The AlgoRhythms Podcast

Welcome to Episode 12 of Season 2: Unlocking Combinatorics and Geometry! This episode provides a detailed exploration of geometry problems commonly encountered in the AMC mathematics competitions. The content covers various geometric concepts, beginning with three-dimensional geometry and providing foundational theorems like Euler's formula and distance and volume formulas for various shapes.

Welcome to Episode 11 of Season 2: Unlocking Combinatorics and Geometry! This episode provides geometric problem-solving techniques commonly used in competitive math, particularly focusing on analytical geometry. The initial section presents an example of solving for lengths using a system of equations derived from geometric properties. The core of the audio outlines fundamental theorems in coordi...

Welcome to Episode 10 of Season 2: Unlocking Combinatorics and Geometry! This episode focuses on geometry problems related to circles and triangles, particularly those encountered in competitive math exams like the AMC 10 and AMC 12. It begins with an example of area calculation within a triangle, illustrating a numerical approach. The audio then transitions to discussing key theorems regarding ci...

Welcome to Episode 9 of Season 2: Unlocking Combinatorics and Geometry! This episode discusses various methods for calculating area ratios in triangles, particularly those that appear on the AMC 10 and AMC 12 exams. Several theorems are presented, explaining how area ratios relate to side length ratios based on shared heights, shared bases, or shared angles. The audio includes solved problems demo...

Welcome to Episode 8 of Season 2: Unlocking Combinatorics and Geometry! This episode 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 defi...

Welcome to Episode 7 of Season 2: Unlocking Combinatorics and Geometry! This episode 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...

Welcome to Episode 6 of Season 2: Unlocking Combinatorics and Geometry! This episode 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...

Welcome to Episode 5 of Season 2: Unlocking Combinatorics and Geometry! This episode 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 inters...

Welcome to Episode 4 of Season 2: Unlocking Combinatorics and Geometry! 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 requir...

Welcome to Episode 3 of Season 2: Unlocking Combinatorics and Geometry! 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...

Welcome to Episode 2 of Season 2: Unlocking Combinatorics and Geometry! This episode 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 candie...

Welcome to Episode 1 of Season 2: Unlocking Combinatorics and Geometry! This episode 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 li...

Welcome to Episode 12 of Season 1: Unlocking Number Theory and Algebra! 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 examp...

Welcome to Episode 11 of Season 1: Unlocking Number Theory and Algebra! 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 import...

Welcome to Episode 10 of Season 1: Unlocking Number Theory and Algebra! 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 Favori...

Welcome to Episode 9 of Season 1: Unlocking Number Theory and Algebra! 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 invo...

Welcome to Episode 8 of Season 1: Unlocking Number Theory and Algebra! This video presents solutions to various math problems, primarily focusing on polynomial equations. Vieta's Formula 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 prop...

Welcome to Episode 7 of Season 1: Unlocking Number Theory and Algebra! 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 Americ...

Welcome to Episode 6 of Season 1: Unlocking Number Theory and Algebra! This episode dives into various number theory problems, primarily focusing on the application of modular arithmetic and the Chinese Remainder Theorem. Many problems involve finding remainders, solving congruences, or determining properties of integers. The solutions demonstrate techniques for simplifying complex problems into s...

Welcome to Episode 5 of Season 1: Unlocking Number Theory and Algebra! This episode focuses on Euler's totient function and Fermat's Little Theorem, illustrating their applications through examples and problem-solving. The author employs modular arithmetic and prime factorization to solve a problem involving finding the largest integer with specific properties. The text also includes an...

Welcome to Episode 4 of Season 1: Unlocking Number Theory and Algebra! This episode focuses on unit digits in number theory problems. Specific examples illustrate how to find patterns in unit digits of powers and sums, using techniques like modular arithmetic. The audio introduces modular arithmetic as a tool for solving these problems, showing how to find remainders when dividing by a specific nu...

Welcome to Episode 3 of Season 1: Unlocking Number Theory and Algebra! This episode presents solutions to several math problems involving number bases and other mathematical concepts. Specific problems include finding remainders after division, converting between different number bases (such as base 10, base 8, and base 11), and identifying palindromic numbers. Each problem is clearly stated, foll...

Welcome to Episode 2 of Season 1: Unlocking Number Theory and Algebra! In this episode, we delve into the fascinating world of palindromes—numbers that read the same forwards and backwards. We'll start by defining palindromes and sharing illustrative examples before diving into problem-solving techniques that involve these unique numbers. Our discussion includes calculating probabilities of p...

Welcome to Season 1: Unlocking Number Theory and Algebra! In Episode 1 of this season, join us as we dive into the fascinating world of number theory! In this episode, we explore essential concepts like prime and composite numbers, divisibility rules, and methods for finding least common multiples (LCMs) and greatest common divisors (GCDs). Through clear explanations and real-world examples, we br...