JONAS'S GARDEN

Competitive Programming Math Notes Guide

4 min read

1. Core Philosophy (Important)

When writing CP math notes:

  • prioritize clarity over beauty
  • formulas must be scannable
  • examples must be runnable mentally
  • each topic should be self-contained
  • avoid long paragraphs
  • always show the closed form

Think like this:

“If I forget this during a contest, can I recover in 10 seconds?”

If not → rewrite the note.

2. Basic Obsidian Math Syntax

Inline math

Use for small expressions inside text.

The complexity is $O(n \log n)$.

Block math (use this for formulas)

Always prefer block math for identities.

$$
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
$$

Rule:

  • inline → small symbols
  • block → important formulas

3. Recommended Note Structure

For every math topic, use this structure:

# Topic Name

## Idea
(short intuition)

## Formula
(main identities)

## Example
(concrete numbers)

## CP Usage
(when it appears in problems)

## Notes
(edge cases / tricks)

This structure is extremely effective for revision.

4. Example: Sum Formulas (Model Note)

You can copy this style directly.

Sum Formulas

Idea

Many summations of the form

have closed-form polynomial expressions of degree (k+1).
These formulas are heavily used in time complexity analysis and counting problems.

Core Formulas

Sum of first n integers

Sum of squares

Sum of cubes

Example

Compute:

Using the formula:

CP Usage

Common when:

  • nested loop analysis
  • prefix sum reasoning
  • counting pairs
  • combinatorics simplification

Notes

  • Always reduce to closed form when possible.
  • Prevent overflow using long long in C++.
  • Often appears in problems with arithmetic progressions.

5. Example: Arithmetic Progression

Arithmetic Progression

Definition

An arithmetic progression (AP) is a sequence where the difference between consecutive terms is constant.

General form:

Sum of First n Terms

Alternative form:

where (l) is the last term.

Example

Sequence: (2, 5, 8, 11, 14)

  • (a = 2)
  • (d = 3)
  • (n = 5)

CP Usage

Shows up in:

  • pattern construction
  • greedy math problems

6. Example: Set Theory Note Style

Set Theory

Basic Definitions

A set is a collection of distinct elements.

Notation:

  • membership: (x \in A)
  • not in: (x \notin A)
  • subset: (A \subseteq B)

Important Operations

Union

Intersection

Difference

Inclusion–Exclusion Principle

For two sets:

For three sets:

|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| - |A \cap B \cap C| $$ --- ### CP Usage Extremely common in: - counting problems - bitmask problems - probability questions --- # 7. Example: Logic Notes --- ## Logic ### Common Operators | Symbol | Meaning | | ---------- | ------- | | ( \land ) | AND | | ( \lor ) | OR | | ( \lnot ) | NOT | | ( \oplus ) | XOR | --- ### Important Identities **De Morgan’s Laws**

\lnot (A \land B) = \lnot A \lor \lnot B

\lnot (A \lor B) = \lnot A \land \lnot B

--- ### CP Usage - bit manipulation - boolean DP - condition simplification --- # 8. Example: Logarithms --- ## Logarithms ### Definition

\log_b a = c \iff b^c = a

--- ### Important Rules **Product rule**

\log_b(xy) = \log_b x + \log_b y

\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y

\log_b(x^k) = k \log_b x

\log_b a = \frac{\log_c a}{\log_c b}

--- ### CP Usage - complexity analysis - binary search reasoning - divide and conquer depth --- # 9. High-Value Obsidian Tips Jonas, these separate average notes from elite ones. ## Use callouts for key formulas ``` > [!important] Must Remember > > $$ > \sum_{i=1}^{n} i = \frac{n(n+1)}{2} > $$ ``` --- ## Keep formulas isolated Bad: ``` The sum is $$...$$ which is useful. ``` Good: formula on its own line. --- ## Create one master note Create: ``` CP Math Cheatsheet.md ``` Inside, link to: - Sum formulas - Number theory - Logarithms - Combinatorics - etc. This becomes your contest revision weapon. ---

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