AoPS Wiki:Article of the Day/Archive

2/3/08 - 2/29/08

AotD will be back in March as Article of the Week!


Law of Tangents

The Law of Tangents is a useful trigonometric identity that, along with the law of sines and law of cosines, can be used to determine angles in a triangle. Note that the law of tangents usually cannot determine sides, since only angles are involved in its statement.... [more]


Law of Sines

The Law of Sines is a useful identity in a triangle, which, along with the law of cosines and the law of tangents can be used to determine sides and angles. The law of sines can also be used to determine the circumradius, another useful function... [more]


Law of Cosines

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The Law of Cosines is a theorem which relates the side-lengths and angles of a triangle. It can be derived in several different ways, the most common of which are listed in the "proofs" section ... [more]


Euler's totient function

Euler's totient function $\phi(n)$ applied to a positive integer $n$ is defined to be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. $\phi(n)$ is read "phi of n."... [more]


There was no AotD for January 1st, 2008.


Expected value

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Given an event with a variety of different possible outcomes, the expected value is what one should expect to be the average outcome if the event were to be repeated many times. Note that this is not the same as the "most likely outcome."

More formally, we can define expected value as follows: if we have an event $Z$ whose outcomes have a discrete probability distribution, the... [more]


Introductory modular arithmetic

Modular arithmetic is a special type of arithmetic that involves only integers. This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using... [more]


Rational approximation of famous numbers

Rational approximation is the application of Dirichlet's theorem which shows that, for each irrational number $x\in\mathbb R$, the inequality $\left|x-\frac pq\right|<\frac 1{q^2}$ has infinitely many solutions. On the other hand, sometimes it is useful to know that $x$ cannot be approximated by rationals too well, or, more precisely, that $x$ is not a Liouvillian number, i.e., that for some power $M<+\infty$, the inequality [more]


Power set

The power set of a given set $S$ is the set $\mathcal{P}(S)$ of all subsets of that set This is denoted, other than by the common $\math{P}(S)$ (Error compiling LaTeX. Unknown error_msg), by $2^{S}$ (which has to do with the number of elements in the power set of a... [more]



A function is a rule that maps one set of values to another set of values. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its input value, and squares it to get an output value. One can... [more]


There was no AotD for January 23rd, 2008.



A permutation of a set of $r$ objects is any rearrangement (linear ordering) of the $r$ objects. There are $r!$ (the factorial of $r$) permutations of a set with $r$ distinct objects.

One can also consider permutations of infinite sets. In this case, a permutation of a set $S$ is simply a bijection between $S$ and... [more]


Euclidean algorithm

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The Euclidean algorithm (also known as the Euclidean division algorithm or Euclid's algorithm) is an algorithm that finds the greatest common divisor (GCD) of two elements of a Euclidean domain, the most common of which is the nonnegative integers $\mathbb{Z}{\geq 0}$, without factoring... [more]


On vacation.


Math books

These math books are recommended by Art of Problem Solving administrators and members of the <url>index.php AoPS-MathLinks Community</url>.

Levels of reading and math ability are loosely defined as follows:

  • Elementary is for elementary school students up through possibly early middle school.
  • Getting Started is recommended for students grades 6 to 9.
  • Intermediate is recommended for students grades 9 to 12.
  • Olympiad is recommended for high school students... [more]



For a real function $f(x)$ and some value $c$, $\lim_{x\rightarrow c} f(x)$ (said, "the limit of $f$ at $x$ as $x$ goes to $c$) equals $L$ iff for every $\epsilon > 0$ there exists a $\delta$ such that if $0<|x-c|<\delta$, then $|f(x)-L|< \epsilon$... [more]


American Mathematics Competitions

The American Mathematics Competitions (AMC) consist of a series of increasingly difficult tests for students in middle school and high school. The AMC sets the standard in the United States for talented high school students of mathematics. The AMC curriculum is both comprehensive and modern. AMC exams are so well designed that some top universities such as MIT now ask students for their AMC scores. "AMC" is also used as an abbreviation for American Math Contest, used to refer to the AMC 8, AMC 10, and AMC 12... [more]



The study of energy is known as physics. Everything concerning energy in some form or the other is covered by physics.

Physics as was known till the end of the nineteenth century is known now as classical physics. It is broadly classified into the following branches:



United States of America Mathematical Olympiad

The United States of America Mathematical Olympiad (USAMO) is the third test in a series of exams used to challenge bright students on the path toward choosing the team that represents the United States at the International Mathematics Olympiad (IMO).

The USAMO is administered by the American Mathematics Competitions (AMC). Art of Problem Solving (AoPS) is a proud sponsor of the AMC and of the recent expansion of USAMO participants from around 250 to around 400. [more]

12/13/07 - 1/2/08

AotD is on vacation.


Leonhard Euler

Leonhard Euler (1707-1735, pronounced Oiler) was a famous Swiss mathematician and physicist. He made numerous contributions to many fields of mathematics and science. Euler is often considered to be one of the greatest mathematicians of all time, along with Isaac Newton, Archimedes, and Carl Friedrich Gauss.

Euler was born on April 15, 1707 in Basel, Switzerland. Euler's parents were Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor's


Zermelo-Fraenkel Axioms

The Zermelo-Fraenkel Axioms are a set of axioms that compiled by Ernst Zermelo and Abraham Fraenkel that make it very convenient for set theorists to determine whether a given collection of objects with a given property describable by the language of set theory could be called a set. As shown by paradoxes such as Russell's Paradox, some restrictions must be put on which collections to call sets.

This axiom establishes the... [more]


Joining an ARML team

Team selection for the American Regions Mathematics League varies from team to team.

Florida ARML sends two teams to ARML each year. The selection criteria for the Florida ARML team takes into consideration several factors:

  • AMC and AIME performance
  • Past AMC, AIME, and USAMO scores
  • Past ARML performance
  • FAMAT-designated competitions
  • An annual statewide tryout test

The organizers are Jason Wiggins of... [more]


Zorn's Lemma

Zorn's Lemma is a set theoretic result which is equivalent to the Axiom of Choice.

Let $A$ be a partially ordered set.

We say that $A$ is inductively ordered if every totally ordered subset $T$ of $A$ has an upper bound, i.e., an element $a \in A$ such that for all $x\in T$, $x \le a$. We say that $A$ is strictly inductively ordered if every totally ordered subset $T$ of $A$ has a least upper bound, i.e., an upper bound $a$ so that if $b$ is an upper bound of $T$, then $a \le b$.

An element $m \in A$ is maximal if the relation $a \ge m$ implies $a=m$. (Note that a set may have several maximal... [more]


There was no AotD for December eighteenth.


Diophantine equation

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A Diophantine equation is an multi-variable equation for which integer solutions (or sometimes natural number or whole number solutions) are to be found.

Finding the solution or solutions to a Diophantine equation is closely tied to modular arithmetic and number theory. Often, when a Diophantine equation has infinitely many solutions, parametric form is used to express the relation between the variables of the equation.

Diophantine equations are named for the ancient Greek/Alexandrian mathematician Diophantus.

A Diophantine equation in the form $ax+by=c$ is known as a linear combination. If two relatively prime integers $a$ and $b$ are written in this form with $c=1$, the equation will have an infinite number of solutions. More generally, there will always be an... [more]


Fibonacci sequence

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The Fibonacci sequence is a sequence of integers in which the first and second terms are both equal to 1 and each subsequent term is the sum of the two preceding it. The first few terms are $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...$.

The Fibonacci sequence can be written recursively as $F_1 = F_2 = 1$ and $F_n=F_{n-1}+F_{n-2}$ for $n \geq 3$. This is the simplest nontrivial... [more]


Cauchy-Schwarz inequality

The Cauchy-Schwarz Inequality (which is known by other names, including Cauchy's Inequality, Schwarz's Inequality, and the Cauchy-Bunyakovsky-Schwarz Inequality) is a well-known inequality with many elegant applications... [more]


Rearrangement inequality

The Rearrangement Inequality states that, if $A=\{a_1,a_2,\cdots,a_n\}$ is a permutation of a finite set (in fact, multiset) of real numbers and $B=\{b_1,b_2,\cdots,b_n\}$ is a permutation of another finite set of real numbers, the quantity $a_1b_1+a_2b_2+\cdots+a_nb_n$ is maximized when ${A}$ and ${B}$ are similarly sorted (that is, if $a_k$ is greater than or equal to exactly ${i}$ of the other members of $A$, then ${b_k}$ is also greater than or equal to exactly ${i}$ of the other members of $B$). Conversely, $a_1b_1+a_2b_2+\cdots+a_nb_n$ is minimized when $A$ and $B$ are oppositely sorted (that is, if $a_k$ is less than or equal


There was no AotD for December thirteenth.


Trigonometric identities

Trigonometric identities are used to manipulate trigonometry equations in certain ways. Here is a list of them:

The six basic trigonometric functions can be defined using [more]



MATHCOUNTS is a large national mathematics competition and mathematics coaching program that has served millions of middle school students since 1984. Sponsored by the CNA Foundation, National Society of Professional Engineers, the National Council of Teachers of Mathematics, and others, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.

MATHCOUNTS curriculum... [more]



A polynomial is a function in one or more variables that consists of a sum of variables raised to nonnegative, integral powers and multiplied by coefficients (usually integral, rational, real or complex, but in abstract algebra often coming from an arbitrary field).

For example, these are... [more]


Pascal's identity

Pascal's identity is a common and useful theorem in the realm of combinatorics dealing with combinations (also known as binomial coefficients), and is often used to reduce large, complicated combinations.

Pascal's identity is also known as Pascal's rule, Pascal's formula, and occasionally... [more]


Canada/USA Mathcamp

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The Canada/USA Mathcamp is an intensive 5-week-long summer program for mathematically talented high school students, designed to inspire and motivate mathematically talented high school students by exposing them to the beauty and variety of mathematics, to impart valuable knowledge and skills for the pursuit of mathematics in high school, university, and beyond, and to provide a supportive and fun environment for interaction among students who love mathematics.

The environment of Mathcamp tends to be relaxed in terms of rules; in fact it officially only has four rules (generally involving common sense and respect). Originally it had been stricter, though by 1997 most of the mentors found the structure too... [more]


Prime number

A prime number (or simply prime) is a positive integer $p>1$ whose only positive divisors are 1 and itself. Note that $1$ is usually defined as being neither prime nor composite because it is its only factor among the natural numbers. The Sieve of Eratosthenes is a relatively quick method for... [more]



The discovery of the branch of mathematics known as calculus was motivated by two classical problems: how to find the slope of the tangent line to a curve at a point and how to find the area bounded by a curve. What is surprising is that these two problems are fundamentally connected and, together with the notion of limits, can be used to analyse instantaneous rates of change, accumulations of change, volumes of irregular solids, and... [more]



Pi is an irrational number (in fact, transcendental number, as proved by Lindeman in 1882) denoted by the greek letter $\pi$.

Pi is the ratio of the circumference (perimeter) of a given circle to its diameter. It is approximately equal to 3.141592653. The number pi is one of the most important constants in all of mathematics and appears in some of the most surprising places, such as in the sum $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$. Some common... [more]


Isaac Newton

Isaac Newton (1643 – 1727) was a famous British physicist and mathematician. His most famous work in mathematics was the compilation of calculus.

Isaac Newton was born on January 4, 1643 in Lincolnshire, England. Newton was born very shortly after the death of his father. He did very well... [more]



Awards.gif This article was also a AoPSWiki word of the week

Logarithms and exponents are very closely related. In fact, they are inverse functions. This means that logarithms can be used to reverse the result of exponentiation and vice versa, just as addition can be used to reverse the result of subtraction. Thus, if we have $a^x=b$, then taking the logarithm with base $a$ on both sides will give us $x=\log_{a}b$.

We would read this as "the logarithm of b, base a, is x". For example, we know that $3^4=81$. To express the same fact... [more]


American Invitational Mathematics Examination

The American Invitational Mathematics Examination (AIME) is the second exam in the series of exams used to challenge bright students on the path toward choosing the team that represents the United States at the International Mathematics Olympiad (IMO). While most AIME participants are high school students, some bright middle school students also qualify each year.

High scoring AIME students are invited to take the prestigious United States of America Mathematics Olympiad (USAMO).

The AIME is administered by... [more]



The subject of mathematical inequalities is tied closely with optimization methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in mathematics Olympiads.

Inequalities are arguably a branch of... [more]


Asymptote (geometry)

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An asymptote is a line or curve that a certain function approaches.

Linear asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique).

The vertical asymptote can be found by finding values of $x$ that make the function undefined, generally because it results in a division by zero, which is undefined... [more]


Complex number

The complex numbers arise when we try to solve equations such as $x^2 = -1$.

We know (from the trivial inequality) that the square of a real number cannot be negative, so this equation has no solutions in the real numbers. However, it is possible to define a number, $i$, such that $i = \sqrt{-1}$. If we add this new number to the reals, we will have solutions to $x^2 = -1$. It turns out that in the system that results... [more]



The notion of a set is one of the fundamental notions in mathematics that is difficult to precisely define. Of course, we have plenty of synonyms for the word "set," like collection, ensemble, group, etc., but those names really do not define the meaning of the word set; all they can do is replace it in various sentences. So, instead of defining what sets are, one has to define what can be done with them or, in other words, what axioms the sets satisfy. These axioms are chosen to agree with our intuitive concept of a set, on one hand, and to allow various, sometimes quite sophisticated, mathematical constructions on the other hand. For the full collection...[more]