<![CDATA[MKLab - PROBLEM SOLVING AND COMPETITIONS]]> https://mklab.gr/ Sun, 13 Sep 2026 11:12:52 +0000 MyBB <![CDATA[1988 IMO Question Six]]> https://mklab.gr/showthread.php?tid=1924 Wed, 09 Sep 2026 23:35:12 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1924 Author: Ethan Horsfall
Published: January 5, 2020, in Cantor’s Paradise

The article explains the celebrated Problem 6 of the 1988 International Mathematical Olympiad, often regarded as one of the hardest IMO problems ever posed. The problem asks: if positive integers $a,b$ satisfy $\frac{a^2+b^2}{ab+1}=x$ for some integer $x$, prove that $x$ must be a perfect square. 

The proof uses elementary algebra but develops a powerful descent argument. Writing the equation as $a^2+b^2=x(ab+1)$ and treating it as a quadratic in one variable gives $b^2-xab+(a^2-x)=0$. If $b$ is one integer root, Vieta’s formulas provide another integer root, allowing one solution to be replaced by a smaller related solution. The argument first establishes bounds such as $a,b\geq\sqrt{x}$, and then repeatedly applies this root-switching process.

 If both variables always remained strictly larger than $\sqrt{x}$, the descent would continue indefinitely, contradicting the existence of a smallest positive solution. Therefore eventually one obtains $a=\sqrt{x}$ or $b=\sqrt{x}$. Since $a$ and $b$ are integers, $\sqrt{x}\in\mathbb Z$, and consequently $x$ is a perfect square. This ingenious method became a classic example of Vieta jumping, combining quadratic equations, Vieta’s formulas and infinite descent to solve an apparently very difficult Diophantine problem using only elementary mathematics.

Key takeaways: The central equation is $\frac{a^2+b^2}{ab+1}=x$; interpreting it as a quadratic creates a second integral solution through Vieta’s formulas; repeatedly moving to a smaller solution produces an infinite-descent argument; and the process ultimately forces $a=\sqrt{x}$ or $b=\sqrt{x}$, proving that $x$ is a perfect square.

ARTICLE / ARTICLE [PDF]]]> Author: Ethan Horsfall
Published: January 5, 2020, in Cantor’s Paradise

The article explains the celebrated Problem 6 of the 1988 International Mathematical Olympiad, often regarded as one of the hardest IMO problems ever posed. The problem asks: if positive integers $a,b$ satisfy $\frac{a^2+b^2}{ab+1}=x$ for some integer $x$, prove that $x$ must be a perfect square. 

The proof uses elementary algebra but develops a powerful descent argument. Writing the equation as $a^2+b^2=x(ab+1)$ and treating it as a quadratic in one variable gives $b^2-xab+(a^2-x)=0$. If $b$ is one integer root, Vieta’s formulas provide another integer root, allowing one solution to be replaced by a smaller related solution. The argument first establishes bounds such as $a,b\geq\sqrt{x}$, and then repeatedly applies this root-switching process.

 If both variables always remained strictly larger than $\sqrt{x}$, the descent would continue indefinitely, contradicting the existence of a smallest positive solution. Therefore eventually one obtains $a=\sqrt{x}$ or $b=\sqrt{x}$. Since $a$ and $b$ are integers, $\sqrt{x}\in\mathbb Z$, and consequently $x$ is a perfect square. This ingenious method became a classic example of Vieta jumping, combining quadratic equations, Vieta’s formulas and infinite descent to solve an apparently very difficult Diophantine problem using only elementary mathematics.

Key takeaways: The central equation is $\frac{a^2+b^2}{ab+1}=x$; interpreting it as a quadratic creates a second integral solution through Vieta’s formulas; repeatedly moving to a smaller solution produces an infinite-descent argument; and the process ultimately forces $a=\sqrt{x}$ or $b=\sqrt{x}$, proving that $x$ is a perfect square.

ARTICLE / ARTICLE [PDF]]]> <![CDATA[Perspectives on mathematics competitions]]> https://mklab.gr/showthread.php?tid=1706 Thu, 20 Aug 2026 18:26:49 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1706 Perspectives on mathematics competitions and their relationship with mathematics education
Authors: María Falk de Losada & Peter James Taylor
Journal:ZDM – Mathematics Education, Vol. 54, pp. 941–959
Published: 8 August 2022
DOI: 10.1007/s11858-022-01404-z 

The article surveys the history, educational role and mathematical philosophy behind mathematics competitions and Olympiad-style problem solving. The authors argue that competitions should not be viewed merely as mechanisms for identifying gifted students. Rather, they constitute a form of “design science” in mathematics education: competition organizers deliberately design problems, difficulty levels, time constraints and formats to encourage particular kinds of mathematical thinking. Good competition problems can stimulate creativity, abstraction, generalization, proof and sustained problem solving, while also providing teachers with high-quality material for classrooms, mathematics clubs and enrichment activities. Modern mathematical competitions can be traced particularly to the Hungarian Eötvös Competition of 1894, which helped establish a culture in which challenging problems played an important role in mathematical education. 
Q`
A central theme is the distinction between mathematics as theory building and mathematics as problem solving. The authors use Leonhard Euler and Paul Erdős as historical models of the problem-solving tradition. Euler's work on subjects such as the Königsberg bridges, number theory, Latin squares and polyhedra resembles the style of many modern competition problems: elementary to formulate but requiring ingenuity to solve. Erdős continued this tradition through his lifelong emphasis on posing and solving beautiful problems. Competitions therefore give students a limited but authentic experience of mathematical research—confronting unfamiliar questions for which no standard algorithm is immediately available. They also develop reasoning, argumentation, proof and particularly visual mathematical thinking, rather than simply testing memorized techniques. 

The authors also stress that competitions have substantial value for mathematics education research. Large international contests generate enormous datasets that can be used to investigate gender differences, risk-taking, mathematical ability, problem-solving strategies and long-term educational outcomes. For example, many former International Mathematical Olympiad participants subsequently pursued mathematics-related careers, and several IMO gold medallists—including Timothy Gowers, Terence Tao and Maryam Mirzakhani—later became Fields Medallists. However, the authors caution against equating competition success with mathematical talent: some highly gifted students perform poorly in competitive settings, while excessive emphasis on ranking may discourage them. The greatest educational value of competitions therefore lies not in medals but in creating opportunities for students to experience the pleasure, creativity and intellectual challenge of genuine mathematical problem solving. 

Key takeaways
  • Mathematics competitions are educational tools, not merely talent-selection mechanisms.
  • Their strongest contribution is developing problem solving, creativity, reasoning, proof and visualization.
  • Competition mathematics continues a historical problem-solving tradition represented especially by Euler and Erdős.
  • Competition problems and the enormous datasets generated by large contests can contribute substantially to mathematics education and educational research. 
Central message: a well-designed mathematics competition can expose students to something ordinary classroom exercises often cannot—the experience of facing an unfamiliar problem, experimenting with ideas and eventually discovering a solution. In that sense, competitions can offer students a small-scale experience of what it actually means to do mathematics. 


ARTICLE]]> Perspectives on mathematics competitions and their relationship with mathematics education
Authors: María Falk de Losada & Peter James Taylor
Journal:ZDM – Mathematics Education, Vol. 54, pp. 941–959
Published: 8 August 2022
DOI: 10.1007/s11858-022-01404-z 

The article surveys the history, educational role and mathematical philosophy behind mathematics competitions and Olympiad-style problem solving. The authors argue that competitions should not be viewed merely as mechanisms for identifying gifted students. Rather, they constitute a form of “design science” in mathematics education: competition organizers deliberately design problems, difficulty levels, time constraints and formats to encourage particular kinds of mathematical thinking. Good competition problems can stimulate creativity, abstraction, generalization, proof and sustained problem solving, while also providing teachers with high-quality material for classrooms, mathematics clubs and enrichment activities. Modern mathematical competitions can be traced particularly to the Hungarian Eötvös Competition of 1894, which helped establish a culture in which challenging problems played an important role in mathematical education. 
Q`
A central theme is the distinction between mathematics as theory building and mathematics as problem solving. The authors use Leonhard Euler and Paul Erdős as historical models of the problem-solving tradition. Euler's work on subjects such as the Königsberg bridges, number theory, Latin squares and polyhedra resembles the style of many modern competition problems: elementary to formulate but requiring ingenuity to solve. Erdős continued this tradition through his lifelong emphasis on posing and solving beautiful problems. Competitions therefore give students a limited but authentic experience of mathematical research—confronting unfamiliar questions for which no standard algorithm is immediately available. They also develop reasoning, argumentation, proof and particularly visual mathematical thinking, rather than simply testing memorized techniques. 

The authors also stress that competitions have substantial value for mathematics education research. Large international contests generate enormous datasets that can be used to investigate gender differences, risk-taking, mathematical ability, problem-solving strategies and long-term educational outcomes. For example, many former International Mathematical Olympiad participants subsequently pursued mathematics-related careers, and several IMO gold medallists—including Timothy Gowers, Terence Tao and Maryam Mirzakhani—later became Fields Medallists. However, the authors caution against equating competition success with mathematical talent: some highly gifted students perform poorly in competitive settings, while excessive emphasis on ranking may discourage them. The greatest educational value of competitions therefore lies not in medals but in creating opportunities for students to experience the pleasure, creativity and intellectual challenge of genuine mathematical problem solving. 

Key takeaways
  • Mathematics competitions are educational tools, not merely talent-selection mechanisms.
  • Their strongest contribution is developing problem solving, creativity, reasoning, proof and visualization.
  • Competition mathematics continues a historical problem-solving tradition represented especially by Euler and Erdős.
  • Competition problems and the enormous datasets generated by large contests can contribute substantially to mathematics education and educational research. 
Central message: a well-designed mathematics competition can expose students to something ordinary classroom exercises often cannot—the experience of facing an unfamiliar problem, experimenting with ideas and eventually discovering a solution. In that sense, competitions can offer students a small-scale experience of what it actually means to do mathematics. 


ARTICLE]]> <![CDATA[The Pros and Cons of Math Competitions [AoPS]]]> https://mklab.gr/showthread.php?tid=1705 Thu, 20 Aug 2026 18:21:22 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1705 Source: Art of Problem Solving (AoPS)

The article argues that mathematics competitions such as MATHCOUNTS and the American Mathematics Competitions can be extremely valuable because they turn mathematics into an engaging intellectual challenge. Much as sports motivate students to improve physically, contests can motivate them to become better problem solvers and eventually develop a genuine interest in mathematics itself. Competitions also teach broader life skills: students learn that improvement requires practice, experience both success and failure, and become accustomed to performing under pressure. 

However, competitions can also have negative effects when they are poorly designed or pursued too intensely. Contests that emphasize speed, memorization, or routine classroom techniques may give students a distorted view of mathematics, whereas the most important skill should be the ability to think deeply about difficult problems. Students who are repeatedly given problems far beyond their current ability may become discouraged rather than challenged. Excessive preparation can also cause burnout, potentially leading students not merely to dislike competitions but to lose interest in mathematics altogether. 

Perhaps the article's strongest argument in favor of competitions is their social dimension. Mathematically talented students may feel isolated in ordinary classrooms, but contests allow them to meet others with similar interests and abilities, forming communities that provide friendship, inspiration, and intellectual stimulation. The overall conclusion is therefore not that competitions are inherently good or bad, but that they should be used intelligently: the objective should be to cultivate long-term mathematical curiosity and problem-solving ability, rather than simply producing winners. 

Key takeaways
  • Competitions can motivate students to explore mathematics beyond the school curriculum.
  • The best contests reward reasoning and problem solving, not merely speed or memorization.
  • Excessive difficulty, pressure, and training can cause discouragement and burnout.
  • One of their greatest benefits is creating a community of mathematically interested students who might otherwise feel isolated. 

ARTICLE]]> Source: Art of Problem Solving (AoPS)

The article argues that mathematics competitions such as MATHCOUNTS and the American Mathematics Competitions can be extremely valuable because they turn mathematics into an engaging intellectual challenge. Much as sports motivate students to improve physically, contests can motivate them to become better problem solvers and eventually develop a genuine interest in mathematics itself. Competitions also teach broader life skills: students learn that improvement requires practice, experience both success and failure, and become accustomed to performing under pressure. 

However, competitions can also have negative effects when they are poorly designed or pursued too intensely. Contests that emphasize speed, memorization, or routine classroom techniques may give students a distorted view of mathematics, whereas the most important skill should be the ability to think deeply about difficult problems. Students who are repeatedly given problems far beyond their current ability may become discouraged rather than challenged. Excessive preparation can also cause burnout, potentially leading students not merely to dislike competitions but to lose interest in mathematics altogether. 

Perhaps the article's strongest argument in favor of competitions is their social dimension. Mathematically talented students may feel isolated in ordinary classrooms, but contests allow them to meet others with similar interests and abilities, forming communities that provide friendship, inspiration, and intellectual stimulation. The overall conclusion is therefore not that competitions are inherently good or bad, but that they should be used intelligently: the objective should be to cultivate long-term mathematical curiosity and problem-solving ability, rather than simply producing winners. 

Key takeaways
  • Competitions can motivate students to explore mathematics beyond the school curriculum.
  • The best contests reward reasoning and problem solving, not merely speed or memorization.
  • Excessive difficulty, pressure, and training can cause discouragement and burnout.
  • One of their greatest benefits is creating a community of mathematically interested students who might otherwise feel isolated. 

ARTICLE]]> <![CDATA[The Pros and Cons of Math Tournaments]]> https://mklab.gr/showthread.php?tid=1704 Thu, 20 Aug 2026 18:17:02 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1704 The Pros and Cons of Math Tournaments: A Reflection on My Eight-Year Competition Math Career
Author: Catherine Cossaboom — GLeaM
Published: May 31, 2021; updated May 17, 2022. 

Drawing on eight years of participation in competitions such as MATHCOUNTS, AMC/AIME, Georgia ARML, and Math Prize for Girls, the author argues that mathematical tournaments can be extremely valuable when they emphasize creativity rather than memorization or raw speed. Good competition problems expose students to ideas beyond the normal school curriculum—particularly number theory, combinatorics, geometry, and creative problem solving—and encourage them to construct solutions from basic principles. This process can build persistence, confidence, independent learning, time-management skills, and enthusiasm for mathematics and STEM. The author considers the social dimension equally important: teams, tournaments, camps, and math circles can create strong communities and lifelong friendships. 

However, competitions can also distort a student's perception of mathematics. Poorly designed contests may reward rapid calculation, memorized tricks, or speed rather than genuine mathematical insight. More seriously, students may begin to identify their mathematical ability—and even their self-worth—with rankings and scores. Strict time limits, comparison with stronger competitors, parental expectations, and the assumption that progress should always be linear can produce frustration and burnout. The author stresses that competition mathematics is not representative of mathematical research, where difficult problems may take months or years rather than minutes. Students who dislike competitive environments may therefore thrive much more in exploratory activities such as math circles, research projects, or summer programs. 

The article also discusses inclusion. Although competition scoring itself is relatively objective, the surrounding community can still contain discrimination; the author describes her experiences as a girl competing in environments where female participation at the highest levels was sometimes below 10%. Nevertheless, her overall assessment remains strongly positive: the greatest benefit of mathematical competitions was not trophies or rankings but the community, intellectual curiosity, confidence, and friendships they created. The central message is therefore not that students should avoid competition, but that competitions should be treated as one way of exploring mathematics rather than as a test of whether someone is “good at math.” 

Key takeaways
  • Best benefit: develops creative problem-solving, persistence, and exposure to mathematics beyond the school curriculum.
  • Biggest danger: confusing competition performance with mathematical ability or personal worth.
  • Good competitions reward insight, whereas excessive speed, memorization, and routine calculation undermine their educational value.
  • Mathematical competitions are only one form of mathematics; slower exploratory mathematics, research, camps, and math circles may suit many students better.

ARTICLE]]> The Pros and Cons of Math Tournaments: A Reflection on My Eight-Year Competition Math Career
Author: Catherine Cossaboom — GLeaM
Published: May 31, 2021; updated May 17, 2022. 

Drawing on eight years of participation in competitions such as MATHCOUNTS, AMC/AIME, Georgia ARML, and Math Prize for Girls, the author argues that mathematical tournaments can be extremely valuable when they emphasize creativity rather than memorization or raw speed. Good competition problems expose students to ideas beyond the normal school curriculum—particularly number theory, combinatorics, geometry, and creative problem solving—and encourage them to construct solutions from basic principles. This process can build persistence, confidence, independent learning, time-management skills, and enthusiasm for mathematics and STEM. The author considers the social dimension equally important: teams, tournaments, camps, and math circles can create strong communities and lifelong friendships. 

However, competitions can also distort a student's perception of mathematics. Poorly designed contests may reward rapid calculation, memorized tricks, or speed rather than genuine mathematical insight. More seriously, students may begin to identify their mathematical ability—and even their self-worth—with rankings and scores. Strict time limits, comparison with stronger competitors, parental expectations, and the assumption that progress should always be linear can produce frustration and burnout. The author stresses that competition mathematics is not representative of mathematical research, where difficult problems may take months or years rather than minutes. Students who dislike competitive environments may therefore thrive much more in exploratory activities such as math circles, research projects, or summer programs. 

The article also discusses inclusion. Although competition scoring itself is relatively objective, the surrounding community can still contain discrimination; the author describes her experiences as a girl competing in environments where female participation at the highest levels was sometimes below 10%. Nevertheless, her overall assessment remains strongly positive: the greatest benefit of mathematical competitions was not trophies or rankings but the community, intellectual curiosity, confidence, and friendships they created. The central message is therefore not that students should avoid competition, but that competitions should be treated as one way of exploring mathematics rather than as a test of whether someone is “good at math.” 

Key takeaways
  • Best benefit: develops creative problem-solving, persistence, and exposure to mathematics beyond the school curriculum.
  • Biggest danger: confusing competition performance with mathematical ability or personal worth.
  • Good competitions reward insight, whereas excessive speed, memorization, and routine calculation undermine their educational value.
  • Mathematical competitions are only one form of mathematics; slower exploratory mathematics, research, camps, and math circles may suit many students better.

ARTICLE]]> <![CDATA[Advice on mathematics competitions [Tao]]]> https://mklab.gr/showthread.php?tid=1703 Thu, 20 Aug 2026 18:07:36 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1703 Advice on mathematics competitions 
by [Terence Tao]
Terence Tao presents mathematics competitions as an enjoyable and valuable part of a young mathematician’s development, but warns against confusing competition mathematics with mathematics itself. Olympiads provide excitement, intellectual challenge, interaction with talented peers, and—at higher levels—opportunities for national and international travel. They also demonstrate that mathematics can be pursued for creativity and problem solving rather than merely for grades and examinations. Tao himself recalls his high-school competition experience very positively. 
At the same time, Tao stresses that Olympiad problems are quite different from university-level mathematics and especially from mathematical research. Competition problems are deliberately designed to have elegant, relatively self-contained solutions, whereas research usually requires much slower work: reading existing literature, experimenting with special cases, applying known techniques, searching for counterexamples, and repeatedly pursuing approaches that may fail. Olympiad training can make someone extremely good at executing clever individual steps, but it does not replace the patience and broader knowledge required for research.
There is also a change in mathematical culture. Olympiad mathematics tends to emphasize classical subjects such as Euclidean geometry, elementary number theory and combinatorics, while undergraduate and graduate mathematics increasingly uses abstract structures and modern theories. Tao points out, however, that classical mathematics remains underneath much of modern mathematics: elementary number theory feeds into algebra and modern number theory, while classical geometry provides intuition for algebraic and differential geometry. His central recommendation is therefore simple: enjoy competitions, but do not allow competition preparation to replace a broad mathematical education. 
Key takeaways
  • Olympiads are excellent training, but not a model of mathematical research.
  • Mathematical research requires persistence, experimentation, reading, and tolerance of long periods without a solution.
  • Strong competition students should eventually expand beyond tricks and problem-solving techniques into theory, abstraction, and mathematical literature.
  • Tao's most important warning is that the apparently “boring” parts of mathematical education—definitions, theory, systematic study, and foundational knowledge—often become more useful in the long run than competition techniques. 
A useful one-sentence summary of Tao's message would be:
Mathematical competitions are a wonderful sport for the mind, but becoming a mathematician requires learning to play a much larger and slower game.
ARTICLE
]]> Advice on mathematics competitions 
by [Terence Tao]
Terence Tao presents mathematics competitions as an enjoyable and valuable part of a young mathematician’s development, but warns against confusing competition mathematics with mathematics itself. Olympiads provide excitement, intellectual challenge, interaction with talented peers, and—at higher levels—opportunities for national and international travel. They also demonstrate that mathematics can be pursued for creativity and problem solving rather than merely for grades and examinations. Tao himself recalls his high-school competition experience very positively. 
At the same time, Tao stresses that Olympiad problems are quite different from university-level mathematics and especially from mathematical research. Competition problems are deliberately designed to have elegant, relatively self-contained solutions, whereas research usually requires much slower work: reading existing literature, experimenting with special cases, applying known techniques, searching for counterexamples, and repeatedly pursuing approaches that may fail. Olympiad training can make someone extremely good at executing clever individual steps, but it does not replace the patience and broader knowledge required for research.
There is also a change in mathematical culture. Olympiad mathematics tends to emphasize classical subjects such as Euclidean geometry, elementary number theory and combinatorics, while undergraduate and graduate mathematics increasingly uses abstract structures and modern theories. Tao points out, however, that classical mathematics remains underneath much of modern mathematics: elementary number theory feeds into algebra and modern number theory, while classical geometry provides intuition for algebraic and differential geometry. His central recommendation is therefore simple: enjoy competitions, but do not allow competition preparation to replace a broad mathematical education. 
Key takeaways
  • Olympiads are excellent training, but not a model of mathematical research.
  • Mathematical research requires persistence, experimentation, reading, and tolerance of long periods without a solution.
  • Strong competition students should eventually expand beyond tricks and problem-solving techniques into theory, abstraction, and mathematical literature.
  • Tao's most important warning is that the apparently “boring” parts of mathematical education—definitions, theory, systematic study, and foundational knowledge—often become more useful in the long run than competition techniques. 
A useful one-sentence summary of Tao's message would be:
Mathematical competitions are a wonderful sport for the mind, but becoming a mathematician requires learning to play a much larger and slower game.
ARTICLE
]]> <![CDATA[Problem solving 101]]> https://mklab.gr/showthread.php?tid=537 Sun, 21 Jun 2026 16:07:17 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=537 Problem solving 101
BY  Steven Muirhead

Summary

The article “Problem Solving 101” by Stephen Muirhead explains that mathematical problem solving is not a mysterious talent possessed only by geniuses, but a skill developed through practice and the use of effective techniques. It presents several powerful strategies: building intuition by testing simple cases and changing parameters, identifying and exploiting patterns or symmetry, and using extreme cases to prove ideas or construct solutions. 

Through examples involving geometry, algebra, clocks, games, irrational numbers, and tournaments, the article shows that many difficult-looking problems become manageable when their hidden structure is recognized. The main message is that successful problem solving comes from patience, experimentation, and applying simple but powerful mathematical ideas rather than waiting for sudden inspiration.

ARTICLE]]> Problem solving 101
BY  Steven Muirhead

Summary

The article “Problem Solving 101” by Stephen Muirhead explains that mathematical problem solving is not a mysterious talent possessed only by geniuses, but a skill developed through practice and the use of effective techniques. It presents several powerful strategies: building intuition by testing simple cases and changing parameters, identifying and exploiting patterns or symmetry, and using extreme cases to prove ideas or construct solutions. 

Through examples involving geometry, algebra, clocks, games, irrational numbers, and tournaments, the article shows that many difficult-looking problems become manageable when their hidden structure is recognized. The main message is that successful problem solving comes from patience, experimentation, and applying simple but powerful mathematical ideas rather than waiting for sudden inspiration.

ARTICLE]]> <![CDATA[How to Solve Math Problems [Tiwari]]]> https://mklab.gr/showthread.php?tid=450 Wed, 17 Jun 2026 14:21:26 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=450 How to Solve Math Problems: Polya’s Method and Beyond
by Gaurav Tiwari

Summary

The article explains that mathematical problem solving is not about memorizing formulas but about recognizing patterns, understanding structures, and applying a systematic approach. It presents George Pólya’s four-step method: understand the problem, devise a plan, carry out the plan, and look back to verify and learn from the solution. 
The article emphasizes useful strategies such as working backward, simplifying problems, drawing diagrams, searching for patterns, using contradiction, induction, and making analogies. It also highlights that becoming a strong problem solver requires practice, developing a mental library of solved problems, accepting difficulties, and writing solutions clearly. Ultimately, mathematical problem solving is presented as a skill that improves through deliberate practice, reflection, and learning from each challenge.

ARTICLE]]> How to Solve Math Problems: Polya’s Method and Beyond
by Gaurav Tiwari

Summary

The article explains that mathematical problem solving is not about memorizing formulas but about recognizing patterns, understanding structures, and applying a systematic approach. It presents George Pólya’s four-step method: understand the problem, devise a plan, carry out the plan, and look back to verify and learn from the solution. 
The article emphasizes useful strategies such as working backward, simplifying problems, drawing diagrams, searching for patterns, using contradiction, induction, and making analogies. It also highlights that becoming a strong problem solver requires practice, developing a mental library of solved problems, accepting difficulties, and writing solutions clearly. Ultimately, mathematical problem solving is presented as a skill that improves through deliberate practice, reflection, and learning from each challenge.

ARTICLE]]> <![CDATA[Mathnet]]> https://mklab.gr/showthread.php?tid=52 Sun, 17 May 2026 14:51:01 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=52 MathNet is an online platform developed by researchers at the Massachusetts Institute of Technology (MIT) that hosts more than 30,000 mathematics olympiad problems from 47 countries and 17 languages. The website allows users to explore problems in algebra, geometry, number theory, and combinatorics drawn from national and international mathematical competitions, many of which are accompanied by detailed solutions. 

The “Examples” section showcases representative problems of varying difficulty levels and origins, highlighting the breadth and diversity of the collection. In addition, MathNet serves as a research resource for evaluating the mathematical reasoning capabilities of artificial intelligence systems, providing tools for problem solving, retrieval of similar mathematical problems, and assessment of Retrieval-Augmented Generation (RAG) techniques.



https://mathnet.mit.edu/]]> MathNet is an online platform developed by researchers at the Massachusetts Institute of Technology (MIT) that hosts more than 30,000 mathematics olympiad problems from 47 countries and 17 languages. The website allows users to explore problems in algebra, geometry, number theory, and combinatorics drawn from national and international mathematical competitions, many of which are accompanied by detailed solutions. 

The “Examples” section showcases representative problems of varying difficulty levels and origins, highlighting the breadth and diversity of the collection. In addition, MathNet serves as a research resource for evaluating the mathematical reasoning capabilities of artificial intelligence systems, providing tools for problem solving, retrieval of similar mathematical problems, and assessment of Retrieval-Augmented Generation (RAG) techniques.



https://mathnet.mit.edu/]]> <![CDATA[The ‘trickonometry’ of math olympiad inequalities]]> https://mklab.gr/showthread.php?tid=50 Fri, 25 Apr 2025 12:50:24 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=50
https://www.andreinc.net/2025/03/17/the-...equalities]]>
https://www.andreinc.net/2025/03/17/the-...equalities]]>