Unlocking Combinatorics And Geometry - Path Counting, Bijections, And Stars And Bars The AlgoRythms podcast

7:38

This audio explores several key geometric concepts and theorems. It begins by demonstrating how to calculate the area of a rhombus using area expressions and algebraic manipulation. The audio then defines important triangle properties such as cevians, angle bisectors, incenters, medians, and centroids, providing a clear definition for each. Furthermore, it presents and illustrates key theorems like the Median Theorem and Stewart's Theorem, showing their application in solving various geometry problems, primarily sourced from the AMC math competition. The examples walk through step-by-step solutions, utilizing these concepts and theorems to find unknown lengths, ratios, and areas of triangles and quadrilaterals.…

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Unlocking Combinatorics and Geometry - Advanced Area Calculation Methods 8:00

14 days ago8:00

8:00

This podcast focuses on various methods for calculating the area of polygons, particularly triangles. It starts by introducing the familiar base-height formula for triangle area but then presents more advanced theorems, including the use of the semi-perimeter and inradius, Heron's Formula, and the circumradius. Additional theorems are provided for finding the area of a trapezoid and a rhombus. The audio concludes with examples which apply these methods, demonstrating their practical use in competitive mathematics contexts.…

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Unlocking Combinatorics and Geometry - Similar and Special Triangles 7:31

21 days ago7:31

7:31

This audio focuses on the concept of similar triangles. It begins by outlining fundamental definitions and theorems related to triangle congruence, such as SSS, SAS, ASA, and AAS. The podcast then transitions to exploring properties of similar triangles, including angle and side ratios. Several worked-out problems illustrate how to apply these concepts to calculate areas, lengths, and other geometric properties in figures containing similar triangles. This podcast also provides specific characteristics of equilateral triangles.…

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Unlocking Combinatorics and Geometry - Angles and Polygons 10:11

28 days ago10:11

10:11

This audio introduces fundamental concepts related to angles, starting with definitions of straight and right angles. It then covers complementary and supplementary angles and explores properties of interior and exterior angles in polygons. The audio further explains relationships between angles formed by transversals intersecting parallel lines and theorems concerning angles within circles. Finally, it provides a reference list of common 2D geometric shapes and their properties, concluding with example problems applying these principles to triangles and quadrilaterals.…

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Unlocking Combinatorics and Geometry - Probability Fundamentals for AMC 10 and 12 5:47

5 weeks ago5:47

5:47

The audio focuses on fundamental concepts within probability. It introduces the definition of conditional probability and Bayes' Theorem, illustrating their application through worked examples involving dice rolls and event dependencies. The audio then explores expected value, linearity of expectation, and problems requiring the calculation of expected values in various scenarios, such as sequences and random selections. Finally, it touches upon geometric probability, using coordinate systems and areas to determine probabilities based on spatial conditions. Overall, the audio offers a structured introduction to key probability principles with illustrative problems relevant to mathematics competitions.…

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Unlocking Combinatorics and Geometry - Binomials, Identities, and Recursion 6:30

6 weeks ago6:30

6:30

This audio source focuses on binomials and recursion, two fundamental concepts in mathematics often encountered in these competitions. The audio introduces the binomial theorem and related identities, such as Pascal's Identity and the Hockey Stick Identity, providing formulas and examples. It then transitions to recursion, defining it and illustrating its principles through problems involving counting possibilities with specific constraints, such as climbing stairs or forming subsets. The purpose of this audio is to equip students with the theoretical understanding and problem-solving techniques necessary to tackle AMC-level questions related to these topics.…

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Unlocking Combinatorics and Geometry - Path Counting, Bijections, and Stars and Bars 7:37

7 weeks ago7:37

7:37

This audio explores methods for counting and arranging items under various constraints. It introduces fundamental combinatorial concepts such as calculating the number of paths on a grid and determining the arrangements of letters or objects. Several example problems illustrate these principles, including distributing candies, arranging dice rolls, and placing objects in a row. This material utilizes formulas and step-by-step solutions to explain how to solve these types of combinatorial questions. Overall, the podcast provides an introduction to basic techniques in enumerative combinatorics.…

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Unlocking Combinatorics and Geometry - Combinatorics Principles and Sets 6:43

8 weeks ago6:43

6:43

This podcast on combinatorics introduces fundamental principles for counting, such as casework and complementary counting. It provides formulas and theorems for permutations with repeated elements and combinations. Several examples and problems illustrate how to calculate the number of ways to arrange letters in words, form license plates, select integers with specific properties, and analyze sets. The podcast also defines the Principle of Inclusion and Exclusion (PIE) and offers examples of its application in set theory problems. Overall, the material covers basic combinatorial techniques and their use in solving various counting scenarios.…

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Unlocking Number Theory - Floor Functions and Recursive Sequences 13:09

9 weeks ago13:09

13:09

Welcome to Episode 12 of Season 1: Unlocking Number Theory! This video contains mathematical problems and solutions. The problems cover a range of topics including logarithms, floor functions, and recursive sequences. Each problem is sourced, indicating where it originated. Detailed solutions are provided, often utilizing algebraic manipulation and problem-solving strategies. The examples are geared towards advanced mathematics competitions like the AMC 10 and AMC 12. The content is designed to help students prepare for these types of contests.…

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Unlocking Number Theory - Inequalities 7:31

10 weeks ago7:31

7:31

Welcome to Episode 11 of Season 1: Unlocking Number Theory! This video covers inequalities and functional equations, along with telescoping series, for math competition preparation. It introduces key theorems like the AM-GM and Cauchy-Schwarz inequalities and demonstrates their use in problem-solving. The text also explores various functional equation strategies, emphasizing the importance of substituting values. Telescoping series techniques, including partial fraction decomposition, are explained and illustrated through examples. The concepts are reinforced with practice problems sourced from past math competitions like AIME and HMMT. The video is designed to equip students with tools and strategies for tackling advanced mathematical problems in contests.…

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Unlocking Number Theory - Equation Solutions 10:59

11 weeks ago10:59

10:59

Welcome to Episode 10 of Season 1: Unlocking Number Theory! This episode contains mathematical problems and solutions aimed at students preparing for the AMC 10 and AMC 12 competitions. It focuses on various problem-solving techniques, including factoring, manipulating equations, and utilizing substitutions. Specific examples cover topics like difference of squares, Simon's Favorite Factoring Trick, systems of equations, and polynomial roots. Solutions are provided step-by-step, showing the algebraic manipulation required to find the answers. The problems are sourced from various math competitions, including the AMC and AIME.…

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Unlocking Number Theory - Logarithmic Problem Solving 15:40

12 weeks ago15:40

15:40

Welcome to Episode 9 of Season 1: Unlocking Number Theory! This episode presents a series of logarithm problems and their solutions, demonstrating various techniques for solving logarithmic equations. Key rules and formulas are introduced, including the change of base formula. The problems cover a range of complexity, from basic logarithmic identities to more advanced applications involving systems of equations and summations. Solutions are provided step-by-step, showing the algebraic manipulation required to find the answers. The problems are sourced from various math competitions, including the AMC and AIME.…

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Unlocking Number Theory - Advanced Polynomials and Vieta's 9:11

13 weeks ago9:11

9:11

Welcome to Episode 8 of Season 1: Unlocking Number Theory! This video presents solutions to various math problems, primarily focusing on polynomial equations. Vieta's Theorem is frequently applied to find relationships between polynomial coefficients and their roots. The problems explore concepts such as finding the sum of roots, determining the smallest integer with specific properties, and solving for unknown coefficients in polynomial expressions. The solutions demonstrate the use of algebraic manipulation and factoring techniques. Several problems originate from math competitions like the AMC and PUMaC.…

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Unlocking Number Theory - Arithmetic and Geometric Sequences and Polynomials 16:40

14 weeks ago16:40

16:40

Welcome to Episode 7 of Season 1: Unlocking Number Theory! This episode presents mathematical problems and solutions focusing on arithmetic, geometric, and arithmetic-geometric sequences. Key concepts covered include relevant formulas, such as the sum of arithmetic and geometric series. Examples illustrate problem-solving techniques using these formulas within the context of the American Invitational Mathematics Examination (AIME). Additional topics explored are Vieta's Theorem and its application to polynomials. The text ultimately aims to equip readers with the skills needed to solve advanced mathematical problems involving sequences and polynomials.…

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Unlocking Number Theory - Chinese Remainder's Theorem and Word Problems 11:01

15 weeks ago11:01