June 15th
Location: Gates Hall, Room 4101

6. Fourier series and applications to elementary problems.

Group: blue
Time and Date: 8.30-10 AM, Thursday, June 16th
Location: Scaife Hall, Room 125

7. Fourier series and applications to elementary problems.

Group: black
Time and Date: 1.15-2.45 PM, Thursday, June 16th
Location: Wean Hall, Room 8220

8. Sequences, series of real numbers and inequalities.

Group: blue
Time and Date: 8.30-10 AM, Tuesday, June 21st
Location: Scaife Hall, Room 125

9. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: blue
Time and Date:  8.30-10 AM, Wednesday, June 22nd
Location: Scaife Hall, Room 125

10. Problems in computational geometry.

Group: red 2
Time and Date: 8.30-10 AM, Thursday, June 23rd
Location: Gates Hall, Room 4102

11. Problems in computational geometry.

Group: red 1
Time and Date: 10.15-11.45 AM, Thursday, June 23rd
Location: Gates Hall, Room 4101

12. Algebraic integers and applications.

Group: blue
Time and Date: 1.15-2.45 PM, Tuesday, June 28th
Location: Scaife Hall, Room 125

13. Algebraic integers and applications.

Group: red 2
Time and Date:  8.30-10 AM, Wednesday, June 29th
Location: Gates Hall, Room 4102

14. Algebraic integers and applications.

Group: red 1
Time and Date:  10.15-11.45 AM, Wednesday, June 29th
Location: Gates Hall, Room 4101

15. Irreducible polynomials in one variable and applications.

Group: blue
Time and Date: 8.30-10 AM, Friday, July 1st
Location: Scaife Hall, Room 125

PHILOSOPHY. Beyond MOP.

Groups: black, green, red 1 & 2
Time and date: 1.15-2.45 PM, Friday July 1st
Location: Stever Dorm




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The William Lowell Putnam competition 2015 results (University of Pittsburgh)

The results of the 2015 William Lowell Putnam Mathematical Competition came out! The winner is Massachusetts Institute of Technology (MIT) followed by Carnegie Mellon University (CMU) and Princeton University in third place. There were 4275 contestants from over 500 universities in USA and Canada.

The University of Pittsburgh official Team (Stefan Ivanovici, Mingzhi Tian and Matthew Smylie) ranked 24th nationally and two of its members, Stefan and Ming, made it to top 460. In fact, Stefan made it to the top 200. Also, Derek Orr delivered a fine performance and he managed to make it to top 460 as well. Moreover, this news appeared on department’s front page.

This is the best performance as a team since 2002. Congratulations to all Pitt participants! It has been a great pleasure for George Sparling and I to work with such a group of talented students here at Pitt and this huge accomplishment is the result of two years of training. On the other hand, we want to mention and thank other faculty and graduate students involved in this year’s Putnam seminar: Gregory Constantine, Bogdan Ion, Kiumars Kaveh and Irina-Roxana Popescu. Moreover, we would like to acknowledge Bogdan’s involvement in previous year’s Putnam seminar. Some of these performances are listed below.

Pitt Team rank:

  • 2014-108th
  • 2013-98th
  • 2002-17th
  • 2001-18th
  • 2000-23rd
  • 1998-13th (best ever performance!)
  • 1997-15thpitt_logo
    putnam team at the ceremony

    From left to right: Jonathan Rubin (undergraduate director), Stefan Ivanovici, yours truly, Matthew Smylie and Derek Orr

    Putnam Team and the coaches April 2016

    From left to right: Derek Orr, Matthew Smylie, yours truly, Stefan Ivanovici and George Sparling

    Putnam group 2015

    From left to right: Derek Orr, Stefan Ivanovici, Matt Smylie, yours truly, Alex Mang, Jack Hafer, George Sparling, Tommy Bednar and Mark Poulson

    williamputnamMAA formuale

Initial value problems, arcsin power series expansion and Euler’s zeta(2)

In this post we shall discuss an ODE with initial conditions to derive a power series expansion for \arcsin^2(x) near x=0. Let us consider the following initial value problem

\displaystyle (1-x^2)y^{{\prime}{\prime}}-xy^{\prime}-2=0, y(0)=y^{\prime}(0)=0.

It is quite clear that the function g:(-1, 1)\to\mathbb{R}, \displaystyle g(x)=\arcsin^2(x) satisfies the ODE above and since the coefficients are variable it is natural to look for a power series solution \displaystyle y(x)=\sum_{n=0}^{\infty}a_{n}x^n. Differentiation term by term yields

\displaystyle y^{\prime}(x)=\sum_{n=0}^{\infty}(n+1)a_{n+1}x^n and \displaystyle y^{{\prime}{\prime}}(x)=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n.

Since a_{0}=y(0)=0, a_{1}=y^{\prime}(0)=0, our initial value problem will be equivalent to

\displaystyle 2a_{2}+6a_{3}x+\sum_{n=2}^{\infty}((n+2)(n+1)a_{n+2}-n^2a_{n})x^n=2.

Clearly, a_{2}=1, a_{3}=0, and for n\geq 2, \displaystyle a_{2n+1}=0, a_{n+2}=\frac{n^2}{(n+2)(n+1)}a_{n}. We easily obtain that for n\geq 2,

\displaystyle a_{2n+1}=0 and \displaystyle a_{2n}=\frac{(2^{n-1}(n-1)!)^2}{(2n)(2n-1)\ldots 4\cdot 3}=\frac{1}{2}\frac{2^{2n}}{n^2\binom{2n}{n}}.

Therefore, this implies

y(x)=\boxed{\displaystyle \arcsin^2(x)=\frac{1}{2}\sum_{n=1}^{\infty}\frac{(2x)^{2n}}{n^2\binom{2n}{n}}}, |x|\leq 1.

The above formula serves as a good ingredient in evaluating series involving the central binomial coefficient like

\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2\binom{2n}{n}}=\frac{\pi^2}{18}, \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^3\binom{2n}{n}}=\frac{2}{5}\zeta(3), \sum_{n=1}^{\infty}\frac{1}{n^4\binom{2n}{n}}=\frac{17}{3456}\pi^4.

This series converges at x=1 by Raabe’s test, and for x\in (-1, 1) it is uniformly convergent by the Weierstrass M-test.

Now, substitute x=\sin t, 0<t<\frac{\pi}{2}, and we get

\displaystyle t^2=\sum_{n=1}^{\infty}\frac{2^{2n-1}}{n^2\binom{2n}{n}}\sin^{2n}t.

Integrating from 0 to \frac{\pi}{2}, we have

\displaystyle \frac{\pi^3}{24}=\sum_{n=1}^{\infty}\frac{2^{2n-1}}{n^2\binom{2n}{n}}\int_0^{\frac{\pi}{2}}\sin^{2n}t.

On the other hand, Wallis’ formula (integral form) tells us that \displaystyle\int_0^{\frac{\pi}{2}}\sin^{2n}tdt=\frac{\pi}{2^{2n+1}}\binom{2n}{n}, and thus we finally obtain Euler’s celebrated

\displaystyle\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.

Remark. The ideas presented above have been generalized in this short paper.