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Algebra

Algebra is one of the fundamental areas of olympiad mathematics.
However, it does not simply coincide with the algebra studied in the first years of high school: that forms the foundation, but many problems require more advanced tools and techniques.

In olympiad competitions, algebra often appears in the form of manipulation of expressions, study of polynomials, inequalities, and properties of complex numbers.
Learning to recognize hidden structures within expressions and transform them appropriately is one of the most important skills.

Below are some of the main topics to know.

Fundamental topics

  • Polynomials
  • Inequalities and means
  • Complex numbers

Operations on polynomials

A polynomial is an algebraic expression consisting of sums of powers of a variable multiplied by coefficients.

With polynomials one can perform the usual algebraic operations:

  • addition and subtraction
  • multiplication
  • powers

These operations form the basis of many olympiad problems.

Polynomial division

Similarly to integers, polynomials can also be divided.

Among the most important results are:

  • the Remainder Theorem
  • Ruffini’s rule

In particular, the Remainder Theorem states that the remainder of the division of a polynomial $P(x)$ by $(x-a)$ is equal to $P(a)$.

Bézout’s theorem for polynomials

Bézout’s theorem for polynomials states that a polynomial $P(x)$ is divisible by $(x-a)$ if and only if

$$ P(a) = 0 $$

This result is a fundamental tool in the study of polynomial roots.

Factorization

Many problems can be solved by correctly factoring a polynomial.

Learning to recognize special products, factorization techniques, and other decomposition strategies is essential for simplifying expressions and proving identities.

Fundamental means

Among the most important tools in olympiad algebra are the means of positive numbers:

  • arithmetic mean
  • geometric mean
  • harmonic mean

These quantities are related by important inequalities.

AM–GM inequality

One of the most frequently used inequalities states that for positive numbers

$$ \text{arithmetic mean} \geq \text{geometric mean} $$

This result is very useful for proving estimates and solving optimization problems.

Arithmetic and geometric sequences

Arithmetic and geometric sequences and series frequently appear in olympiad problems.

Their formulas allow one to quickly compute sums of many terms and recognize recurring structures in expressions.

Algebraic form

A complex number has the form

$$ z = a + bi $$

where $a$ and $b$ are real numbers and $i$ is the imaginary unit with the property

$$ i^2 = -1 $$

Complex numbers allow us to naturally describe the roots of many polynomials.

Trigonometric and exponential form

A complex number can also be written in trigonometric form or exponential form, using its modulus and argument.

These representations are particularly useful for computing powers and roots.

Powers and roots

Thanks to De Moivre’s formula, it is possible to compute powers and $n$-th roots of complex numbers easily.

A particularly important case is that of the $n$-th roots of unity, which appear in many olympiad problems.

Where to start preparing

A large part of the basic tools of algebra is studied in the first years of high school.
Before exploring olympiad topics in depth, it is therefore useful to have a solid command of school-level algebra (in particular polynomials and factorization).

To continue your preparation, you can use the following resources.

Senior Pills

One of the best available resources is the Senior Pills, a collection of video lectures from the Senior training camp of the Mathematics Olympiads.

🎬 http://olimpiadi.dm.unibo.it/il-senior-in-pillole/

These lectures cover many of the fundamental topics of olympiad algebra and represent an excellent starting point for reaching a more advanced level of preparation.

Further information about the Senior training camp is available on the page Senior Training Camp Lessons.

To study further, several volumes from the U Math series are very useful:

In the Resources section of the website you can find further information about the U Math series and other useful materials for training.