AMO flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Working backwards

A strategy that starts from the stated final result and undoes each operation in reverse order, using the inverse operation, until the unknown starting value is found.

Guess and check table

A table listing each trial guess, the resulting value, and the difference from the target, so the next guess can be adjusted in the correct direction.

Parity

Whether a number is even or odd. Parity rules let you rule out impossible answers, since odd plus odd is even and any product with an even factor is even.

Pattern step

The constant amount added from one term of a sequence to the next; term n equals the first term plus (n − 1) times the step.

Two-type count problem

A puzzle with two kinds of items sharing a total count and total value; assuming one type throughout and dividing the value gap by the per-swap difference gives the other count.

Age difference invariant

The gap between two people's ages never changes over time, so expressing both ages from one variable and using that fixed gap solves age puzzles.

Arithmetic series sum

The sum of evenly spaced terms equals the first plus the last, multiplied by the number of terms, divided by two.

Division algorithm form

Every division can be written as total equals divisor times quotient plus remainder, where the remainder is at least zero and less than the divisor.

Fixed-perimeter rectangle

Among rectangles with the same perimeter, the one closest to a square encloses the greatest area; a long thin rectangle encloses the least.

Two-set overlap

When two groups overlap, the total equals the first group plus the second group minus the overlap, plus anyone in neither group.

Combined work rate

Each worker's rate is one divided by their solo time; adding rates gives the team rate, and the team time is one divided by that sum.

Clock hand speed

The minute hand sweeps six degrees per minute and the hour hand half a degree per minute, so the gap between them changes by 5.5 degrees per minute.

Average recovery

Since average equals total divided by count, the total can be recovered by multiplying the average by the count, which solves changed-average puzzles.

Counting grid paths

On a grid with only right and up moves, the number of routes to a cell equals the sum of the routes to the cell to its left and the cell below.

Invariant

A quantity that stays the same while other values change, such as a total or a difference; identifying it usually unlocks the solution.

Successive percent change

Percent changes applied one after another multiply rather than add, so a 20 percent increase followed by a 20 percent decrease leaves 96 percent of the original.

Ratio parts method

Convert a ratio into equal parts; one part equals the total divided by the sum of the ratio terms, and each share is that part times its term.

Mixture concentration

The amount of pure substance equals concentration times volume; in mixture problems track the pure amount, which is conserved when solutions are combined.

Average speed

Total distance divided by total time for the whole trip. It equals the plain average of two leg speeds only when both legs take equal time.

Divisibility by 9

A number is divisible by 9 exactly when the sum of its digits is divisible by 9; the same digit-sum test with 3 checks divisibility by 3.

Pigeonhole principle

If more items than boxes are filled, at least one box must hold two or more items; with n boxes, n + 1 items guarantee a repeat.

Elimination method

Solving a system by adding or subtracting the equations, after scaling if needed, so one variable cancels and the other can be found.

Absolute value equation

An equation of the form the absolute value of x minus a equals d has two solutions, a plus d and a minus d, when d is positive.

Perpendicular slopes

Two nonvertical lines are perpendicular exactly when their slopes multiply to negative one, so each slope is the negative reciprocal of the other.

Exterior angle theorem

An exterior angle of a triangle equals the sum of the two interior angles that are not adjacent to it.

Inequality sign flip

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

Discriminant

The value b squared minus 4ac in a quadratic equation; positive gives two real roots, zero gives one, and negative gives no real roots.

Pythagorean triples

Whole-number right triangle sides such as 3-4-5, 5-12-13, and 8-15-17, along with every whole-number multiple of these sets.

Similar figure scaling

When two figures are similar with scale factor k, corresponding lengths scale by k while areas scale by k squared and volumes by k cubed.

Geometric sequence

A sequence in which each term is the previous term multiplied by a fixed ratio r; the nth term is the first term times r to the n minus one.

Distance formula

The distance between two points equals the square root of the sum of the squared horizontal difference and the squared vertical difference.

Painted cube counting

In a cube painted on all faces and cut into small cubes, corners show three painted faces, edge pieces two, face centers one, and interior cubes none.