SOF IMO flashcards

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

Divisibility by 9

A number is divisible by 9 exactly when the sum of its digits is divisible by 9; repeatedly summing digits gives the digital root, which equals the remainder modulo 9.

Divisibility by 4

A number is divisible by 4 exactly when its last two digits form a number divisible by 4, since 100 is a multiple of 4 and higher place values vanish modulo 4.

Divisibility by 11

A number is divisible by 11 when the alternating sum of its digits (first minus second plus third, and so on) is 0 or a multiple of 11, including negatives.

Sum of first n integers

The sum 1 + 2 + … + n equals n(n+1)/2. This closed form underlies counting pairs, handshakes, and many series problems in the reasoning section.

Sum of first n odd numbers

The sum 1 + 3 + 5 + … + (2n−1) equals n², because the nth square is formed by adding the nth odd number to the previous square.

Sum of first n squares

The sum 1² + 2² + … + n² equals n(n+1)(2n+1)/6. It appears in problems counting stacked objects or lattice points in triangular arrays.

Remainder theorem

When a polynomial P(x) is divided by (x − a), the remainder equals P(a). This converts division problems into simple substitution and sign checks.

Trailing zeros of n factorial

The count of trailing zeros in n! equals floor(n/5) + floor(n/25) + floor(n/125) + …, because each factor 5 paired with a factor 2 creates one zero.

Permutation formula

The number of ordered arrangements of r objects chosen from n distinct objects is nPr = n!/(n−r)!. Order matters, so AB and BA count as different arrangements.

Combination formula

The number of unordered selections of r objects from n distinct objects is nCr = n!/(r!(n−r)!). Order does not matter, so AB and BA count as one selection.

Multiset permutation count

If n objects contain p identical of one kind, q of another, and so on, the number of distinct arrangements is n! divided by the product of the repeated factorials.

Circular arrangement count

The number of distinct ways to seat n distinct objects around a circle is (n−1)!, because rotations are considered identical and one position can be fixed.

Inclusion-exclusion principle

For two sets, |A ∪ B| = |A| + |B| − |A ∩ B|. For three sets, subtract pairwise intersections and add back the triple intersection to avoid double counting.

Pigeonhole principle

If more than k·n objects are placed into n boxes, at least one box contains k+1 or more objects. It proves existence without constructing an explicit example.

Handshake count

In a group of n people where each pair shakes hands once, the total number of handshakes is n(n−1)/2, the number of unordered pairs among n items.

Ratio cross-multiplication

A proportion a:b = c:d holds exactly when ad = bc. Cross-multiplication converts ratio equations into linear equations that are easy to solve.

Partitive division of totals

To split a total T in the ratio a:b:c, the shares are T·a/(a+b+c), T·b/(a+b+c), and T·c/(a+b+c); each share equals T times its fraction of the whole.

Average speed formula

For equal distances at speeds x and y, the average speed is 2xy/(x+y), the harmonic mean, not the arithmetic mean (x+y)/2; the arithmetic mean overstates the result.

Work rate addition

If one worker finishes a job in a days and another in b days, together they finish it in ab/(a+b) days, since their rates 1/a and 1/b add.

Clock angle formula

At h hours and m minutes, the angle between the hour and minute hands is |30h − 5.5m| degrees; if the result exceeds 180, subtract it from 360.

Relative speed

When two objects move toward each other, their closing speed is the sum of their speeds; when moving in the same direction, it is the difference of their speeds.

Simple interest formula

Simple interest equals principal × rate × time (I = PRT/100 when the rate is a percentage). Unlike compound interest, it is computed only on the original principal.

Compound interest formula

The amount after t years at rate r compounded annually is P(1 + r)^t. For n compoundings per year, the amount is P(1 + r/n)^(nt).

Profit and loss percentage

Profit percent equals (selling price − cost price)/cost price × 100. Loss percent uses the same denominator, cost price, never the selling price.

Successive percentage change

Two successive changes of a% and b% produce a net change of a + b + ab/100 percent. The cross term matters: a 10% rise then a 10% fall nets a 1% decrease.

Number of subsets

A set with n elements has 2ⁿ subsets in total, of which 2ⁿ − 1 are non-empty. The empty set is always counted as a subset.

Logical converse and contrapositive

For the statement 'if P then Q', the converse is 'if Q then P' (not equivalent), while the contrapositive 'if not Q then not P' is logically equivalent to the original.

Syllogism validity

A categorical syllogism is valid only if the conclusion follows necessarily from the premises; conclusions that merely seem plausible or typical are not logically forced.

Venn diagram two-set regions

Two overlapping sets split the universe into four regions: only A, only B, both, and neither. Assigning counts to these regions solves most set-counting problems.

Symmetric difference count

The number of elements in exactly one of two sets is |A| + |B| − 2|A ∩ B|, obtained by subtracting the overlap twice from the union total.

Odd and even parity rules

Sum or difference of two integers is even exactly when both have the same parity; the product is even if either factor is even. Parity arguments rule out impossible outcomes.

Unit digit cycles

Powers of 2, 3, 7, and 8 repeat unit digits in cycles of 4; powers of 4 and 9 cycle in 2; powers of 5 and 6 always end in 5 and 6.