All formulas, one page

FORMULAS TO KNOW COLD

• PEMDAS: Parentheses, Exponents, Mul/Div, Add/Sub.
• Arithmetic series: sum = (first + last) / 2 × count.
• Sum 1+2+…+n = n(n+1)/2. (1+2+…+100 = 5050.)
• Sum 1+3+5+…+(2n−1) = n² (sum of first n odd numbers).
• Distributive: a × b + a × c = a × (b + c).
• Distributing minus over parens: a − (b + c) = a − b − c.
• Average ↔ total: total = average × count.
• Deviations from the mean sum to zero.
• Keep-fractions: after p% off → multiplier (1 − p/100).
• Median position: (n+1)/2 if odd; average positions n/2 and n/2+1 if even.
• Powers of 10: 10³ = 1,000; 10⁶ = million; 10⁹ = billion.

FACTS TO KNOW COLD

• Primes ≤ 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
• The only even prime is 2. 1 is NOT prime.
• To test if n is prime: trial-divide by primes ≤ √n. Beyond that, no factors can hide.
• Divisibility by 3 or 9: digit sum.
• Divisibility by 4: last 2 digits.
• Divisibility by 8: last 3 digits.
• Divisibility by 11: alternating digit sum.
• Divisibility by 7: chop last digit, double it, subtract from what's left.
• Divisibility by 13: chop last digit, multiply by 4, add to what's left.
• 1001 = 7 · 11 · 13 — split into 3-digit groups, alternating sum tests all three at once.
• Difference of squares: a² − b² = (a − b)(a + b).
• Divisor count formula: for n = p^a · q^b · …, count = (a+1)(b+1)·…
• Perfect-square ↔ odd number of divisors. Every other number has an even count.
• Perfect-square divisors of n: count divisors using only EVEN exponents.
• Divisors pair up: d ↔ n/d, product = n.
• GCD(a, b) × LCM(a, b) = a × b (two positive integers — fails for three or more).
• Euclidean algorithm: gcd(a, b) = gcd(b, a mod b). Replace the bigger with the leftover until you hit 0.
• Units-digit cycles: 2/3/7/8 → period 4; 4/9 → period 2; 0/1/5/6 → stuck.
• Digit sum ≡ N (mod 9).

CONVERSIONS TO MEMORIZE

• 1/2 = 0.5 = 50%; 1/3 ≈ 0.333 = 33⅓%; 2/3 ≈ 0.667 = 66⅔%
• 1/4 = 0.25 = 25%; 3/4 = 0.75 = 75%
• 1/5 = 0.2 = 20%; 2/5 = 0.4 = 40%; 3/5 = 0.6 = 60%; 4/5 = 0.8 = 80%
• 1/6 ≈ 0.167; 5/6 ≈ 0.833
• 1/8 = 0.125 = 12.5%; 3/8 = 0.375; 5/8 = 0.625; 7/8 = 0.875
• 1/9 ≈ 0.111; 1/11 ≈ 0.0909; 1/12 ≈ 0.0833
• KCF for division: Keep, Change, Flip — keep the first, change ÷ to ×, flip the second.
• Telescoping product: (1−1/2)(1−1/3)…(1−1/N) = 1/N.
• Telescoping sum: 1/(n(n+1)) = 1/n − 1/(n+1).
• +25% then −20% returns to start (because 1.25 × 0.8 = 1).
• +1/n undoes with −1/(n+1). +25% (=+1/4) undoes with −20% (=−1/5).

FORMULAS / FACTS TO KNOW COLD

• D = S × T (and the two cousins). Same as Work = Rate × Time.
• Average speed = total distance ÷ total time. NEVER the average of the speeds (unless times are equal).
• 1 hour = 3600 seconds = 60 minutes.
• 1 mile = 5280 feet. 1 km = 1000 m. 1 yard = 3 feet.
• 1 mph ≈ 1.467 ft/s (so 60 mph = 88 ft/s).
• Equal-distance round trip avg speed: 2ab / (a+b) (harmonic mean).
• Equal-time legs avg speed: (a+b)/2 (simple average).
• Together-time (A alone takes a, B alone takes b): T = ab / (a+b).
• Same-direction closing speed: faster − slower.
• Opposite-direction closing speed: sum of speeds.
• Exponential growth: V(n) = V₀ · rⁿ. Compound interest: V₀ · (1 + p/100)ⁿ.

FORMULAS TO KNOW COLD

• Linear: y = mx + b. Slope m = rise/run.
• Arithmetic sequence: a_n = a_1 + (n−1)d. Sum = n(a_1 + a_n)/2.
• Geometric sequence: a_n = a_1 · r^(n−1).
• Difference of squares: a² − b² = (a+b)(a−b). (Mental math: 51·49 = 50²−1 = 2499.)
• Square of binomial: (a±b)² = a² ± 2ab + b².
• Sum 1+2+…+n = n(n+1)/2.
• Sum 1+3+5+…+(2n−1) = n².
• Sum of squares 1²+2²+…+n² = n(n+1)(2n+1)/6.
• Sum of cubes 1³+2³+…+n³ = [n(n+1)/2]².
• Sum of n consecutive integers = n × middle.
• Average of an arithmetic sequence = (first + last) / 2.
• Exponent rules: x^a·x^b = x^a+b; x^a/x^b = x^a−b; (x^a)^b = x^ab; x^−a = 1/x^a; x⁰ = 1.
• Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … (each = sum of previous two).
• Digit-counts for pages: 1–9 → 9, 10–99 → 180, 100–999 → 2700.

FORMULAS TO KNOW COLD

• Rectangle: A = lw, P = 2(l + w).
• Square: A = s², P = 4s, diagonal = s√2.
• Triangle: A = ½ b h. Pythagorean: a² + b² = c².
• Equilateral triangle (side s): A = (√3/4) s². Height = (√3/2) s.
• Parallelogram: A = b h (h ⊥ base).
• Trapezoid: A = ½ (b₁ + b₂) h.
• Regular hexagon (side s): A = (3√3/2) s² (= 6 equilateral triangles of side s).
• Circle: A = π r², C = 2π r. Sector: π r² · θ/360. Arc: 2π r · θ/360.
• Cube: SA = 6 s², V = s³. Space diagonal = s√3.
• Box (a × b × c): SA = 2(ab + bc + ca), V = abc. Space diagonal = √(a² + b² + c²).
• Cylinder: V = π r² h, SA = 2π r² + 2π r h.
• Cone (radius r, height h): V = ⅓ π r² h.
• Sphere: V = (4/3) π r³, SA = 4π r².
• Distance between two points: √((x₂−x₁)² + (y₂−y₁)²).
• Equation of a circle: (x−a)² + (y−b)² = r² (center (a,b), radius r).
• Pick's Theorem (lattice polygons): A = I + B/2 − 1.
• 13-14-15 triangle: area = 84 (splits into 5-12-13 + 9-12-15).
• Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, 20-21-29.
• Special right triangles: 45-45-90 sides 1:1:√2. 30-60-90 sides 1:√3:2.
• Polygon interior angle sum: (n−2) · 180°.
• Regular polygon angles: interior = (n−2)·180/n; exterior = 360/n. Pentagon 108, hexagon 120, octagon 135, n-gon 9 → 140, 10-gon → 144.
• Inscribed right triangle: if a triangle inscribed in a circle has a side equal to the diameter, the opposite angle is 90°.
• Tangent ⊥ radius: a tangent line meets the radius at the touch-point at 90°.

QUADRILATERAL AREAS

• square s² · rectangle / parallelogram b·h
• rhombus ½·d₁·d₂ (the diagonals — not the sides!)
• trapezoid ½(b₁+b₂)·h (average the parallel sides, times height)

FORMULAS TO KNOW COLD

• Multiplication principle: independent steps multiply.
• Factorial: n! = n·(n−1)·…·2·1. 5! = 120. 6! = 720. 7! = 5040. (0! = 1 by convention.)
• Permutations (order matters): P(n, k) = n! / (n−k)!.
• Combinations (order doesn't matter): C(n, k) = n! / (k! · (n−k)!).
• Symmetry: C(n, k) = C(n, n−k).
• Handshake formula: C(n, 2) = n(n−1) / 2.
• Subsets of an n-element set: 2ⁿ.
• Circular arrangements: (n−1)! (rotations of the same arrangement count once).
• Letters with repeats: n! / (d₁! · d₂! · …).
• Stars and bars: identical N items among k distinct recipients (zeros allowed) = C(N+k−1, k−1).
• Probability: favorable / total (with equally-likely outcomes).
• Independent events: P(A and B) = P(A) · P(B).
• Mutually exclusive: P(A or B) = P(A) + P(B).
• Complement: P(not A) = 1 − P(A).
• Inclusion-exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|.
• Lattice paths (right/up): C(m+n, m).
• Pigeonhole (guarantee N from k): k(N−1) + 1 items.

FACTS TO KNOW

• Contrapositive: 'If A then B' ≡ 'If not B then not A'.
• Converse (not equivalent): 'If B then A'.
• Inverse (not equivalent): 'If not A then not B'.
• De Morgan: NOT(A and B) = (NOT A) or (NOT B). NOT(A or B) = (NOT A) and (NOT B).
• Counterexample disproves a 'for all'. One specific example with A and not-B kills 'A → B'.
• One example doesn't prove a 'for all'. You'd need a general argument.
• For 'there exists' claims: one example proves; you'd need to check every case to disprove.

THE SEVEN HABITS

1. Read the question twice. Underline the question word. What does it actually ask?
2. Circle the tiny words. “inclusive,” “except,” “positive,” “non-negative,” “at most,” “different,” “distinct.”
3. Match the units. Cents vs dollars. Minutes vs hours. Inches vs feet.
4. Plug your answer back in. Does it satisfy every condition the problem stated?
5. Don’t trust mental math past 3 digits. Write it on scratch.
6. Re-read your final letter. The bubble has to match your work.
7. Two-minute rule. If a problem’s taking too long, mark it and move on.

USE THE ANSWERS

The answer choices are part of the problem. Ask: what do they have in common? What’s DIFFERENT about each?

• All whole numbers → the answer is a whole number.
• Three choices obviously too big or too small → cross them off without solving.
• Choices differ by units digit → find the units digit; you’re done.
• Choices differ by parity → find the parity first.