**University of West Timisoara, Romania**- 2003

### Professional Preparation

**University of West Timisoara, Romania**- 1980

**University of West Timisoara, Romania**- 1980

### Research Areas

##### Research Interests

1. Enhancing students interests and skills in mathematics.

I am the Director and Founder of AwesomeMath, which includes a
summer program for students training to succeed at the Olympiad level, a
correspondence-based lecture series for students continuing education,
called AwesomeMath Year-round, and Mathematical Reflections, a free
online journal focused primarily on mathematical problem solving. The
main purpose of this initiative is to give students an opportunity to
engage in meaningful learning activities and explore in detail areas in
advanced mathematics. AwesomeMath's primary focus is on problem-solving.
I use it as a tool to enhance students' interest and skills in
mathematics.

I believe that there are two major parts in significant mathematics teaching and learning: higher concepts (introducing and developing new topics) and applying those concepts creatively to concrete problems (bringing life to the new topics). These two areas rely on each other, but we focus primarily on the latter. I feel that certain advanced mathematics topics are best introduced to young students by motivating the concepts through problems that encourage investigation.

I am actively involved with mathematics competitions at the secondary and undergraduate levels. I write and contribute questions for the American Mathematics Competitions examinations as well as the International Mathematics Olympiad and the W. L. Putnam competition.

I would like to involve undergraduate students in my research. My area of research is an ideal entrance point to research for talented students, since it does not require a lot of background while offering abundance of open problems. In addition to the problems I can suggest, the students can also easily make their own conjectures, experiment by using Computer Algebra Systems, attack special cases, generalize and transfer ideas from one case to another, and learn and use various techniques while trying to resolve the problems. Thus they would experience first hand all the phases and subtleties of doing original research.

2. Diophantine Analysis, with emphasis on Quadratic Diophantine
Equations.

This research area focuses especially on the study of the general
Pell's equation, which is connected to problems from various domains of
mathematics and science, such as Thue's Theorem, Hilbert's Tenth
Problem, Euler's Concordant Forms, Einstein's Homogeneous Manifolds,
Hecke Groups, and so forth. I have obtained numerous original results
such as:

- In the cases when the equation is solvable, I found an elegant explicit form for the solutions. I then extended a result of D. T. Walker [The American Mathematical Monthly, vol. 74, no. 5, 1966, 504-513].
- I proved partial results about the equation and formulated conjectures regarding its solvability.
- I obtained results about the equation , including those regarding the LMM algorithm of finding the fundamental solutions to the general Pell's equation based on continued fractions.
- I devised two original methods of solving the equation .
- I proved that numerous important quadratic equations have infinitely many solutions in integers.

I devoted special attention to the Diophantine representability of several interesting sequences of positive integers. I introduced the concept of r-Diophantine representability of a sequence of positive integers and studied in an original manner the equations by employing the special Pell's equation . I have extensively studied the Diophantine representability of the Fibonacci, Lucas, and Pell sequences using methods of investigation different from and simpler than the ones already found in the literature. I also studied the problem of Diophantine representability of generalized Lucas sequences, which I introduced, finding conditions under which their general solution is a linear combination with rational coefficients of the classical Fibonacci and Lucas sequences.

I found numerous applications of the results above. For
example, I determined conditions under which the numbers and are
simultaneously perfect squares for infinitely many positive integers n.
I discovered special properties of triangular numbers, such as proving
that any positive rational number r, where is irrational, can be written
as the ratio of two triangular numbers. I extended some results
pertaining to triangular numbers to polygonal numbers.

### Publications

**Publication**

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### Appointments

**Associate Professor**

*University of Texas at Dallas (UTD), Richardson TX*[2005–Present]

Science/Mathematics Education Department

**Visiting Scholar**

*University of Wisconsin-Whitewater (UWW), Whitewater, Wl*[2003–2004]

**Visiting Scholar**

*Tokai University, Tokyo, Japan*[2003–2003]

**Director**

*University of Nebraska-Lincoln (UNL), Lincoln, NE*[1998–2003]

**Mathematics Teacher**

*Illinois Mathematics and Science Academy (IMSA), Aurora, IL*[1991–1998]

**Mathematics Teacher**

*Loga Academy, Timisoara, Romania*[1980–1990]

### Projects

##### Keynote speaker at MathCamp in Vancouver

**1998–1998**

##### Keynote speaker at the Gauss private school in Lima, Peru

**2000–2000**

*Keynote speaker at the Gauss private school in Lima, Peru, January, 2000*

##### Consultant to the Iberoamerican Mathematics Olympiad in Puerto Plata, Dominican Republic

**1999–1999**

##### Worked with Brazilian students preparing for their National Olympiad and the IMO at "Farias Brito" School in Fortaleza, Brazil

**1999–2000**

##### Opening lecturer at Canada/USA

**2000–2000**

*Opening lecturer at Canada/USA Math Camp in Toronto, Canada, July, 2000*

### Additional Information

##### Other Student Activities

1995-2002 Director of the Mathematical Olympiad Summer Program (MOSP) and Leader of the US delegation to the IMO. In addition to MOSP teaching responsibilities, oversaw the work of the other instructors and closely supervised their teaching.

1994-2002 Head Coach of the USA Mathematical Olympiad Team. Led the US team to its historic lst place in 1994 when all six American students achieved perfect scores, unique performance in the 45-year history of the IMO. Led Team USA to 2"d place in 1996 and 2001, and 3rd place in 1998, 2000, and 2002 in a field of more than 80 participating countries.

1993-1999 Grading room chair of American Regions Mathematics League (ARML)

1993-1994 and 2003—Present Assistant Coach of the USA IMO Team

1991-1998 Coach of the Chicago Area All-star Mathematics Team

1991 -1998 Co-sponsor of the IMSA chapter of Mu Alpha Theta

1983-1989 Counselor, Romanian Ministry of Education. Designed and implemented specialized programs to optimize the education of gifted middle and high school students, involving close interaction with university professors and senior teachers associated with Romania's best secondary schools. Assistant Coach of the Romanian Mathematics Olympiad Team and Deputy Leader of the Romanian Team for the Mathematical Contest of the Balkan Countries and the IMO.

##### Academic Honors/Awards

2001-2003 On the list of MAA lecturers. The MAA has a program whereby each MAA section may invite at the MAAs expense one speaker from the list to lecture at their annual section meeting.1995 Certificate of Appreciation Presented by President of the MAA for "Outstanding service as Coach of the USA Mathematical

Olympiad Program in preparing the USA Team for its perfect performance in Hong Kong at the 1994 IMO".

1994 First place winner of Edith May Sliffe Award for Distinguished High School Mathematics Teaching, awarded by President of the MAA.

1983 Distinguished Teacher Award bestowed by the Romanian Secretary of Education.

##### Membership in Professional Societies

- American Mathematical Society, since 1995
- Mathematical Association of America, since 1994
- National Council of Teachers of Mathematics, since 1992
- American-Romanian Academy, since 2002