AMO key terms

A quantity or property that stays the same while other things change during a process. Finding one often proves that a certain outcome is impossible or that two configurations must be equivalent.

Whether a whole number is even or odd. Parity reasoning solves problems about pairing, flipping, alternating states, and whether a total can reach a target value.

A quantity that always increases or always decreases with each step of a process. Because it cannot change direction, it limits how many steps are possible.

If more items are placed into fewer containers, at least one container holds two or more items. It proves existence without naming which container or which pair.

A proof strategy that examines the largest, smallest, first, or last case. Choosing an extreme object often forces a contradiction or reveals structure that arbitrary choices hide.

Assume the claim is false, then derive a logical impossibility. Since the assumption fails, the original claim must be true. Common in problems asking to show something cannot happen.

A single valid case that makes a general statement false. One counterexample disproves a claim completely, while many confirming examples never prove it.

A pattern-based guess that has not been proven. In contest work, a conjecture must be tested on edge cases and then justified with a general argument.

Extending a specific observation to a broader rule covering all cases. A valid generalization needs justification, not just agreement with several examples.

A restricted version of a problem, such as using small numbers or symmetric shapes. Solving special cases builds intuition and can expose the general pattern.

Organizing all possible cases in a fixed order so none are missed and none repeat. Often paired with a table to count outcomes reliably.

Splitting a problem into separate situations that together cover every possibility, then solving each. Categories must be mutually exclusive and exhaustive.

Starting from the final result and undoing each operation in reverse order. Useful when the problem states the end value and asks for the starting value.

The operation that cancels another, such as addition with subtraction or multiplication with division. Undoing steps in reverse order is the basis of solving equations.

A branching picture listing choices at each stage. Multiplying along a branch and adding across branches counts outcomes and supports probability reasoning.

If one choice has m options and a second independent choice has n options, the pair has m times n options. It extends to any number of stages.

An arrangement where order matters. The number of orderings of n distinct objects is n factorial, written n!, the product of all whole numbers from 1 to n.

A selection where order does not matter. Counting combinations requires dividing the number of ordered arrangements by the number of ways to order each selection.

Counting the same object more than once. Fix it by dividing by the number of duplicate counts or by choosing a canonical way to represent each object once.

A one-to-one matching between two sets, where every element of each set is paired with exactly one element of the other. A bijection proves the sets have equal size.

Assuming the least helpful arrangement to find a guaranteed bound. It answers questions of the form: how many must be taken so that a certain property is certain?

A classic setting where each pair of people shakes hands once. With n people the total is n(n-1)/2, because each handshake is counted twice.

A problem combining work, speed, or flow over time, often with two agents acting together. Combined rates add, while combined times do not.

A rate expressed per one unit, found by dividing. Unit rates make comparisons between different-sized groups straightforward and are the basis of proportional reasoning.

An equation stating that two ratios are equal. Solving a proportion often uses cross multiplication, which works because multiplying both sides clears the denominators.

A comparison of two quantities by division, written as a fraction or with a colon. Ratios scale by multiplying both parts by the same nonzero number.

The amount left after dividing a whole number by another. Remainders turn repeating patterns into problems about cycles and positions within a cycle.

Arithmetic concerned only with remainders after division by a fixed number. Two numbers are congruent modulo n when their difference is a multiple of n.

The total obtained by adding the digits of a number. Digit sums reveal divisibility by 3 and 9, and they change predictably when numbers are combined.

A quick test for whether one whole number divides another without full division. Examples include last-digit tests for 2 and 5 and digit-sum tests for 3 and 9.

The largest whole number dividing two or more given numbers exactly. It governs how quantities can be split into equal groups with nothing left over.

The smallest nonzero whole number that is a multiple of two or more given numbers. It answers when repeating events will next coincide.

Writing a whole number greater than 1 as a product of primes. The result is unique apart from order, which makes it a powerful tool in divisor problems.

A rule generating successive terms, whether by adding a constant, multiplying, or combining earlier terms. Identifying the rule lets you predict terms far ahead.

A table listing successive differences between terms of a sequence. Repeated difference rows expose polynomial-style patterns and predict the next term.

A property where a figure or expression looks unchanged after a reflection, rotation, or swap. Symmetry reduces casework because symmetric cases behave identically.

Covering a region completely with non-overlapping pieces. Tiling problems often use area, coloring, or parity arguments to prove a covering is impossible.

Assigning colors to cells or positions so that each allowed piece covers a fixed color pattern. Comparing color counts then proves a covering cannot exist.

The amount of surface a two-dimensional figure covers, measured in square units. Area comparisons often settle geometric puzzles that resist direct measurement.

The total distance around the outside of a figure. Figures with equal area can have very different perimeters, which is a frequent source of trick questions.