<![CDATA[MKLab - PROBLEM SOLVING AND CONTESTS]]>

<![CDATA[MKLab - PROBLEM SOLVING AND CONTESTS]]> https://mklab.gr/ Sat, 12 Sep 2026 17:27:34 +0000 MyBB <![CDATA[Introduction to Geometry [Rusczyk]]]> https://mklab.gr/showthread.php?tid=1701 Thu, 20 Aug 2026 17:39:09 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1701 Book:Introduction to Geometry
Author: Richard Rusczyk
Publisher: AoPS Incorporated (Art of Problem Solving)
Original publication: 2006
Edition: 2nd edition, 2009
Length: 557-page textbook + 226-page solutions manual
ISBN: 978-1-934124-08-6
Level: Roughly grades 7–10 / strong middle-school to early high-school mathematics

Summary

Introduction to Geometry is not a conventional school geometry textbook. Richard Rusczyk organizes the material around problem solving and mathematical discovery rather than presenting a theorem followed immediately by routine exercises. Sections typically begin with problems that students are encouraged to attack before reading the explanation. The subsequent discussion develops the relevant ideas through solutions, observations, proofs, and general techniques. This makes the book particularly suitable for students who want to understand why geometric results work and how to recognize when to use them, rather than simply memorize formulas. The book contains more than 900 problems, with complete solutions available in the accompanying solutions manual.

The mathematical coverage is broad enough to serve as a complete introductory geometry course. It develops angles and parallel lines, formal proof, triangle congruence and similarity, right triangles, areas, perpendicular bisectors, altitudes, angle bisectors, medians and concurrency, quadrilaterals, polygons, geometric inequalities, circles, tangents, power of a point, three-dimensional geometry, geometric transformations, coordinate/analytic geometry, and introductory trigonometry. The associated AoPS curriculum goes as far as the Laws of Sines and Cosines and trigonometric identities, while also emphasizing general geometry problem-solving strategies.

A major strength is that elementary ideas are repeatedly pushed into non-routine and competition-style problems. The intended audience includes students preparing for MATHCOUNTS and the AMC 10/12, although competition participation is not necessary to benefit from it. AoPS recommends that students already be comfortable with basic algebra, particularly solving linear equations. Consequently, the book sits somewhere between a rigorous school geometry course and an introduction to mathematical-competition geometry: substantially more challenging than most ordinary secondary-school textbooks, but not yet an olympiad-geometry text.

Main topics

The progression is approximately:
Foundations & triangles
Angles → parallel lines → proofs → congruent triangles → isosceles/equilateral triangles → similarity → right triangles.
Classical Euclidean geometry
Area → perpendicular bisectors → altitudes → angle bisectors → medians → concurrency → Ceva's theorem → quadrilaterals → polygons → geometric inequalities.
Circles
Arcs and sectors → angles in circles → tangent lines → secants → chords → power of a point.
Beyond elementary plane geometry
3-D geometry → transformations → analytic/coordinate geometry → introductory trigonometry → Laws of Sines and Cosines → trigonometric identities → general problem-solving strategies.

What makes the book distinctive
The central philosophy is essentially:
problem → experiment → discover an idea → prove it → use it on harder problems.
That is quite different from the more common:
definition → theorem → worked example → repetitive exercises.

Rusczyk's approach encourages students to develop habits that are extremely useful in higher mathematics: drawing auxiliary lines, looking for similar triangles, translating geometric relationships into algebra, identifying invariants, working backwards from the desired result, and combining several elementary facts rather than searching for a single memorized theorem. AoPS itself describes the course as helping students transition toward a more mature view of mathematics and formal proof.

Who should read it?
I would particularly recommend it for:

strong students aged roughly 12–16;
students who find ordinary school geometry too mechanical;
students preparing for AMC 8/10/12 or MATHCOUNTS;
teachers looking for challenging enrichment problems;
adults wanting to relearn Euclidean geometry from a problem-solving perspective.

It may be frustrating for a complete beginner who wants lots of straightforward examples before attempting problems. The book deliberately expects the reader to struggle productively with unfamiliar questions.

Key takeaways

Geometry is taught as problem solving, not memorization. The student is expected to discover many ideas by attempting problems first.
The coverage is unusually comprehensive: classical Euclidean geometry, proofs, circles, transformations, analytic geometry, 3-D geometry and introductory trigonometry all appear in one course.
The difficulty rises well beyond ordinary textbook exercises. Its 900+ problems make it particularly valuable for mathematically ambitious students and competition preparation.
It is an excellent bridge toward olympiad mathematics, although students aiming specifically at high-level olympiad geometry will eventually need a more advanced text.

AoPS — Introduction to Geometry]]> Book:Introduction to Geometry
Author: Richard Rusczyk
Publisher: AoPS Incorporated (Art of Problem Solving)
Original publication: 2006
Edition: 2nd edition, 2009
Length: 557-page textbook + 226-page solutions manual
ISBN: 978-1-934124-08-6
Level: Roughly grades 7–10 / strong middle-school to early high-school mathematics

Summary

Introduction to Geometry is not a conventional school geometry textbook. Richard Rusczyk organizes the material around problem solving and mathematical discovery rather than presenting a theorem followed immediately by routine exercises. Sections typically begin with problems that students are encouraged to attack before reading the explanation. The subsequent discussion develops the relevant ideas through solutions, observations, proofs, and general techniques. This makes the book particularly suitable for students who want to understand why geometric results work and how to recognize when to use them, rather than simply memorize formulas. The book contains more than 900 problems, with complete solutions available in the accompanying solutions manual.

The mathematical coverage is broad enough to serve as a complete introductory geometry course. It develops angles and parallel lines, formal proof, triangle congruence and similarity, right triangles, areas, perpendicular bisectors, altitudes, angle bisectors, medians and concurrency, quadrilaterals, polygons, geometric inequalities, circles, tangents, power of a point, three-dimensional geometry, geometric transformations, coordinate/analytic geometry, and introductory trigonometry. The associated AoPS curriculum goes as far as the Laws of Sines and Cosines and trigonometric identities, while also emphasizing general geometry problem-solving strategies.

A major strength is that elementary ideas are repeatedly pushed into non-routine and competition-style problems. The intended audience includes students preparing for MATHCOUNTS and the AMC 10/12, although competition participation is not necessary to benefit from it. AoPS recommends that students already be comfortable with basic algebra, particularly solving linear equations. Consequently, the book sits somewhere between a rigorous school geometry course and an introduction to mathematical-competition geometry: substantially more challenging than most ordinary secondary-school textbooks, but not yet an olympiad-geometry text.

Main topics

The progression is approximately:
Foundations & triangles
Angles → parallel lines → proofs → congruent triangles → isosceles/equilateral triangles → similarity → right triangles.
Classical Euclidean geometry
Area → perpendicular bisectors → altitudes → angle bisectors → medians → concurrency → Ceva's theorem → quadrilaterals → polygons → geometric inequalities.
Circles
Arcs and sectors → angles in circles → tangent lines → secants → chords → power of a point.
Beyond elementary plane geometry
3-D geometry → transformations → analytic/coordinate geometry → introductory trigonometry → Laws of Sines and Cosines → trigonometric identities → general problem-solving strategies.

What makes the book distinctive
The central philosophy is essentially:
problem → experiment → discover an idea → prove it → use it on harder problems.
That is quite different from the more common:
definition → theorem → worked example → repetitive exercises.

Rusczyk's approach encourages students to develop habits that are extremely useful in higher mathematics: drawing auxiliary lines, looking for similar triangles, translating geometric relationships into algebra, identifying invariants, working backwards from the desired result, and combining several elementary facts rather than searching for a single memorized theorem. AoPS itself describes the course as helping students transition toward a more mature view of mathematics and formal proof.

Who should read it?
I would particularly recommend it for:

strong students aged roughly 12–16;
students who find ordinary school geometry too mechanical;
students preparing for AMC 8/10/12 or MATHCOUNTS;
teachers looking for challenging enrichment problems;
adults wanting to relearn Euclidean geometry from a problem-solving perspective.

Geometry is taught as problem solving, not memorization. The student is expected to discover many ideas by attempting problems first.
The coverage is unusually comprehensive: classical Euclidean geometry, proofs, circles, transformations, analytic geometry, 3-D geometry and introductory trigonometry all appear in one course.
The difficulty rises well beyond ordinary textbook exercises. Its 900+ problems make it particularly valuable for mathematically ambitious students and competition preparation.
It is an excellent bridge toward olympiad mathematics, although students aiming specifically at high-level olympiad geometry will eventually need a more advanced text.

AoPS — Introduction to Geometry]]> <![CDATA[Problem-Solving Through Problems [Larson]]]> https://mklab.gr/showthread.php?tid=1623 Mon, 17 Aug 2026 17:26:25 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1623 Problem-Solving Through Problems
Author: Loren C. Larson
First published: 1983
Publisher: Springer-Verlag, New York
Series:Problem Books in Mathematics

Loren C. Larson’s Problem-Solving Through Problems is a classic introduction to the art of mathematical problem solving. Rather than organizing mathematics primarily around theories and definitions, Larson organizes the book around techniques that repeatedly appear when attacking unfamiliar problems. The opening chapter develops heuristics such as searching for patterns, drawing diagrams, reformulating a problem, exploiting symmetry, working backwards, considering extreme cases, using contradiction, and generalizing. These strategies are then put to work across induction and the pigeonhole principle, number theory and arithmetic, algebra, series, real analysis, inequalities, and geometry.

The book's strength is its emphasis on learning mathematics by actually solving problems. It contains more than 700 problems, with over one-third worked out in detail, and Larson deliberately selects problems that demonstrate how a relatively small collection of ideas can work in surprisingly different mathematical settings.

The result is somewhere between a textbook, a problem collection, and a handbook of mathematical strategies. It is especially useful for strong undergraduate students, mathematics teachers, and students interested in competitions such as the Putnam. More broadly, the book teaches an important lesson: becoming a better problem solver is less about accumulating isolated tricks and more about developing a repertoire of flexible ideas—and learning to recognize when a familiar idea can unlock an unfamiliar problem.

Key takeaways

Problem solving is learned through practice: the problems themselves are the central teaching mechanism.
Heuristics matter: symmetry, parity, contradiction, extreme cases, pattern recognition, reformulation, and working backwards become reusable tools.
Techniques cross subject boundaries: the same idea may solve problems in algebra, number theory, analysis, or geometry.
Excellent preparation for advanced problem solving: particularly valuable for undergraduates, teachers, and mathematical competition students.

Overall: ★★★★★ — A substantial classic for anyone who wants to move from simply knowing mathematics toward becoming genuinely skilled at solving mathematical problems.

Springer — Problem-Solving Through Problems]]> Problem-Solving Through Problems
Author: Loren C. Larson
First published: 1983
Publisher: Springer-Verlag, New York
Series:Problem Books in Mathematics

Loren C. Larson’s Problem-Solving Through Problems is a classic introduction to the art of mathematical problem solving. Rather than organizing mathematics primarily around theories and definitions, Larson organizes the book around techniques that repeatedly appear when attacking unfamiliar problems. The opening chapter develops heuristics such as searching for patterns, drawing diagrams, reformulating a problem, exploiting symmetry, working backwards, considering extreme cases, using contradiction, and generalizing. These strategies are then put to work across induction and the pigeonhole principle, number theory and arithmetic, algebra, series, real analysis, inequalities, and geometry.

The book's strength is its emphasis on learning mathematics by actually solving problems. It contains more than 700 problems, with over one-third worked out in detail, and Larson deliberately selects problems that demonstrate how a relatively small collection of ideas can work in surprisingly different mathematical settings.

The result is somewhere between a textbook, a problem collection, and a handbook of mathematical strategies. It is especially useful for strong undergraduate students, mathematics teachers, and students interested in competitions such as the Putnam. More broadly, the book teaches an important lesson: becoming a better problem solver is less about accumulating isolated tricks and more about developing a repertoire of flexible ideas—and learning to recognize when a familiar idea can unlock an unfamiliar problem.

Key takeaways

Problem solving is learned through practice: the problems themselves are the central teaching mechanism.
Heuristics matter: symmetry, parity, contradiction, extreme cases, pattern recognition, reformulation, and working backwards become reusable tools.
Techniques cross subject boundaries: the same idea may solve problems in algebra, number theory, analysis, or geometry.
Excellent preparation for advanced problem solving: particularly valuable for undergraduates, teachers, and mathematical competition students.

Overall: ★★★★★ — A substantial classic for anyone who wants to move from simply knowing mathematics toward becoming genuinely skilled at solving mathematical problems.

Springer — Problem-Solving Through Problems]]> <![CDATA[Problem-Solving Strategies [Engel]]]> https://mklab.gr/showthread.php?tid=1622 Mon, 17 Aug 2026 17:24:00 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1622 Problem-Solving Strategies
Author: Arthur Engel
First published: 1997/1998
Publisher: Springer
Series:Problem Books in Mathematics

Arthur Engel’s Problem-Solving Strategies is a substantial training manual for mathematical problem solving, aimed particularly at advanced high-school students, mathematics-competition participants, teachers, and Olympiad coaches. Rather than organizing the material only around conventional subjects, Engel emphasizes general strategies that repeatedly unlock difficult problems. The opening chapters develop especially powerful ideas such as the invariance principle, coloring arguments, the extremal principle, and the pigeonhole (“box”) principle. These are followed by more subject-oriented chapters on combinatorics, number theory, inequalities, induction, sequences, polynomials, functional equations, geometry and games, before a final chapter introduces further techniques such as infinite descent and working backwards.

The central philosophy of the book is that becoming a strong problem solver requires learning to recognize structures and patterns, rather than accumulating isolated tricks. For example, when a problem involves repeated transformations, one should ask whether some quantity remains invariant; when dealing with a finite collection, one can examine an extremal object; and when objects are distributed among categories, the pigeonhole principle may reveal something that must occur. Engel demonstrates these principles through problems drawn from more than twenty national and international competitions, including material suitable for the International Mathematical Olympiad (IMO), Tournament of the Towns, and non-calculus Putnam problems.

A major strength is the enormous amount of practice material. Chapters typically begin with representative examples explaining a strategy and then move to carefully selected exercises, most accompanied by solutions or substantial guidance. Counting examples as problems, the book contains more than 1,300 opportunities for problem-solving practice. This makes it less suitable for passive reading than for sustained study: the real value comes from attempting each problem before consulting Engel’s solution. Difficulty varies considerably, from accessible exercises to problems demanding significant ingenuity. For someone interested in mathematical competitions or in teaching students how to think when the method is not immediately obvious, it remains an exceptionally rich resource.

Key takeaways

Problem-solving principles are reusable. Invariants, extremal arguments, coloring, pigeonhole reasoning, induction and working backwards recur across apparently unrelated mathematical problems.
The book teaches strategy rather than formulas. Its objective is to develop the ability to recognize what kind of reasoning a new problem requires.
Practice is fundamental. With over 1,300 examples and problems available when the worked examples are included, it functions almost as a long-term mathematical training program.
Best suited to serious problem solvers. It is particularly valuable for Olympiad students, competition trainers, mathematics teachers and anyone who enjoys challenging non-routine mathematics.

Overall: ★★★★★ — One of the stronger general-purpose books for developing competition-style mathematical problem-solving skills, especially because it organizes a huge collection of problems around a relatively small set of powerful ideas. Goodreads currently lists it at roughly 4.47/5, which is consistent with its reputation among problem-solving enthusiasts.

Springer — Problem-Solving Strategies · Goodreads — Problem-Solving Strategies]]> Problem-Solving Strategies
Author: Arthur Engel
First published: 1997/1998
Publisher: Springer
Series:Problem Books in Mathematics

Arthur Engel’s Problem-Solving Strategies is a substantial training manual for mathematical problem solving, aimed particularly at advanced high-school students, mathematics-competition participants, teachers, and Olympiad coaches. Rather than organizing the material only around conventional subjects, Engel emphasizes general strategies that repeatedly unlock difficult problems. The opening chapters develop especially powerful ideas such as the invariance principle, coloring arguments, the extremal principle, and the pigeonhole (“box”) principle. These are followed by more subject-oriented chapters on combinatorics, number theory, inequalities, induction, sequences, polynomials, functional equations, geometry and games, before a final chapter introduces further techniques such as infinite descent and working backwards.

The central philosophy of the book is that becoming a strong problem solver requires learning to recognize structures and patterns, rather than accumulating isolated tricks. For example, when a problem involves repeated transformations, one should ask whether some quantity remains invariant; when dealing with a finite collection, one can examine an extremal object; and when objects are distributed among categories, the pigeonhole principle may reveal something that must occur. Engel demonstrates these principles through problems drawn from more than twenty national and international competitions, including material suitable for the International Mathematical Olympiad (IMO), Tournament of the Towns, and non-calculus Putnam problems.

A major strength is the enormous amount of practice material. Chapters typically begin with representative examples explaining a strategy and then move to carefully selected exercises, most accompanied by solutions or substantial guidance. Counting examples as problems, the book contains more than 1,300 opportunities for problem-solving practice. This makes it less suitable for passive reading than for sustained study: the real value comes from attempting each problem before consulting Engel’s solution. Difficulty varies considerably, from accessible exercises to problems demanding significant ingenuity. For someone interested in mathematical competitions or in teaching students how to think when the method is not immediately obvious, it remains an exceptionally rich resource.

Key takeaways

Problem-solving principles are reusable. Invariants, extremal arguments, coloring, pigeonhole reasoning, induction and working backwards recur across apparently unrelated mathematical problems.
The book teaches strategy rather than formulas. Its objective is to develop the ability to recognize what kind of reasoning a new problem requires.
Practice is fundamental. With over 1,300 examples and problems available when the worked examples are included, it functions almost as a long-term mathematical training program.
Best suited to serious problem solvers. It is particularly valuable for Olympiad students, competition trainers, mathematics teachers and anyone who enjoys challenging non-routine mathematics.

Overall: ★★★★★ — One of the stronger general-purpose books for developing competition-style mathematical problem-solving skills, especially because it organizes a huge collection of problems around a relatively small set of powerful ideas. Goodreads currently lists it at roughly 4.47/5, which is consistent with its reputation among problem-solving enthusiasts.

Springer — Problem-Solving Strategies · Goodreads — Problem-Solving Strategies]]> <![CDATA[The IMO Compendium [Djukić]]]> https://mklab.gr/showthread.php?tid=1619 Mon, 17 Aug 2026 17:14:58 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1619 Full title:The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads: 1959–2009
Authors: Dušan Djukić, Vladimir Janković, Ivan Matić & Nikola Petrović
Publication date: 2011
Publisher: Springer New York
Edition: 2nd edition
Series:Problem Books in Mathematics
Length: xiv + 809 pages

The IMO Compendium is an enormous collection of problems associated with the International Mathematical Olympiad, covering the period from the first IMO in 1959 through 2009. Rather than being a conventional textbook, it is essentially a historical archive and advanced problem-solving laboratory. It gathers both problems actually selected for the IMO and many problems proposed during the selection process, making material that was once scattered through old manuscripts and national archives available in a systematic collection. The second edition adds 143 problems beyond the first edition's 1959–2004 coverage. All four authors are themselves former mathematical-olympiad medalists.

The book is particularly valuable because it combines an exceptionally large problem collection with complete solutions. After a short introduction and a section reviewing basic concepts and facts, roughly 320 pages are devoted to problems, followed by more than 440 pages of solutions. The problems span the traditional Olympiad areas—algebra, geometry, number theory and combinatorics—and demand ingenuity rather than advanced university mathematics. This makes the book useful not only for students training seriously for mathematical competitions, but also for teachers looking for challenging classroom material and mathematicians who enjoy elementary problems with surprisingly deep solutions.

Its greatest strength is therefore not that it teaches a fixed collection of techniques step by step, but that it provides an environment in which mathematical problem-solving ability can be developed through sustained practice. Working through the problems chronologically also gives the reader something resembling a 50-year history of Olympiad mathematics, revealing recurring ideas, changing styles and the remarkable creativity possible with relatively elementary mathematics. For anyone seriously interested in Olympiad problem solving, it functions less like a book to read from beginning to end and more like a long-term reference and training manual. The Mathematical Association of America specifically recommended that undergraduate mathematics libraries consider acquiring it.

Key Takeaways

Comprehensive scope: IMO and proposed problems covering 1959–2009.
Huge training resource: more than 800 pages of problems, mathematical background and solutions.
Problem-solving focused: emphasizes creativity, proof and mathematical insight rather than routine calculation.
Broad usefulness: excellent for Olympiad students, competition coaches, mathematics teachers and recreational problem solvers.
Historical value: documents roughly half a century of the development of international mathematical competition problems.
Second edition: adds 143 new problems to the original 1959–2004 collection.

Conclusion:The IMO Compendium is one of the strongest single-volume resources for serious mathematical competition training. It is demanding rather than introductory, but for a reader who wants to develop genuine problem-solving maturity, its combination of hundreds of carefully selected problems and detailed solutions makes it an exceptional reference.

The IMO Compendium — Springer Nature
The IMO Compendium — Goodreads]]> Full title:The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads: 1959–2009
Authors: Dušan Djukić, Vladimir Janković, Ivan Matić & Nikola Petrović
Publication date: 2011
Publisher: Springer New York
Edition: 2nd edition
Series:Problem Books in Mathematics
Length: xiv + 809 pages

The IMO Compendium is an enormous collection of problems associated with the International Mathematical Olympiad, covering the period from the first IMO in 1959 through 2009. Rather than being a conventional textbook, it is essentially a historical archive and advanced problem-solving laboratory. It gathers both problems actually selected for the IMO and many problems proposed during the selection process, making material that was once scattered through old manuscripts and national archives available in a systematic collection. The second edition adds 143 problems beyond the first edition's 1959–2004 coverage. All four authors are themselves former mathematical-olympiad medalists.

The book is particularly valuable because it combines an exceptionally large problem collection with complete solutions. After a short introduction and a section reviewing basic concepts and facts, roughly 320 pages are devoted to problems, followed by more than 440 pages of solutions. The problems span the traditional Olympiad areas—algebra, geometry, number theory and combinatorics—and demand ingenuity rather than advanced university mathematics. This makes the book useful not only for students training seriously for mathematical competitions, but also for teachers looking for challenging classroom material and mathematicians who enjoy elementary problems with surprisingly deep solutions.

Its greatest strength is therefore not that it teaches a fixed collection of techniques step by step, but that it provides an environment in which mathematical problem-solving ability can be developed through sustained practice. Working through the problems chronologically also gives the reader something resembling a 50-year history of Olympiad mathematics, revealing recurring ideas, changing styles and the remarkable creativity possible with relatively elementary mathematics. For anyone seriously interested in Olympiad problem solving, it functions less like a book to read from beginning to end and more like a long-term reference and training manual. The Mathematical Association of America specifically recommended that undergraduate mathematics libraries consider acquiring it.

Key Takeaways

Comprehensive scope: IMO and proposed problems covering 1959–2009.
Huge training resource: more than 800 pages of problems, mathematical background and solutions.
Problem-solving focused: emphasizes creativity, proof and mathematical insight rather than routine calculation.
Broad usefulness: excellent for Olympiad students, competition coaches, mathematics teachers and recreational problem solvers.
Historical value: documents roughly half a century of the development of international mathematical competition problems.
Second edition: adds 143 new problems to the original 1959–2004 collection.

Putnam and Beyond
Authors: Răzvan Gelca & Titu Andreescu
Publication date: August 2007
Publisher: Springer

Putnam and Beyond is an advanced problem-solving textbook designed to bridge the gap between high-school competition mathematics, undergraduate mathematics, and the beginnings of mathematical research. The William Lowell Putnam Mathematical Competition provides the inspiration, but the book is considerably broader than a collection of Putnam problems. Gelca and Andreescu use challenging problems as a way of teaching mathematical ideas and techniques, emphasizing how to discover proofs and develop problem-solving strategies rather than simply memorize theorems. It begins with fundamental proof methods—including contradiction, induction, the pigeonhole principle, ordered sets, and invariants—and then develops increasingly sophisticated mathematics.

The main mathematical areas are algebra, real analysis, geometry and trigonometry, number theory, combinatorics, and probability. Within these areas the reader encounters subjects such as polynomials, linear algebra, inequalities, sequences and series, calculus, differential equations, coordinate geometry, elementary number theory, and combinatorial arguments. A particularly valuable feature is the enormous solutions section: in the first edition, the main chapters occupy roughly the first 320 pages, while solutions begin on page 321 and extend through most of the remainder of the 798-page volume. This reflects the book's philosophy: the problems are the primary vehicle through which mathematics is learned.

The book is best suited to strong undergraduate students, mathematics competition participants, teachers, and anyone wanting to become substantially better at solving unfamiliar mathematical problems. It is not an introductory textbook and can be demanding for readers without a solid foundation in calculus and undergraduate mathematics. For a mathematically mature reader, however, its strength lies precisely in that difficulty: problems encourage experimentation, recognition of hidden structures, and the combination of ideas from different areas. Springer describes it as suitable for self-study by undergraduate and graduate students and for readers wishing to prepare for more advanced mathematics. A substantially expanded second edition appeared in 2017, adding material on quadratic polynomials, functions in real analysis, plane curves, quadratic fields, graph theory and other subjects, with more than 1,300 problems.

Key takeaways

Problem solving is the central subject. Algebra, analysis and number theory provide the setting, but the deeper goal is learning how to think when the solution is not obvious.
It bridges Olympiad and university mathematics. The book helps readers move from clever competition techniques toward rigorous undergraduate and eventually research-level mathematical thinking.
The extensive worked solutions are essential. They allow the reader to compare approaches and turn individual tricks into reusable mathematical techniques.
Best used actively: attempt each problem seriously before reading its solution. Reading the solutions without struggling with the problems removes much of the book's educational value.

Problem solving is the central subject. Algebra, analysis and number theory provide the setting, but the deeper goal is learning how to think when the solution is not obvious.
It bridges Olympiad and university mathematics. The book helps readers move from clever competition techniques toward rigorous undergraduate and eventually research-level mathematical thinking.
The extensive worked solutions are essential. They allow the reader to compare approaches and turn individual tricks into reusable mathematical techniques.
Best used actively: attempt each problem seriously before reading its solution. Reading the solutions without struggling with the problems removes much of the book's educational value.

Overall: ★★★★★ — An excellent choice for a mathematically ambitious undergraduate or teacher who wants a large, challenging collection that develops genuine mathematical problem-solving ability rather than routine computational technique.

Putnam and Beyond — Springer
Putnam and Beyond — Goodreads]]> <![CDATA[Thinking Better [du Sautoy]]]> https://mklab.gr/showthread.php?tid=1401 Wed, 29 Jul 2026 00:55:36 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1401 Thinking Better: The Art of the Shortcut in Math and Life
BY Marcus du Sautoy

Summary

In Thinking Better: The Art of the Shortcut in Math and Life, Oxford mathematician Marcus du Sautoy argues that true problem-solving and success come not from brute-force hard work, but from finding clever shortcuts. Du Sautoy demonstrates how mathematical principles—ranging from geometry and probability to calculus—serve as the ultimate time-saving tools, allowing us to solve complex tasks efficiently so we can focus on bigger challenges. Blending history, science, psychology, and real-world anecdotes from artists and entrepreneurs, the book celebrates human ingenuity and explains how strategic thinking gives us a distinct cognitive edge over raw repetition and artificial intelligence.

BOOK]]> Thinking Better: The Art of the Shortcut in Math and Life
BY Marcus du Sautoy

Summary

In Thinking Better: The Art of the Shortcut in Math and Life, Oxford mathematician Marcus du Sautoy argues that true problem-solving and success come not from brute-force hard work, but from finding clever shortcuts. Du Sautoy demonstrates how mathematical principles—ranging from geometry and probability to calculus—serve as the ultimate time-saving tools, allowing us to solve complex tasks efficiently so we can focus on bigger challenges. Blending history, science, psychology, and real-world anecdotes from artists and entrepreneurs, the book celebrates human ingenuity and explains how strategic thinking gives us a distinct cognitive edge over raw repetition and artificial intelligence.

BOOK]]> <![CDATA[When Less is More: Visualizing Basic Inequalities [Alsina]]]> https://mklab.gr/showthread.php?tid=1347 Sun, 26 Jul 2026 02:57:57 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1347 [Image: 7182320.jpg]

When Less is More: Visualizing Basic Inequalities
BY Claudi Alsina

Summary

In When Less is More: Visualizing Basic Inequalities, authors Claudi Alsina and Roger B. Nelsen explore how geometric visualization serves as a transformative tool for understanding fundamental mathematical inequalities. While traditional mathematics education heavily emphasizes equations and strict identities, the authors contend that inequalities possess a far richer, more varied landscape across fields ranging from classical Euclidean geometry to operations research and financial modeling.

To bridge the gap between abstract algebra and intuitive comprehension, this guide systematically demonstrates how drawing diagrams, geometric figures, and visual proofs can make complex mathematical relationships transparent and intuitive. Rather than relying solely on formal algebraic derivations, the book highlights how visual representations not only establish that two quantities are unequal, but also clearly illustrate the precise magnitude and nature of that disparity. By providing systematic methods for constructing visual arguments,

Alsina and Nelsen equip students, educators, and math enthusiasts with creative problem-solving techniques and fresh pedagogical strategies. Ultimately, the guide demonstrates the power of visual reasoning to cultivate a deeper appreciation for mathematical structures, making abstract inequalities tangible, engaging, and far easier to master.

BOOK]]> [Image: 7182320.jpg]

The Cauchy-Schwarz Master Class:
An Introduction to the Art of Mathematical Inequalities
BY [J. Michael Steele]

Summary

The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele is a problem-driven exploration of one of mathematics’ most powerful ideas: the Cauchy-Schwarz inequality. Rather than presenting formulas to memorise, the book teaches readers how to think like mathematicians by guiding them through carefully chosen problems and showing how elegant proofs can be discovered.

Beginning with the Cauchy-Schwarz inequality, it gradually develops a broad framework of classical inequalities, including AM-GM, Hölder, Hilbert, Hardy, and Minkowski, while introducing key concepts such as convexity, power means, majorization, Schur convexity, and sums of squares. Steele emphasizes intuition, creativity, and proof strategies over mechanical computation, encouraging readers to understand why techniques work and how they can be adapted to new situations.

Although a background in calculus is helpful, much of the material is accessible to advanced undergraduates and motivated self-learners. Rich with exercises, historical insights, and elegant arguments, the book serves as both a rigorous introduction to mathematical inequalities and a guide to developing stronger problem-solving skills that extend across analysis, probability, combinatorics, and many other areas of mathematics.

BOOK]]> [Image: 706377.jpg]

Problems and Solutions in Mathematical Olympiad High School
BY Shi-Xiong Liu

Summary

The series is created in line with the mathematics cognition and intellectual development levels of the students in the corresponding grades. All hot mathematics topics of the competition are included in the volumes and are organized into chapters where concepts and methods are gradually introduced to equip the students with necessary knowledge until they can finally reach the competition level.

In each chapter, well-designed problems including those collected from real competitions are provided so that the students can apply the skills and strategies they have learned to solve these problems. Detailed solutions are provided selectively. As a feature of the series, we also include some solutions generously offered by the members of Chinese national team and national training team.

PART 1 , PART 2 , PART 3]]> <![CDATA[Methods for Solving Elementary Algebra Problems [Chirskii]]]> https://mklab.gr/showthread.php?tid=948 Tue, 07 Jul 2026 19:41:18 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=948 Standard and Non-Standard Methods for Solving Elementary Algebra Problems
BY [V G Chirskii]

Summary

Standard and Non-Standard Methods for Solving Elementary Algebra Problems by Vladimir G. Chirskii and Artem I. Kozko is a comprehensive mathematics textbook focused on developing effective problem-solving techniques in elementary algebra. The book goes beyond routine procedures by presenting both standard methods and creative approaches for solving equations, inequalities, and algebraic problems.

It covers a wide range of topics, including exponential and logarithmic equations, polynomial and rational functions, modulus problems, irrational equations, trigonometric equations, systems with parameters, and problems involving integers. Particular emphasis is placed on techniques such as substitutions, transformations, geometric interpretations, inequality methods, and parameter analysis, helping readers understand not only how to solve problems but also why certain methods work.

Designed for students, mathematics teachers, and enthusiasts interested in mathematical reasoning, the book combines theoretical explanations with numerous exercises drawn from challenging examinations and mathematics Olympiads. It highlights the importance of flexibility in mathematical thinking, showing that many difficult problems can be approached from multiple perspectives rather than through a single fixed algorithm.

By connecting algebra with geometry, inequalities, and analytical ideas, the authors demonstrate how creative problem solving strengthens logical skills and deeper mathematical understanding. This work is valuable because it encourages a problem-solving mindset that remains essential not only in mathematics education but also in scientific and analytical thinking more broadly.

BOOK]]> Standard and Non-Standard Methods for Solving Elementary Algebra Problems
BY [V G Chirskii]

Summary

Standard and Non-Standard Methods for Solving Elementary Algebra Problems by Vladimir G. Chirskii and Artem I. Kozko is a comprehensive mathematics textbook focused on developing effective problem-solving techniques in elementary algebra. The book goes beyond routine procedures by presenting both standard methods and creative approaches for solving equations, inequalities, and algebraic problems.

It covers a wide range of topics, including exponential and logarithmic equations, polynomial and rational functions, modulus problems, irrational equations, trigonometric equations, systems with parameters, and problems involving integers. Particular emphasis is placed on techniques such as substitutions, transformations, geometric interpretations, inequality methods, and parameter analysis, helping readers understand not only how to solve problems but also why certain methods work.

Designed for students, mathematics teachers, and enthusiasts interested in mathematical reasoning, the book combines theoretical explanations with numerous exercises drawn from challenging examinations and mathematics Olympiads. It highlights the importance of flexibility in mathematical thinking, showing that many difficult problems can be approached from multiple perspectives rather than through a single fixed algorithm.

By connecting algebra with geometry, inequalities, and analytical ideas, the authors demonstrate how creative problem solving strengthens logical skills and deeper mathematical understanding. This work is valuable because it encourages a problem-solving mindset that remains essential not only in mathematics education but also in scientific and analytical thinking more broadly.

BOOK]]> <![CDATA[Solving Problems In Point Geometry [Zhang]]]> https://mklab.gr/showthread.php?tid=947 Tue, 07 Jul 2026 19:36:46 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=947 Solving Problems In Point Geometry
BY [ Jingzhong Zhang]

Summary

Solving Problems in Point Geometry: Insights and Strategies for Mathematical Olympiad and Competitions, authored by prominent mathematicians Jingzhong Zhang and Xicheng Peng, introduces an innovative framework designed to overcome the historical challenges of elementary geometry. For over two millennia, students and competitors have wrestled with complex auxiliary graphics and exhausting algebraic equations. Inspired by a conceptual query from Leibniz, the authors present "point geometry," a groundbreaking approach that allows problem-solvers to perform operations directly on points.

This system synthesizes the core advantages of traditional vector, coordinate, and mass point methods into a single, cohesive tool. By utilizing identity-based methods and algebraic techniques like undetermined coefficients, the framework condenses intricate, multi-step geometric proofs into concise equations, unlocking a wide range of mathematical propositions with minimal friction.

While the text serves as a rich guide for high school math enthusiasts and Olympiad contestants, its implications reach far beyond competitive mathematics. Point geometry offers a remarkably clean structure for knowledge representation and automated reasoning, making it highly relevant to the field of artificial intelligence.

Because many of the geometric solutions detailed in the text can be generated systematically via computer programs, this framework provides a blueprint for simplifying automated reasoning models. By removing the traditional learning burdens associated with geometric proofs, this work not only shifts how we teach mathematics but also provides critical mathematical tools that could accelerate the development of more efficient AI reasoning systems.

BOOK]]> Solving Problems In Point Geometry
BY [ Jingzhong Zhang]

Summary

Solving Problems in Point Geometry: Insights and Strategies for Mathematical Olympiad and Competitions, authored by prominent mathematicians Jingzhong Zhang and Xicheng Peng, introduces an innovative framework designed to overcome the historical challenges of elementary geometry. For over two millennia, students and competitors have wrestled with complex auxiliary graphics and exhausting algebraic equations. Inspired by a conceptual query from Leibniz, the authors present "point geometry," a groundbreaking approach that allows problem-solvers to perform operations directly on points.

This system synthesizes the core advantages of traditional vector, coordinate, and mass point methods into a single, cohesive tool. By utilizing identity-based methods and algebraic techniques like undetermined coefficients, the framework condenses intricate, multi-step geometric proofs into concise equations, unlocking a wide range of mathematical propositions with minimal friction.

While the text serves as a rich guide for high school math enthusiasts and Olympiad contestants, its implications reach far beyond competitive mathematics. Point geometry offers a remarkably clean structure for knowledge representation and automated reasoning, making it highly relevant to the field of artificial intelligence.

Because many of the geometric solutions detailed in the text can be generated systematically via computer programs, this framework provides a blueprint for simplifying automated reasoning models. By removing the traditional learning burdens associated with geometric proofs, this work not only shifts how we teach mathematics but also provides critical mathematical tools that could accelerate the development of more efficient AI reasoning systems.

BOOK]]> <![CDATA[Geometric Problems on Maxima and Minima [Andreescu]]]> https://mklab.gr/showthread.php?tid=944 Tue, 07 Jul 2026 19:20:56 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=944 [Image: 978-0-8176-4473-4?as=webp]

Geometric Problems on Maxima and Minima
BY [Titu Andreescu]

Summary

Geometric Problems on Maxima and Minima explores the fascinating world of optimization through the lens of Euclidean geometry, showing how maximum and minimum problems can be solved using creativity, intuition, and a variety of mathematical techniques.

Rather than treating extrema only as applications of calculus, the book presents hundreds of carefully selected geometric problems that reveal deeper connections between geometry, algebra, analysis, combinatorics, and topology. The authors, drawing on their experience as mathematicians and Olympiad coaches, develop a unified approach based on geometric transformations, symmetry, convexity, inequalities, and analytical methods.

The book serves both as a problem-solving guide and as an introduction to the broader ideas behind mathematical optimization. It includes classical and modern examples, ranging from elementary geometric configurations to challenging Olympiad-level problems, while providing detailed hints and solutions that encourage independent reasoning.

By emphasizing elegant methods rather than mechanical calculations, the work demonstrates how seemingly complex extremal problems can often be transformed into simple and insightful arguments. Its lasting value lies in showing how geometric thinking remains a powerful tool for discovering optimal solutions, making it an important resource for students, teachers, researchers, and anyone interested in the beauty of mathematical problem solving.

BOOK]]> [Image: 978-0-8176-4473-4?as=webp]

Problems from the Book
BY Titu Andreescu

Summary

Problems from the Book: Third Edition is a collection designed to inspire deeper mathematical thinking through challenging problems, elegant solutions, and surprising discoveries. Written by Titu Andreescu and Gabriel Dospinescu, the book gathers a wide variety of problems that range from accessible mathematical gems to advanced ideas requiring creativity and insight.

Rather than focusing only on techniques, it emphasizes problem-solving strategies, encouraging readers to explore patterns, develop intuition, and appreciate the beauty of mathematics. The problems are organized by increasing difficulty, making the book useful both for students preparing for major mathematics competitions such as the International Mathematical Olympiad and USA Mathematical Olympiad, as well as for anyone who enjoys exploring mathematical ideas beyond standard classroom exercises.
The third edition expands the collection with additional problems, new examples, and corrections while preserving the original structure that guides readers through progressive levels of challenge. Covering topics across elementary and advanced areas of mathematics, the book demonstrates how creative reasoning, persistence, and a willingness to experiment are central to mathematical discovery.

Its value extends beyond competition training: it serves as a source of lifelong mathematical enjoyment, showing how carefully chosen problems can reveal unexpected connections and develop the habits of thought shared by successful mathematicians. The book ultimately highlights why problem solving remains one of the most powerful ways to experience the creativity and depth of mathematics.

BOOK]]> <![CDATA[Mathematical Olympiad in China (2024) [Liu]]]> https://mklab.gr/showthread.php?tid=939 Tue, 07 Jul 2026 18:49:42 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=939 Mathematical Olympiad in China (2024)
by [Yongming Liu]

Summary

Mathematical Olympiad in China (2024): Problems and Solutions, edited by prominent mathematics professor Bin Xiong and translated by Yongming Liu, serves as a comprehensive collection of advanced competitive mathematics. The text primary focuses on the rigorous selection tests and training materials used by the 2024 Chinese National Training Team, alongside an extensive compilation of prominent domestic and international math competitions held between August 2023 and July 2024.

Readers will find meticulously designed mathematical challenges from global events such as the International Mathematical Olympiad, the American Mathematical Olympiad, and the European Girls' Mathematical Olympiad. What sets this volume apart is that the provided solutions are not merely mechanical answers; instead, they feature ingenious, elegant, and highly creative approaches formulated by elite national team members and top-tier researchers who specialize in Olympiad education.

This publication serves as an invaluable pedagogical resource designed for senior high school students, mathematics educators, university undergraduates, and passionate math enthusiasts. By systematically breaking down highly complex problems, the volume bridges the gap between foundational mathematical theory and innovative problem-solving methodologies.

It emphasizes the importance of cognitive flexibility, logical rigor, and creative thinking when approaching unfamiliar mathematical concepts. Ultimately, this work matters because it preserves and shares the highest levels of global mathematical talent, fostering a deeper curiosity and inspiring the next generation of logical thinkers and problem solvers worldwide.

BOOK]]> Mathematical Olympiad in China (2024)
by [Yongming Liu]

Summary

Mathematical Olympiad in China (2024): Problems and Solutions, edited by prominent mathematics professor Bin Xiong and translated by Yongming Liu, serves as a comprehensive collection of advanced competitive mathematics. The text primary focuses on the rigorous selection tests and training materials used by the 2024 Chinese National Training Team, alongside an extensive compilation of prominent domestic and international math competitions held between August 2023 and July 2024.

Readers will find meticulously designed mathematical challenges from global events such as the International Mathematical Olympiad, the American Mathematical Olympiad, and the European Girls' Mathematical Olympiad. What sets this volume apart is that the provided solutions are not merely mechanical answers; instead, they feature ingenious, elegant, and highly creative approaches formulated by elite national team members and top-tier researchers who specialize in Olympiad education.

This publication serves as an invaluable pedagogical resource designed for senior high school students, mathematics educators, university undergraduates, and passionate math enthusiasts. By systematically breaking down highly complex problems, the volume bridges the gap between foundational mathematical theory and innovative problem-solving methodologies.

It emphasizes the importance of cognitive flexibility, logical rigor, and creative thinking when approaching unfamiliar mathematical concepts. Ultimately, this work matters because it preserves and shares the highest levels of global mathematical talent, fostering a deeper curiosity and inspiring the next generation of logical thinkers and problem solvers worldwide.

BOOK]]> <![CDATA[Lecture Notes on Mathematical Olympiad Courses Volume 2 [Xu]]]> https://mklab.gr/showthread.php?tid=938 Tue, 07 Jul 2026 18:45:45 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=938 [Image: 8290-vol2.cover.jpg]

Lecture Notes on Mathematical Olympiad Courses Volume 2
by [Jiagu Xu]

Summary

Lecture Notes on Mathematical Olympiad Courses (Volume 2): For Senior Section provides a framework for advanced mathematical education rather than treating competitive math as a mere checklist of problem-solving tricks. Developed from training seminars conducted by former mathematics professor Jiagu Xu in Singapore, this volume spans essential and complex core concepts that reach well past standard high school curricula.

The lectures guide readers through advanced number theory, algebraic inequalities, sequences, combinatorics, and functional equations. Each chapter emphasizes foundational principles and underlying structures, using straightforward yet challenging real-world competition questions from around the globe to illustrate how abstract mathematical theories are operationalized.

By integrating conceptual depth with practical application, this book is tailored to build analytical resilience and rigorous logical reasoning. It functions as an adaptable guide, serving equally well as a self-study manual for independent students, a text for targeted olympiad preparation, or a reference framework for educators and academic researchers.

Complete with detailed solutions to testing problems, the material balances accessible instruction with high-level cognitive demands. Ultimately, this foundational text bridges the gap between conventional secondary school learning and modern mathematics, empowering the next generation of logical thinkers to navigate complex problem spaces with confidence and creativity.

BOOK]]> [Image: 8290-vol2.cover.jpg]