MathDB

4

Part of 2007 Canada National Olympiad

Problems(1)

Numbers

Source: Canada Mathematical Olympiad 2007

8/1/2007
For two real numbers a a, b b, with ab≠1 ab\neq 1, define the ∗ \ast operation by a∗b=a+b−2ab1−ab. a\ast b=\frac{a+b-2ab}{1-ab}. Start with a list of n≥2 n\geq 2 real numbers whose entries x x all satisfy 0<x<1 0<x<1. Select any two numbers a a and b b in the list; remove them and put the number a∗b a\ast b at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. a. a. Prove that this single number is the same regardless of the choice of pair at each stage. b. b. Suppose that the condition on the numbers x x is weakened to 0<x≤1 0<x\leq 1. What happens if the list contains exactly one 1 1?
algebrapolynomialratiotrigonometryinvariantinductioncombinatorics unsolved