SOF IMO key terms

The property that one whole number can be divided by another with no remainder. Tests based on digit sums, last digits, and alternating sums let you check this quickly without full division.

A whole number greater than 1 whose only positive divisors are 1 and itself. Primes are the building blocks of all integers through factorisation, and recognising them helps in counting and number-theory puzzles.

A whole number greater than 1 that has at least one positive divisor other than 1 and itself. Every composite number can be written as a product of primes in exactly one way apart from order.

Writing a number as a product of prime numbers, for example 84 = 2 x 2 x 3 x 7. It reveals the divisors of the number and simplifies work with fractions, roots, and greatest common divisors.

The largest positive integer that divides each of two or more given integers without a remainder. It is found by comparing prime factorisations or by repeated subtraction and is central to simplifying ratios and fractions.

The smallest positive integer that is a multiple of each of two or more given integers. It equals the product of the numbers divided by their greatest common divisor and is used in problems about repeating events.

Two integers are coprime when their only common positive divisor is 1. Coprime pairs behave like independent cycles when combining repeating patterns, since their least common multiple is simply their product.

The amount left over after dividing one integer by another when the division is not exact. Writing a number as quotient times divisor plus remainder underpins modular reasoning and many digit and cycle problems.

A way of calculating with remainders, where numbers differing by a multiple of a fixed modulus are treated as equal. It simplifies problems about cycles, digit patterns, and large powers ending in particular digits.

Whether an integer is even or odd. Parity arguments quickly rule out impossible configurations, because sums and products of even and odd numbers follow fixed rules that must be preserved in any valid arrangement.

The total obtained by adding the digits of a number. Digit sums give a fast test for divisibility by 3 and 9 and are used in puzzles about rearranging digits while keeping a total fixed.

A comparison of two quantities by division, showing how many times one contains the other. Ratios let you share a total into proportional parts and scale recipes, maps, or mixtures up and down.

A statement that two ratios are equal. In a direct proportion both quantities grow together; in an inverse proportion one grows as the other shrinks. Setting up proportions solves many rate and scaling problems.

A fraction expressed out of 100, written with the percent sign. Percentage change compares a new amount with an original base, and successive percentage changes do not simply add because each applies to a new base.

The sum of a set of values divided by how many values there are. The mean balances deviations above and below it, and knowing the total and count lets you recover it in reverse problems.

A mean in which each value counts in proportion to an assigned weight rather than equally. It is used when groups of different sizes are combined and the overall mean must be found correctly.

The middle value of a set of numbers arranged in order, or the mean of the two middle values when the count is even. It is less affected by extreme values than the mean.

The difference between the largest and smallest values in a data set. It measures how spread out the data are and is often paired with a central value when summarising a list of numbers.

An ordered list of numbers or objects following a rule, indexed by position. Identifying the rule lets you predict later terms, and common types include arithmetic, geometric, and square-number sequences.

A sequence in which each term is obtained by adding a fixed constant, the common difference, to the previous term. Its nth term and total sum have simple formulas depending on the first term and difference.

A sequence in which each term is obtained by multiplying the previous term by a fixed constant, the common ratio. Such sequences model doubling, halving, and growth patterns and appear in ratio and interest problems.

The sum of the terms of a sequence. Recognising a series as arithmetic or geometric lets you total many terms quickly using a formula instead of adding them one by one.

An arrangement of objects in which order matters. The number of orderings of n distinct objects is n factorial, and partial selections are counted with products of decreasing integers.

A selection of objects in which order does not matter. Combinations count groups such as teams or subsets, and they are smaller than permutations of the same size whenever more than one object is chosen.

The product of all whole numbers from 1 up to a given number, written with an exclamation mark, with 0 factorial defined as 1. Factorials count orderings and appear in permutation and combination formulas.

If more objects than containers are distributed among the containers, at least one container must hold more than one object. It proves existence of repeated values or collisions without constructing them explicitly.

A quantity or property that does not change when an allowed operation or move is performed. Finding an invariant proves that certain end states can never be reached from a given starting position.

A proof strategy that examines the largest, smallest, or otherwise most extreme case in a configuration. Assuming such an extreme object exists often forces a contradiction or reveals the structure needed.

Reaching a conclusion by applying valid rules of inference to given statements, so that the conclusion must be true whenever the premises are true. It underlies multi-step puzzles and truth-teller problems.

The statement formed by swapping and negating both parts of a conditional, so that if A then B becomes if not B then not A. A conditional and its contrapositive are logically equivalent.

The statement obtained by swapping the two parts of a conditional, so that if A then B becomes if B then A. A converse is not automatically true when the original statement is true.

A method that assumes the opposite of what is to be proved and derives an impossible consequence, showing the assumption must be false. It is widely used to prove irrationality and infinitude results.

A condition is necessary when it must hold for a statement to be true, and sufficient when its holding guarantees the statement. Saying both means the condition is exactly equivalent to the statement.

For any triangle, the sum of the lengths of any two sides exceeds the length of the third side. It decides whether three given lengths can form a triangle and bounds distances in optimisation problems.

Two figures are congruent when they have exactly the same shape and size, so one can be moved onto the other by translation, rotation, or reflection. Corresponding sides and angles are then equal.

Two figures are similar when they have the same shape, with corresponding angles equal and corresponding lengths in a fixed ratio. Areas of similar figures compare as the square of that ratio.

The total of the interior angles of a polygon with n sides, equal to (n minus 2) times 180 degrees. It follows from splitting the polygon into triangles drawn from a single vertex.

A property of a figure that looks unchanged after a reflection, rotation, or translation. Counting lines of symmetry and rotational positions helps in tiling, folding, and pattern-completion problems.

A ratio comparing two quantities with different units, such as distance per unit time. Combining rates requires care: times add when the same distance is covered in stages, while rates do not simply average.