The AlgoRhythms Podcast

Welcome to Episode 11 of Season 3: Unlocking Advanced Topics! This episode focuses on calculating the area and length of complex geometric shapes. It presents conceptual tricks and remarks for breaking down intricate figures, such as extending lines to form simpler shapes like triangles or dropping altitudes. The episode then provides multiple examples and problems, often originating from math com...

Welcome to Episode 10 of Season 3: Unlocking Advanced Topics! This episode introduces the fundamental concepts of logarithms. It begins by defining what a logarithm is and illustrating its relationship to exponential forms. The source then presents various important logarithmic formulas, including those for multiplication, division, and changes of base, while also discussing natural logarithms inv...

Welcome to Episode 9 of Season 3: Unlocking Advanced Topics! The episode explores mathematical inequalities, beginning with the Trivial Inequality which states that all perfect squares are non-negative. It then introduces the Arithmetic Mean-Geometric Mean (AM-GM) inequality, explaining its application for two or more variables and noting conditions for equality. The episode further expands on Wei...

Welcome to Episode 8 of Season 3: Unlocking Advanced Topics! Today's audio offers foundational understanding of floor and ceiling functions. The episode begins by defining key terms like greatest integer less than or equal to x and fractional part of x. Then it introduces techniques for solving problems involving these functions, such as substitution and graphical analysis. Lastly, the episod...

Welcome to Episode 7 of Season 3: Unlocking Advanced Topics! Today's audio offers a foundational understanding of complex numbers, beginning with their basic definition as numbers expressed in the form a + bi, where 'a' represents the real part and 'bi' the imaginary part. It clarifies the cyclical nature of powers of 'i' and outlines the fundamental operations o...

Welcome to Episode 6 of Season 3: Unlocking Advanced Topics! The audio focuses on geometric trigonometry, specifically detailing how to calculate the area of a triangle using trigonometric functions. It introduces the Law of Sines, illustrating the relationship between a triangle's sides and the sines of its opposite angles, also introducing the circumradius. Furthermore, the document present...

Welcome to Episode 5 of Season 3: Unlocking Advanced Topics! This audio presents a comprehensive overview of trigonometric identities and their applications. It begins by introducing Pythagorean Identities, followed by Double Angle, Addition and Subtraction, Half Angle, Sum to Product, and Product to Sum Identities, each with corresponding formulas. The audio then transitions into graphing trigono...

Welcome to Episode 4 of Season 3: Unlocking Advanced Topics! The audio outlines various foundational concepts within trigonometry. It begins by defining basic trigonometric identities for sine, cosine, and tangent using a right triangle and the mnemonic SOH CAH TOA. Furthermore, the audio expands upon these by introducing more trigonometric identities such as cosecant, secant, and cotangent, which...

Welcome to Episode 3 of Season 3: Unlocking Advanced Topics! This audio  provides a comprehensive introduction to complex numbers. It begins by defining complex numbers and exploring the properties of the imaginary unit 'i', including its cyclical powers. The audio then presents theorems related to complex number operations, such as conjugation and magnitude, and applies these concepts t...

Welcome to Episode 2 of Season 3: Unlocking Advanced Topics! This episode is focused on geometry problems, specifically related to the AMC 10 and AMC 12 competitions. Several examples demonstrate the application of key trigonometric laws, such as the Law of Cosines and the Law of Sines, to solve complex geometric problems involving triangles and quadrilaterals. Detailed steps are provided for find...

Welcome to Episode 1 of Season 3: Unlocking Advanced Topics! This episode introduces basic trigonometry concepts, specifically sine, cosine, and tangent, explaining them as ratios in right triangles. It emphasizes the usefulness of trigonometry in geometry problems, particularly for tests like the AMC 10 and AMC 12, recommending memorization of certain angle values. The audio then demonstrates the...

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...