🏆 Math League & contest prep

These tricks let you skip long computation. Practice until they're automatic.

Fast multiplication

Multiply any number by 11

Two-digit number: add the digits and place the sum in the middle.

Square a number ending in 5

Take the tens digit, multiply it by (tens digit + 1), then attach 25.

Multiply by 25

Multiply by 5

Squares near 50

48² = 2500 − 200 + 4 = 2304  |  53² = 2500 + 300 + 9 = 2809

Difference of squares shortcut

41² − 39² = (80)(2) = 160  |  50² − 49² = (99)(1) = 99

Fractions ↔ Percents — memorize this table

Algebraic expressions — "Scale the equation" trick

Rather than solving for the variable, factor so the target matches the given expression.

Binary and base conversions

Powers of 2 — memorize through 2¹²

2⁰	2¹	2²	2³	2⁴	2⁵	2⁶	2⁷	2⁸	2⁹	2¹⁰	2¹¹	2¹²
1	2	4	8	16	32	64	128	256	512	1024	2048	4096

Binary → decimal

Add the place values (1, 2, 4, 8, 16…) where you see a 1.

Decimal → binary

Repeatedly divide by 2, record remainders, read them bottom to top.

Other bases

Mixed number multiplication

Convert to improper fractions, cross-cancel before multiplying.

Whole × mixed: distribute.   4 × 2¾ = 4×2 + 4×¾ = 8 + 3 = 11

Repeating decimals → fractions

Pattern	Rule	Example
0.ā (1 repeating digit)	digit / 9	0.7̄ = 7/9
0.āb̄ (2 repeating digits)	digits / 99	0.2̄7̄ = 27/99 = 3/11
0.ābc̄ (3 repeating digits)	digits / 999	0.1̄2̄3̄ = 123/999 = 41/333
0.d(repeating) — mixed	(all digits − non-repeating) / (9s then 0s)	0.1̄3̄ = (13−1)/90 = 12/90 = 2/15

Sum of arithmetic sequences

Finding n: n = (last − first) / common difference + 1

Sequence	Shortcut
1 + 2 + … + n	n(n+1)/2
First n odd numbers (1, 3, 5, …)	n²
First n even numbers (2, 4, 6, …)	n(n+1)

Written problem-solving. Know every formula below cold — re-read the question twice before computing.

Geometry — area

Shape	Formula	Notes
Square	A = s²	s = side
Rectangle	A = l × w
Parallelogram	A = b × h	h = perpendicular height
Triangle	A = ½ b × h	any triangle
Trapezoid	A = ½(b₁ + b₂) × h	two parallel bases
Circle	A = πr²	r = radius, not diameter
Sector (pie slice)	A = (θ/360) × πr²	θ in degrees
Rhombus	A = ½ d₁ × d₂	d = diagonals
Equilateral △	A = (√3/4) s²	s = side
Ring (annulus)	A = π(R² − r²)	R = outer, r = inner

Geometry — perimeter & circumference

Shape	Formula
Circle	C = 2πr = πd
Arc length	L = (θ/360) × 2πr
Rectangle	P = 2(l + w)
Regular polygon (n sides)	P = n × s

Geometry — 3D (volume & surface area)

Shape	Volume	Surface area
Cube	s³	6s²
Rectangular prism	l × w × h	2(lw + lh + wh)
Cylinder	πr²h	2πr² + 2πrh
Cone	⅓ πr²h	πr² + πrl (l = slant)
Sphere	4/3 πr³	4πr²
Pyramid	⅓ × base area × h

Special right triangles

Pythagorean triples — know these cold

Triple	Key multiples
3, 4, 5	6-8-10 · 9-12-15 · 12-16-20 · 15-20-25
5, 12, 13	10-24-26
8, 15, 17	16-30-34
7, 24, 25
20, 21, 29

Coordinate plane

Concept	Formula
Distance	d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint	M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope	m = (y₂−y₁)/(x₂−x₁) = rise/run
Slope-intercept	y = mx + b
Parallel lines	Equal slopes (m₁ = m₂)
Perpendicular lines	m₁ × m₂ = −1 (negative reciprocals)

Permutations & combinations

Concept	Formula	When to use
Permutation (order matters)	P(n,r) = n! / (n−r)!	Arranging r items from n
Combination (order doesn't matter)	C(n,r) = n! / [r!(n−r)!]	Choosing r from n
Factorial	n! = n×(n−1)×…×1	5!=120, 6!=720, 7!=5040
Arrange n items	n!	All orderings
With repeated items	n! / (r₁! × r₂! × …)	e.g., letters in MISSISSIPPI
Circular arrangements	(n−1)!	Seats around a table

Handy values: C(5,2)=10 · C(6,2)=15 · C(7,2)=21 · C(10,2)=45 · C(n,2)=n(n−1)/2

Probability

Concept	Formula
Basic probability	P = favorable outcomes / total outcomes
Complement	P(not A) = 1 − P(A)
A or B (mutually exclusive)	P(A) + P(B)
A or B (can overlap)	P(A) + P(B) − P(A and B)
A and B (independent)	P(A) × P(B)
At least one	1 − P(none)   ← almost always easier
Geometric probability	favorable area / total area

Number theory

Divisibility by…	Rule
2	Last digit even
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
5	Last digit 0 or 5
6	Divisible by both 2 and 3
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
11	Alternating digit sum divisible by 11

Sequences & series

Type	nth term	Sum of first n terms
Arithmetic	a₁ + (n−1)d	n/2 × (a₁ + aₙ)
Geometric	a₁ × r^(n−1)	a₁(1−rⁿ)/(1−r)
1 + 2 + … + n		n(n+1)/2
1² + 2² + … + n²		n(n+1)(2n+1)/6
1³ + 2³ + … + n³		[n(n+1)/2]²

Angles & polygons

Concept	Formula
Sum of interior angles (n-gon)	(n−2) × 180°
Each interior angle (regular)	(n−2) × 180° / n
Each exterior angle (regular)	360° / n
Sum of exterior angles (any)	Always 360°
Exterior angle of triangle	= sum of two non-adjacent interior angles
Diagonals in n-gon	n(n−3)/2

Polygon names: 3 triangle · 4 square · 5 pentagon · 6 hexagon · 7 heptagon · 8 octagon · 9 nonagon · 10 decagon · 12 dodecagon

Percent, ratio & proportion

Concept	Formula
Part of a whole	Part = (Percent/100) × Whole
Percent change	(new − old) / old × 100
Percent increase	new = old × (1 + r)
Percent decrease	new = old × (1 − r)
Proportion	a/b = c/d → ad = bc
Simple interest	I = P × r × t

Statistics

Term	Definition
Mean	Sum ÷ count
Median	Middle value when sorted
Mode	Most frequent value
Range	Max − Min
Weighted average	(w₁v₁ + w₂v₂ + …) / (w₁ + w₂ + …)

Miscellaneous formulas

Concept	Formula
Diagonal of rectangle	√(l² + w²)
Diagonal of cube	s√3
Heron's formula (triangle, sides a b c)	A = √[s(s−a)(s−b)(s−c)] where s=(a+b+c)/2
Distance = Rate × Time	d = rt
Handshake problem (n people)	C(n,2) = n(n−1)/2
Euler's formula (polyhedra)	V − E + F = 2
Sum of angles in a 5-pointed star	180°
Similar figures (scale k)	lengths × k · areas × k² · volumes × k³
Difference of squares	a² − b² = (a+b)(a−b)
Perfect square	(a ± b)² = a² ± 2ab + b²

Unit conversions