22nd Brasil 2000


A1. A piece of paper has top edge AD. A line L from A to the bottom edge makes an angle x with the line AD. We want to trisect x. We take B and C on the vertical ege through A such that AB = BC. We then fold the paper so that C goes to a point C' on the line L and A goes to a point A' on the horizontal line through B. The fold takes B to B'. Show that AA' and AB' are the required trisectors.
A2. Let s(n) be the sum of all positive divisors of n, so s(6) = 12. We say n is almost perfect if s(n) = 2n - 1. Let mod(n, k) denote the residue of n modulo k (in other words, the remainder of dividing n by k). Put t(n) = mod(n, 1) + mod(n, 2) + ... + mod(n, n). Show that n is almost perfect iff t(n) = t(n-1).
A3. Define f on the positive integers by f(n) = k² + k + 1, where 2^k is the highest power of 2 dividing n. Find the smallest n such that f(1) + f(2) + ... + f(n) ≥ 123456.
B1. An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for 3/2 minutes and red for 1 minute. For which v can a car travel at a constant speed of v m/s without ever going through a red light?
B2. X is the set of all sequences a₁, a₂, ... , a₂₀₀₀ such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The distance between two members a and b of X is defined as the number of i for which a_i and b_i are unequal. Find the number of functions f : X → X which preserve distance.
B3. C is a wooden cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?

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To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.