Mandelbrot Team Round Topics

Round One topics: parity, Pigeonhole Principle, standard geometric shapes such as a regular pentagon

We know that integers may be classified as either odd or even, that the sum of

two odd numbers is an even number, and other related facts. We could instead split all integers

into three categories: the threeven numbers (multiples of three), the oddup numbers (one more

than a multiple of three), and the oddown numbers (one less than a multiple of three). Thus 9

is threeven, 19 is oddup, and 29 is oddown. The sum of two oddup numbers gives an oddown

number, the sum of an oddup and an oddown number is threeven, and so on.

Given five integers, three of them must have the same parity, either all odd or all even. For

if only two (or less) of them were odd and just two (or less) of them were even, then we would

have at most four numbers. This is an application of the Pigeonhole Principle.

Round Two topics: probability, standard deck of cards, optimal strategy in a game

Weighted probabilities are computed as illustrated in the following example.

Cecil goes grocery shopping on Mondays with probability ⁴/₅ if it is sunny, but with probability

⅓ if it rains. This Monday there is a 75% chance of showers. To compute the likelihood that

Cecil goes shopping, we write ¼ * ⁴/₅ + ¾ * ⅓ = ⁹/₂₀ ; the probability of sunshine times the probability of shopping given sunshine, plus a similar expression to account for the possibility of rain.

Round Three topics: iteration, floor function, sequences, patterns, working with fractions, (induction)

Let ƒ(x) be a function. We may iterate this function beginning with some

initial value by plugging the initial value into ƒ(x), recording the result, then plugging the

result into ƒ(x), recording the new result, etc. For example, if ƒ(x) = x²-8 and we begin with

initial value 3, we obtain ƒ(3) = 1, then ƒ(1) = -7, then ƒ(-7) = 41, and so on.

The expression  (referred to as floor of x) denotes the greatest integer not larger than x.

Thus  = 3,  = 6, = 0, and  = 11.

Round Four topics: Law of Sines/Cosines, trigonometric identities, adding fractions with variables, angles

The Law of Cosines states that c² = a² + b² - 2ab*cos(C), and similarly for the

other angles. On the other hand, the Law of Sines asserts that sin(A)/a = sin(B)/b = sin(C)/c.

There are many other useful trig identities, including the angle addition and subtraction rules,

such as cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y), double angle formulas, such as sin(2x) = 2*sin(x)*cos(x), and relationships like cos(x - 90º) = sin(x) or sin(x - 90º) = -cos(x).