20 essential problems across all core topics — build your foundation for competition math
🔢Number Theory
➗Arithmetic
📐Geometry
🔣Algebra
📊Statistics
🎲Probability
🔄Ratios
🎯Counting
STUDY GUIDE
Core Concepts & Key Formulas
Topic 01 · Number Theory
Factors, Divisibility & Sum of Divisors
A factor (divisor) of n is any integer that divides n evenly. To find all factors, check every integer from 1 up to √n. For perfect squares, the square root is also a factor.
Always evaluate expressions in this exact order: Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right).
Multiplication and Division: same priority, left-to-right
Addition and Subtraction: same priority, left-to-right
Topic 03 · Fractions
Fraction Operations
To add/subtract fractions, find the LCD (Least Common Denominator). To multiply, multiply numerators and denominators directly. To divide, multiply by the reciprocal.
Percent means "per hundred." Three key types: finding a percent of a number, finding what percent one number is of another, and finding a number given its percent.
Part = Percent × Whole
Percent = Part ÷ Whole × 100
Whole = Part ÷ Percent
✏️ Example
40% of 80 is what percent of 64?
40% × 80 = 32. Then 32/64 × 100 = 50
Answer: 50%
Topic 05 · Geometry
Area & Perimeter Formulas
Area measures the space inside a 2D shape. Perimeter is the total distance around a shape. These are foundational formulas you must memorize cold.
Triangle: A = ½ × base × height
Rectangle: A = length × width
Rectangle Perimeter: P = 2(l + w)
Circle: A = πr², C = 2πr
📌 Memorize
Area of triangle = ½bh (NOT b×h alone)
For π problems, use π ≈ 22/7 unless told otherwise
Perimeter uses ALL sides added together
Topic 06 · Ratios & Proportions
Setting Up Proportions
A ratio compares two quantities. A proportion states that two ratios are equal. Cross-multiply to solve: if a/b = c/d, then ad = bc.
If 5 books cost $15:
Unit rate = $15 ÷ 5 = $3 per book
8 books = 8 × $3 = $24
📌 Memorize
Unit rate method: find cost/value of 1 item first
Cross-multiplication: a/b = c/d → ad = bc
Topic 07 · Statistics
Mean, Median, Mode, Range
Four measures describe a data set. Mean = average, Median = middle value when sorted, Mode = most frequent value, Range = max − min.
Mean = sum of values ÷ count
Median = middle value (sorted list)
Mode = most frequently occurring
Range = max − min
✏️ Example
Data: {12, 18, 9, 15, 6}
Mean = (12+18+9+15+6) ÷ 5 = 60 ÷ 5 = 12
Answer: Mean = 12
Topic 08 · Probability
Basic Probability
Probability of an event = (favorable outcomes) ÷ (total outcomes). Always between 0 and 1. Write as a fraction in simplest form.
P(event) = favorable outcomes / total outcomes
P(A or B) = P(A) + P(B) [mutually exclusive]
P(complement) = 1 − P(event)
✏️ Example
Bag: 3 red, 5 blue, 2 green marbles. P(blue)?
P(blue) = 5/(3+5+2) = 5/10
Answer: 1/2
Topic 09 · Algebra
Solving Linear Equations
To solve for x, isolate x by using inverse operations. Whatever you do to one side, do to the other. Goal: get x alone on one side.
3x − 7 = 14
3x = 14 + 7 = 21
x = 21 ÷ 3 = 7
📌 Memorize
Move constants by adding/subtracting to both sides
Move coefficients by multiplying/dividing both sides
Always CHECK your answer by substituting back
Topic 10 · Counting & Combinatorics
Factorials & Arrangements
The number of ways to arrange n distinct objects in a row is n! (n factorial). For choosing r items from n: combinations = C(n,r) = n! / (r!(n-r)!).