Books by Amir Parvardi
Self-published, 2023
The Olympiad Algebra Book comes in two volumes. The first volume, dedicated to Polynomials and Tr... more The Olympiad Algebra Book comes in two volumes. The first volume, dedicated to Polynomials and Trigonometry, is a collection of lesson plans containing 1220 beautiful problems, around two-thirds of which are polynomial problems and one-third are trigonometry problems. The second volume of The Olympiad Algebra Book contains 1220 Problems on Functional Equations and Inequalities, and I hope to finish it before the end of Summer 2023. I hope I can finish collecting the FE and INEQ problems by June 29th, as a reminder of 1220 Number Theory Problems published as the first 1220 set of the J29 Project. The current volumes has 843 Polynomial problems and 377 Trigonometry questions, the last 63 of which are bizarre, ancient spherical geometry problems that were not included in the previous draft (Kaywañan Monster Trigonometry).
This book is supposed to be a problem bank for Algebra, and it forms the resource for the first series of the KAYWAÑAN Algebra Contest. I suggest you start with Polynomials, and before you get bored or exhausted, also start solving Trigonometry problems. If you find these problems easy and not challenging enough, the Spherical Trigonometry lessons and problems are definitely going to be a must try!
Independently published, 2018
This challenging book contains fundamentals of elementary number theory as well as a huge number ... more This challenging book contains fundamentals of elementary number theory as well as a huge number of solved problems and exercises. The authors, who are experienced mathematical olympiad teachers, have used numerous solved problems and examples in the process of presenting the theory. Another point which has made this book self-contained is that the authors have explained everything from the very beginning, so that the reader does not need to use other sources for definitions, theorems, or problems. On the other hand, Topics in Number Theory introduces and develops advanced subjects in number theory which may not be found in other similar number theory books; for instance, chapter 5 presents Thue's lemma, Vietta jumping, and lifting the exponent lemma (among other things) which are unique in the sense that no other book covers all such topics in one place. As a result, this book is suitable for both beginners and advanced-level students in olympiad number theory, math teachers, and in general whoever is interested in learning number theory.
Independently published, May 29, 2018
Functional equations, which are a branch of algebraic problems used in mathematical competitions,... more Functional equations, which are a branch of algebraic problems used in mathematical competitions, appear in recent olympiads very frequently. The current book is the first volume in a series of books on collections of solved problems in functional equations. This volume contains 175 problems on the subject, including those used in latest mathematical olympiads (2017 - 2018) around the world. The basic concepts of functional equations and techniques of problem solving have been briefly discussed in the preamble of the book.
Teaching Documents by Amir Parvardi
Here are 407 missing problems that did not fit into the pool of 1220 problems in the first volume... more Here are 407 missing problems that did not fit into the pool of 1220 problems in the first volume of The Olympiad Algebra Book dedicated to Polynomials and Trigonometry. The majority of the questions are chosen from American competitions such as AIME (American Invitational Mathematics Examination), HMMT (Harvard-MIT Math Tournament), CHMMC (Caltech Harvey Mudd Math Competition), and PUMaC (Princeton University Math Competition). All of the AIME problems are copyright © Mathematical Association of America, and they can be found on the Contests page on the Art of Problem Solving website. In this document, the links to the problems posted on AoPS forums are embedded (if existent).
This problem set was created on August, 2012. It contains problems of different levels in number ... more This problem set was created on August, 2012. It contains problems of different levels in number theory and is a good source for practicing regional and international olympiads.
Crated on June, 2011. Problems are taken from IMO, IMO Shortlist/Longlist, and some other famous ... more Crated on June, 2011. Problems are taken from IMO, IMO Shortlist/Longlist, and some other famous math competitions.
Various Number Theory Problems posted on AoPS on June 16, 2011
Created on January, 2011. Contains medium-to-hard problems for practicing geometry. Solving these... more Created on January, 2011. Contains medium-to-hard problems for practicing geometry. Solving these problems is suggested for preparing for international olympiads such as IMO, APMO, etc.
Created on June, 2011. Contains functional equation problems in algebra and is suggested for stud... more Created on June, 2011. Contains functional equation problems in algebra and is suggested for students who are preparing for international/national math olympiads.
Crated on January, 2011. This was the very first problem set I had ever created.
Created on March, 2011. A good source of polynomial problems in algebra.
Created on February, 2011. This problem set contains 53 interesting problems on trigonometry.
Created on June, 2011. Problems are taken from IMO, IMO Shortlist/Longlist, and some other famous... more Created on June, 2011. Problems are taken from IMO, IMO Shortlist/Longlist, and some other famous math competitions.
Papers by Amir Parvardi
Let p be a prime number. In [1], Booker and Pomerance find an integer y with 1 < y ≤ p such th... more Let p be a prime number. In [1], Booker and Pomerance find an integer y with 1 < y ≤ p such that all non-zero residue classes modulo p can be written as a square-free product of positive integers up to y. Let us denote by y(p) the smallest such y. Booker and Pomerance show in their paper that except for p = 5 and 7, we have y(p) ≤ y and some better upper bounds were conjectured. Later, Munsch and Shparlinski [7] proved those conjectures with even better localization. Their work was done as the same time as ours, but with fairly more complicated methods in the proof. We were seeking to find a solution for the problem using Pólya-Vinogradov inequality or at most its improvement, the Burgess bound on character sums. That being said, we removed the condition in the problem that the product has to be square-free. We proved that for m > p √ , each residue class b of (Z/pZ)× can be written as a product of elements of the set {1, 2, . . . ,m} modulo p. In fact, we showed that the numb...
Lifting The Exponent Lemma is a powerful method for solving exponential Diophantine equations. It... more Lifting The Exponent Lemma is a powerful method for solving exponential Diophantine equations. It is pretty well-known in the Olympiad folklore (see, e.g., [3]) though its origins are hard to trace. Mathematically, it is a close relative of the classical Hensel’s lemma (see [2]) in number theory (in both the statement and the idea of the proof). In this article we analyze this method and present some of its applications. We can use the Lifting The Exponent Lemma (this is a long name, let’s call it LTE!) in lots of problems involving exponential equations, especially when we have some prime numbers (and actually in some cases it “explodes” the problems). This lemma shows how to find the greatest power of a prime p – which is often ≥ 3 – that divides a± b for some positive integers a and b. The proofs of theorems and lemmas in this article have nothing difficult and all of them use elementary mathematics. Understanding the theorem’s usage and its meaning is more important to you than ...
Canadian Mathematical Bulletin
For any prime p, let $y(p)$ denote the smallest integer y such that every reduced residue class (... more For any prime p, let $y(p)$ denote the smallest integer y such that every reduced residue class (mod p) is represented by the product of some subset of $\{1,\dots ,y\}$ . It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue (mod p); we prove that $y(p) \ll _\varepsilon p^{1/(4 \sqrt e)+\varepsilon }$ , thus strengthening Burgess’ classical result. This result is of intermediate strength between two other results, namely Burthe’s proof that the multiplicative group (mod p) is generated by the integers up to $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$ , and Munsch and Shparlinski’s result that every reduced residue class (mod p) is represented by the product of some subset of the primes up to $O_\varepsilon (p^{1/(4 \sqrt e)+\varepsilon })$ . Unlike the latter result, our proof is elementary and similar in structure to Burgess’ proof for the least quadratic nonresidue.
In this paper, we define finitely additive, probability and modular functions over semiring-like ... more In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. Then with the help of those observations, we also generalize some classical results in probability theory such as Boole's Inequality, the Law of Total Probability, Bayes' Theorem, the Equality of Parallel Systems, and Poincaré's Inclusion-Exclusion Theorem. While we prove that modular functions over a couple of celebrated semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring (Id(D), +, ·), where D is a Dedekind domain. Finally, we prove that under suitable conditions a function f is finitely additive iff it is modular and f (0) = 0.
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Books by Amir Parvardi
This book is supposed to be a problem bank for Algebra, and it forms the resource for the first series of the KAYWAÑAN Algebra Contest. I suggest you start with Polynomials, and before you get bored or exhausted, also start solving Trigonometry problems. If you find these problems easy and not challenging enough, the Spherical Trigonometry lessons and problems are definitely going to be a must try!
Teaching Documents by Amir Parvardi
Papers by Amir Parvardi
This book is supposed to be a problem bank for Algebra, and it forms the resource for the first series of the KAYWAÑAN Algebra Contest. I suggest you start with Polynomials, and before you get bored or exhausted, also start solving Trigonometry problems. If you find these problems easy and not challenging enough, the Spherical Trigonometry lessons and problems are definitely going to be a must try!
Taken from The Olympiad Algebra Book (Vol I): 1220 Polynomials & Trigonometry Problems: https://www.academia.edu/101938068/The_Olympiad_Algebra_Book_Vol_I
You can take the auto-graded test on Google Forms: https://forms.gle/kncm2yQFUQ6R4rCh6