AMO cheat sheet

Draw a picture or table for every problem: pictures turn hidden quantities into countable objects and expose the relationship between the numbers.

Working backwards: when the end result and the steps are known, undo each step in reverse order (addition undoes subtraction, multiplication undoes division).

Guess-and-check is systematic: record each guess in a table, compare to the target, and adjust in the direction the error points.

Pattern rules: find how much each term changes (the step), then extend by repeating that step; term n = first term + (n − 1) × step.

A sum of consecutive numbers can be paired from the ends; e.g. 1+2+…+10 pairs to five 11s, giving 55.

Coin and stamp puzzles: fix the number of items, then swap one item type for another and track how the total value changes.

Odd/even parity check: a sum of two odd numbers is even, odd + even is odd, and a product is odd only when all factors are odd.

Check reasonableness: estimate first with rounded numbers, then compare the exact answer to the estimate before writing it down.

Chicken-and-rabbit (two-type) problems: if all were type A, the count difference from the actual total, divided by the per-swap difference, gives the number of type B.

Age problems: the age difference between two people never changes; express both ages in terms of one variable and use the fixed difference.

Area and perimeter: for a fixed perimeter, the rectangle closest to a square has the largest area; for fixed area, the square has the least perimeter.

Digit puzzles: use place value, e.g. a two-digit number ab equals 10a + b, so reversing digits changes the value by 9(a − b).

Sum of an arithmetic sequence = (first + last) × number of terms ÷ 2; count terms as (last − first) ÷ step + 1.

Distribution with remainder: total = divisor × quotient + remainder, with 0 ≤ remainder < divisor; test divisors against the stated remainder.

Tree-planting on a line: number of trees = number of gaps + 1 when both ends are planted, and gaps = length ÷ spacing.

Overlapping sets: total = A + B − both + neither; drawing two overlapping circles prevents double counting.

Work problems: rate = 1 ÷ time for each worker; combined rate is the sum of individual rates, and combined time = 1 ÷ combined rate.

Fraction of a fraction: multiply numerators and denominators; 'of' signals multiplication, so 2/3 of 3/4 = 1/2.

Unit conversion in rate problems: keep units attached and cancel them, e.g. km/h × h = km.

Clock problems: the minute hand moves 6° per minute, the hour hand 0.5° per minute, so the hands separate at 5.5° per minute.

Average = total ÷ count; the total is recovered as average × count, which is the key step in 'changed average' puzzles.

Consecutive-integer sums: for n consecutive integers the average is the middle term, so sum = n × middle.

Counting paths on a grid: the number of routes equals the sum of routes to the cell above and to the left (Pascal-style addition).

Invariant reasoning: identify what stays constant (total, difference, product) while other quantities change, and solve from that constant.

Percent change: new = original × (1 ± rate); successive changes multiply, so +20% then −20% gives ×0.96, a net 4% loss.

Ratio sharing: convert ratios to parts; one part = total ÷ sum of ratio terms, then multiply by each term.

Mixture problems: track the amount of the pure substance, not the total volume; pure amount = concentration × volume.

Simple interest I = P × r × t; compound amount A = P(1 + r)^t, with r as a decimal per period.

Speed with two legs: average speed = total distance ÷ total time, never the average of the two speeds unless times are equal.

Divisibility rules: by 3 or 9 use the digit sum; by 4 use the last two digits; by 8 use the last three digits; by 11 alternate the digit sums.

Pigeonhole principle: with n boxes, placing n + 1 items guarantees some box holds at least two items.

Counting with restrictions: count all arrangements, then subtract the forbidden ones, or fix the restricted position first and multiply choices.

Linear equations: isolate the variable by applying the same operation to both sides; check by substituting back into the original equation.

Systems of two equations: substitution when one variable is already isolated, elimination when coefficients are opposites or easily matched.

Absolute value equations |x − a| = d give two solutions x = a + d and x = a − d when d > 0, and none when d < 0.

Exponent rules: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 for a ≠ 0.

Slope = (y₂ − y₁) ÷ (x₂ − x₁); parallel lines share a slope, perpendicular slopes multiply to −1.

Triangle angle facts: interior angles sum to 180°, exterior angle equals the sum of the two remote interior angles.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.

Counting with permutations and combinations: arrangements of n distinct objects = n!; choosing r from n ignores order and equals n! ÷ (r!(n − r)!).

Quadratic factoring: for x² + bx + c find two numbers with product c and sum b, then write (x + p)(x + q).

Quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a; the discriminant b² − 4ac gives two, one, or no real roots.

Pythagorean theorem a² + b² = c² for right triangles; common triples are 3-4-5, 5-12-13, 8-15-17, and their multiples.

Distance between points = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Similar figures: corresponding sides are proportional and areas scale by the square of the scale factor.

Function notation: f(a) means substitute a for x; the domain excludes values making a denominator zero or a square root negative.

Arithmetic vs geometric sequences: arithmetic adds a constant d, geometric multiplies by a constant r; nth term is a + (n−1)d or a·r^(n−1).

Three-dimensional counting: volume = base area × height; surface area sums face areas, and cube-painting problems count faces by position (corner, edge, face, interior).