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The AlgoRhythms Podcast
The AlgoRhythms Team
36 episodes
2 months ago
Welcome to Episode 12 of Season 3: Unlocking Advanced Topics! This episode introduces mathematical concepts related to probability. It begins by defining states within probability problems and outlines steps for solving them, including assigning variables and setting up equations. The episode then presents examples to illustrate these concepts, progressing from basic probability scenarios, like coin flips, to more complex problems involving recursion and the expected value of states. Solution...
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Education
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Welcome to Episode 12 of Season 3: Unlocking Advanced Topics! This episode introduces mathematical concepts related to probability. It begins by defining states within probability problems and outlines steps for solving them, including assigning variables and setting up equations. The episode then presents examples to illustrate these concepts, progressing from basic probability scenarios, like coin flips, to more complex problems involving recursion and the expected value of states. Solution...
Show more...
Education
Episodes (20/36)
The AlgoRhythms Podcast
Unlocking Advanced Topics - Probability States
Welcome to Episode 12 of Season 3: Unlocking Advanced Topics! This episode introduces mathematical concepts related to probability. It begins by defining states within probability problems and outlines steps for solving them, including assigning variables and setting up equations. The episode then presents examples to illustrate these concepts, progressing from basic probability scenarios, like coin flips, to more complex problems involving recursion and the expected value of states. Solution...
Show more...
2 months ago
12 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Area and Length of Complex Shapes
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 competitions like AIME and AMC, that illustrate these techniques. These examples often involve squar...
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2 months ago
12 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Logarithms
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 involving Euler's number. Finally, it touches upon logarithms in the context of geometric progressio...
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2 months ago
11 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Inequalities and Optimization Techniques
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 Weighted AM-GM Inequality, providing methods for visualizing and applying it to maximize or minimize...
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2 months ago
12 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Floor and Ceiling Functions
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 episode includes a series of practice problems for students to apply their understanding of floor and ceilin...
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3 months ago
8 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Complex Numbers: Foundations and Operations
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 of addition, subtraction, and multiplication for these numbers. Furthermore, the audio introduces the concept of complex conjugates a...
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3 months ago
6 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Geometric Trigonometry: Triangle Area and Laws of Sines & Cosines
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 presents the Law of Cosines, which provides a formula to find the length of a side of a triangle given the le...
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3 months ago
10 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Algebraic Trigonometry: Identities, Graphs, and Problems
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 trigonometric functions and defining their periods, providing visual examples for sine, cosine, and tang...
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3 months ago
8 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics- Trigonometric Identities and Values
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 are reciprocals of the initial three functions. Explore a table of important trigonometric value...
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4 months ago
14 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Complex Numbers Basics
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 to solve various challenging problems typical of the AIME (American Invitational Mathematics Examinatio...
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4 months ago
11 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Law of Cosines and Law of Sines
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 finding side lengths and angles within geometric figures, including using concepts like the Pythagore...
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4 months ago
12 minutes

The AlgoRhythms Podcast
Unlocking Advanced Topics - Trigonometric Principles
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 application of these principles through solved example problems from past AMC contests, illustra...
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4 months ago
8 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - 3D Geometry
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.
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4 months ago
15 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - Coordinate Geometry and Shoelace Theorem
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 coordinate geometry, including the distance formula, properties of parallel and perpendicular lines, an...
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5 months ago
8 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - Circles
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 circles, including properties of tangents, intersecting circles, and various applications of the Po...
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5 months ago
11 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - Geometry Area Ratios
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 demonstrating the application of these theorems to complex geometric figures and area calculations. T...
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5 months ago
6 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - Triangle Medians, Cevians, and Areas
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 definition for each. Furthermore, it presents and illustrates key theorems like the Median Theorem an...
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5 months ago
7 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - Advanced Area Calculation Methods
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 theorems are provided for finding the area of a trapezoid and a rhombus. The audio concludes with exa...
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6 months ago
8 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - Similar and Special Triangles
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 how to apply these concepts to calculate areas, lengths, and other geometric properties in figur...
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6 months ago
7 minutes

The AlgoRhythms Podcast
Unlocking Combinatorics and Geometry - Angles and Polygons
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 intersecting parallel lines and theorems concerning angles within circles. Finally, it provides a refer...
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6 months ago
10 minutes

The AlgoRhythms Podcast
Welcome to Episode 12 of Season 3: Unlocking Advanced Topics! This episode introduces mathematical concepts related to probability. It begins by defining states within probability problems and outlines steps for solving them, including assigning variables and setting up equations. The episode then presents examples to illustrate these concepts, progressing from basic probability scenarios, like coin flips, to more complex problems involving recursion and the expected value of states. Solution...