(redacted)
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English
December 19, 2016
Ms. (redacted)
Concept Paper on Olympiad Mathematics
I. Introduction
Olympiad mathematics refers to “the problems that appear in the International
Mathematical Olympiad (IMO) or equivalent competitions” (Chen, 2016) and the background
theory involved in solving these problems. Such problems are labeled as Olympiad problems.
Working on Olympiad problems and studying topics in Olympiad mathematics is a popular
academic extracurricular among mathematically-inclined students. (Holton, 2010)
According to Mark Saul of the Mathematical Assocation of America, “the Olympiad
problem is an introduction, a glimpse into the world of mathematics not afforded by the usual
classroom situation.” (Andreescu & Gelca, 2000) As such, Olympiad mathematics focuses on
the topics outside the regular high-school curriculum. Olympiad problems' solutions, however,
must not require mathematical concepts or methods covered only in the undergraduate level or
above. The four major branches of Olympiad mathematics are Algebra, Number Theory,
Combinatorics, and Geometry. Olympiad problems may require in-depth understanding of
theorems and techniques in these subjects (IMO, n.d.)
Three forms of mathematics popular among mathematically-talented high school
students are Olympiad mathematics, accelerated curricular learning (“learning undergraduate/
graduate materials such algebra and analysis”), and mathematical research (“working on open
problems and conjectures”). (Chen, 2016) Another form to be distinguished from Olympiad
mathematics is computational competition mathematics (solving problems with a desired
numerical answer in a short time limit) and is colloquially known as short-answer problem
solving.
This paper seeks to compare and contrast the nature and purpose of Olympiad
mathematics with two other forms of mathematics: curricular mathematics (both secondary and
post-secondary) and short-answer problem solving.
II. Olympiad Mathematics vs Curricular Mathematics
Although Olympiad mathematics is meant to give high-performing mathematics students
an opportunity to explore mathematics beyond the high school curriculum (Gillespie, 2013), they
both are based on established mathematical theorems and aim to impart logic (Rusczyk, n.d.-c)
and problem-solving (Lehoczky and Rusczyk, 2006). Olympiad mathematics also involves
similar rigor and depth as post-secondary curricular mathematics and “may be put in context
with topics which will be encountered after high school,” - undergraduate mathematics subjects
such as differential equations and linear algebra. (Shulman, 2016)
Olympiad and curricular mathematics differ in their skill emphasis, content focus, and
predictability.
Olympiad mathematics requires the student to formulate a proof and create new
mathematical arguments. The student, therefore, gains extensive skill in mathematical writing,
argumentation, and reasoning. He also gains deep conceptual understanding and improved
intuitive thought. (Tao, n.d.)
On the other hand, curricular mathematics, from the primary to
post-secondary level, requires students to “remember and carry out methods that others have
previously discovered.” (Shulman, 2016). Instructional emphasis is thus placed on the reliable
application of well-known algorithms and standard formulas to arrive at answers, to be
reproduced in examinations. (Lehoczky & Ruczyk, 2006)
Discrete mathematics (number theory and combinatorics) is emphasized in Olympiad
mathematics, while it is virtually ignored in the school curriculum except for students majoring in
Computer Science. The standard high school curriculum covers Geometry as well as Calculus
and its pre-requisites. (Rusczyk, n.d.-a) At the undergraduate level, the most common subjects
are Calculus (including its advanced variants) and Linear Algebra.
Discrete mathematics is
occasionally offered as an elective but is rarely taken. In Olympiad mathematics, however,
approximately half of the problems may be categorized as discrete mathematics. (Patrick, n.d.)
Olympiad mathematics also covers several other topics not found in the school curriculum such
as collinearity, inequalities, and functional equations (Andreescu & Gelca, 2000) as well as proof
techniques such as proof by contradiction and mathematical induction.
As it focuses on the application of formulae, curricular mathematics is predictable and
routine. Olympiad mathematics, however, moves “towards creative solutions to unconventional
problems,” and “best reflect mathematical ingenuity.” (Andreescu & Gelca, 2000) Olympiad
problems typically do not have a single solution, and the solution is not immediately clear to the
student. The student must explore the problem and identify key conjectures to solve the
problem. As a result, familiarity with theorems and memorization of formulae are not sufficient to
consistently solve Olympiad problems. (Lehoczky & Rusczyk, 2006).
III. Olympiad Mathematics vs Short-answer Problem Solving
Several mathematics competitions such as MATHCOUNTS, the American Mathematics
Competition 10/12, and most countries’ first selection competition for the IMO require students
to provide a numerical answer to a large set of problems. The student has approximately one to
ten minutes for each question, and solutions / proofs are neither required nor considered.
Solving these problems and learning background material behind these are referred to as shortanswer problem solving.
Olympiad mathematics and most short-answer problem solving are at a level above the
standard high school curriculum and are aimed at establishing a problem-solving culture around
the world. (Andreescu & Gelca, 2000) They both are part of the “competition circuit”: students
encounter them outside the classroom when participating in and preparing for periodic
competitions (Shulman, 2016); therefore, they contain topics outside the regular high school
curriculum. To some degree, they both are unpredictable, because in the more difficult shortanswer problems (such as those in the American Invitational Mathematics Examination (AIME)),
the solution path is not evident - students must examine possible approaches and pursue
various paths before arriving at an answer.
The two main differences between Olympiad mathematics and short-answer problem
solving are the utility of memorized tricks and the problem-solving mindset involved.
In many short-answer competitions, learning artificial, standard tricks specific to a certain
class of problems would be useful. (Less Wrong, 2010) It is common for students to have
formula sheets of computational mathematics tricks. (Rusczyk, n.d.-c) Olympiad problems, on
the other hand, are solved not by recalling standard tricks but by gaining a deep understanding
on “what is going on” in the problem. Olympiad mathematics is considered closer to higher
mathematics and mathematical research compared to short-answer problem solving, which may
inadvertently encouraged mass memorization of standard tricks, absent in higher mathematics.
(Rusczyk, n.d.-a)
The time-constrained format of short-answer competitions also yields an “emphasis on
being quick, at the expense of deep thought.” (Less Wrong, 2010) Students may opt to forego
rigor and “solving the problem fully” in favor of quickly obtaining the numerical answer, as they
do not gain credit for formulating a rigorous and mathematically-correct solution. Several tricks
such as systematic guess-and-check, looking at special cases, and “engineer’s
induction” (guessing the formula based on small cases) would greatly help in short-answer
competitions. In Olympiad competitions, however, these would not completely solve the
problem, as a complete proof is required. (Holton, 2010) As it removes the possibility of
obtaining credit a problem without completely solving it or obtaining the main idea, Olympiad
mathematics better trains “the ability to think about and solve complex problems.” (Rusczyk,
n.d.-b)
IV. Conclusion
The relationship (similarities and differences) of Olympiad mathematics with curricular
mathematics (Figure 1) and short-answer problem solving (Figure 2) are shown in the following
Venn Diagrams.
Figure 1. Olympiad mathematics vs curricular mathematics
Figure 2. Olympiad mathematics vs short-answer problem solving
V. Bibliography
Andreescu, T., Boreico, I., Mushkarov, O., & Nikolov, N. (2012). Topics in functional equations.
Plano, TX: XYZ Press.
Andreescu, T., & Gelca, R. (2000). Mathematical Olympiad challenges. Boston: Birkhäuser.
Gillespie, M. (2013, June 17). Olympiad vs. research mathematics. Retrieved December 15,
2016, from http://www.mathematicalgemstones.com/misc/olympiad-vs-higher-math/
Holton, D. A. (2010). A first step to Mathematical Olympiad problems. Hackensack, NJ: World
Scientific.
Lehoczky, S., & Rusczyk, R. (2006). The art of problem solving. Alpine, CA: AoPS.
Less Wrong. (2010, October 11). Great Mathematicians on Math Competitions and "Genius"
Retrieved December 15, 2016, from http://lesswrong.com/lw/2v1/
great_mathematicians_on_math_competitions_and/
Patrick, D. (n.d.). Articles - Art of Problem Solving. Retrieved December 15, 2016, from
www.artofproblemsolving.com/articles/discrete-math
Rusczyk, R. (n.d.-a). Articles - Art of Problem Solving. Retrieved December 15, 2016, from
http://www.artofproblemsolving.com/articles/calculus-trap
Rusczyk, R. (n.d.-b). Articles - Art of Problem Solving. Retrieved December 15, 2016, from
http://www.artofproblemsolving.com/articles/competitions-pros-cons
Rusczyk, R. (n.d.-c). Articles - Art of Problem Solving. Retrieved December 15, 2016, from
http://www.artofproblemsolving.com/articles/what-is-problem-solving
Shulman, R. (2016, August 24). When Math And Creativity Combine: U.S. Wins Math Olympiad
For Second Consecutive Year. Retrieved December 15, 2016, from http://
www.huffingtonpost.com/robyn-shulman-/when-math-and-creativity-_b_11669968.html
Tao, T. (n.d.-b). Advice on mathematics competitions. Retrieved December 15, 2016, from
https://terrytao.wordpress.com/career-advice/advice-on-mathematics-competitions/