Problems 10-19

Nic McPhee
University of Minnesota, Morris

[Problem 10] [Problem 11] [Problem 12] [Problem 13] [Problem 14] [Problem 15] [Problem 16] [Problem 17] [Problem 18] [Problem 19]

Problem 10

The multiplication of two complex numbers

(a + b * i) * (x + y * i) = a*x - b*y + (b*x + a*y) * i

appears to require 4 multiplications (a*x, b*y, b*x, a*y). If additions are free, however, can this be accomplished using just 3 multiplications? Just 2?

Problem 11

A microbe either disintigrates or splits into two perfect copies of itself. If the probability of splitting is p, what is the probability that one microbe will produce a colony that lasts forever?

Problem 12

Assume we have n distinct points on the plane. Show that there exist three points determining an angle a such that 0 <= a <= PI/n.

Problem 13

Devise an experiment that only uses tosses of a fair coin, but which has probability of success of 1/3. Do the same for any probability of success p, where 0 <= p <= 1.

Problem 14

A certain traffic light is programmed to be green for 30 seconds, and then red for 30 seconds. How much time, on average, will a vehicle wait at this light?

Problem 15

If the sum of a set of positive integers is n, what is the biggest its product can be?

Problem 16

Assuming friendship is symmetric, prove that at any party there are two people having the same number of friends present.

Problem 17

Find the positive value of x that maximizes

2-x + 2-1/x

Problem 18

Show that N distinct, non-collinear points determine at least N lines.

Problem 19

Find a set of positive numbers that sum to 1000 and have maximum product.

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