Notes for curious people.

[Manganese] Colorful Black

A systematic overview of manganese chemistry, from the element and its coordination compounds to manganates, permanganates, laboratory reactions, and analytical applications.

2027 Strong Foundation Mathematics: AM-GM Inequalities

This article compiles 22 typical examples of mean inequality, covering denominator substitution, incremental substitution, matching, etc., local inequalities and the best estimate under product conditions. It is suitable for systematic review of the strong foundation plan and the basic stage of mathematics competitions.

Basic Organic Chemistry: Overview of Stereochemistry

This article introduces the basic research objects of stereochemistry, sorting out the hierarchical relationship between the composition, structure, configuration and conformation of molecules, and sorting out the basic judgment methods of molecular collinearity and coplanarity from the aspects of bond length, bond angle, central atom hybridization and lone pair of electrons.

2027 Strong Foundation Mathematics: Algebraic Manipulation

This article organizes 15 typical example questions around algebraic deformation, covering identities, factorization, symmetry processing and common variable substitutions. It focuses on showing how to simplify the complex through structural observation. It is suitable for the review of the strong foundation plan and the basic stage of mathematics competitions.

Basic Organic Chemistry L9-3

These are study notes for Basic Organic Chemistry L9-3: Do You Really Understand Substitution and Elimination Reactions of Alcohols?.

Peijian Education 2022 Labor Day Problem-Solving Workshop (May 1, Part II)

This article is the second part of Peijian Education's 2022 May 1st question brushing class (5.1). It continues to explain the conjugate, module and complex plane geometry applications of complex numbers through typical examples, including real number determination, four-point concircle, barycentric concircle and Ptolemy's inequality; at the same time, it organizes common methods for high-level joint test such as trigonometric identity transformation, triple angle formula, trigonometric function minimum and equation, complex number replacement to solve higher-order equations and Jensen's inequality.

Basic Organic Chemistry L9-2

These are study notes for Basic Organic Chemistry L9-2: Properties, Acid–Base Behavior, and Nucleophilicity of Alcohols.

Perkin Reaction

Perkin reaction mechanism, stereoselectivity, special cases of coumarin synthesis and layering exercises.

Peijian Education 2022 Labor Day Problem-Solving Workshop (May 1, Part I)

This article is the review notes for complex numbers in Peijian Education's 2022 May 1st question brushing class (5.1). It systematically sorts out the three forms of complex numbers, algebra, trigonometric, and exponent, as well as geometric applications (area, cocircle, similarity), and demonstrates in detail the combination of numbers and shapes, rotation skills, unit root summation, and argument principal value processing through 10 typical example questions.

Orbital Mechanics: Final Topics

This article systematically introduces several core topics in celestial motion, including the solution and small quantity approximation of Lagrangian points, hyperbola and parabolic orbit problems, calculus derivation of Kepler's first law, and detailed answers to several typical exercises.

Peijian Education 2022 Labor Day Problem-Solving Workshop (April 30)

Orbital Mechanics: Applications of Universal Gravitation

Orbital Mechanics: The Law of Universal Gravitation

A panoramic view of the laws of universal gravitation: from Kepler's three laws to the three major cosmic velocities, supplemented by detailed explanations of special topics and examples such as spherical shell gravity, double and triple stars, and celestial energy.

Class Review 1

The Hypergeometric Distribution and Its Properties

Starting from the definition, the expectation and variance of the hypergeometric distribution are completely deduced with the help of two combination identities (reduced order formula and Vandermonde identity), and every step is well documented.

A New-Definition Sequence Problem from the 2025 Haidian Grade 11 Final

Given a positive integer $n(n\ge 3)$, if the sequence $A_n:a_1,a_2,...,a_n$ satisfies the following two properties at the same time, then the sequence $A_n$ is called

Introduction to Basic Calculus: Applications II

Use calculus as a tool to solve physical problems: from the conservation of mechanical energy, simple harmonic motion and elastic potential energy, to the radius of curvature, the principle of virtual work and the variable mass rope, to the kinematics of the micro-element method and the moment of inertia of rigid bodies, and the differential equations and integral ideas in series mechanics.

Introduction to Basic Calculus: Applications I

This article introduces the application of calculus in physics, covering center-of-mass calculations (cones, hemispheric shells, hemispheres), conservative force potential energy and equilibrium stability criteria, Taylor small quantity expansion and its application in classic problems such as electric dipole forces, relativistic kinetic energy reduction, and non-uniform buoyancy.

Introduction to Basic Calculus: Elementary Integration

Starting from the mass integral, this article systematically introduces the concepts of indefinite and definite integrals, commonly used formulas and substitution methods, and demonstrates the practical application of integrals in competition physics through physical examples such as center of gravity calculation, n-dimensional sphere volume derivation, and resistance motion.

Introduction to Basic Calculus: Limits and Derivatives

The calculus covered in this article serves physics competition and focuses on practicality rather than rigor

Focal Radius Formulas for Conic Sections

The focal chord length of a conic section is often asked in questions. This article uses straight line parametric equation to derive: - Ellipse - Hyperbola - Parabola The Focus Cho

AM-GM Inequality Exercises

Small Blueprint "Mean Inequality and Cauchy's Inequality" Chapter 1 After-Class Exercises (Personal Answers)

Install crDroid on OnePlus Ace3Pro

[!WARNING] The following tutorial is only for OnePlus Ace3Pro to install crDroid 12 (corvette) Irregular gaming, relatives shed tears Readers are responsible for any hardware damag

Reverse Recursion and Absolute-Value Sequence Problems

Question (I) enumerates the possible values of $a_4$; question (II) proves that if there is a maximum value, it must contain $0$; question (III) uses proof by contradiction to prove that the positive term sequence has no upper bound: assuming there is an upper bound $M$, we can derive two adjacent items, one large and one small. After three steps of recursion, the large term increases and the small term $2\\times$ is exceeded. Continuous repetition will eventually exceed $M$, contradictory.

Hermite-Hadamard Inequality

2026 Beijing Chemistry Qualifiers

In the 2026 High School Chemistry Olympiad Beijing Regional Qualifiers, there are no difficult element inferences. It is all about the application of Basic Chemistry Theory and The

2026 Inorganic Chemistry Summer Training (I)

Hydrogen Storage Technology (39th Session) 6.1 High-pressure hydrogen storage. Compressed hydrogen is currently one of the most commonly used hydrogen storage methods. Hydrogen is

A Quotient Sequence from Arithmetic and Geometric Progressions

Introduction It is known that $\{a_n\}$ is an infinite arithmetic sequence with the first term being $a$ and a tolerance of 2. $\{b_n\}$ is an infinit

Bounded Monotone Sequence Transformations Eventually Repeat

(2024·Xuzhou Simulation) For the sequence $P:a_1,a_2,\cdots,a_n$ where each item is a positive integer, define transformations $T_1$ and $T_1$ to transform the s

Factorization of a^3 + b^3 + c^3 - 3abc

Detailed derivation of the factorization and application of a^3 + b^3 + c^3 - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)