Question 98
A finite collection $C$ of (not necessarily distinct) real numbers is *suitable* if it contains two numbers $a$ and $b$ such that $a+b \neq s+1$, where $s$ is the sum of all numbers in $C$; such numbers $a$ and $b$ form an *eligible* pair.
Fix an integer $n \ge 2$. A number of $n$ pairwise distinct real numbers are written on a board. A step consists in choosing an eligible pair $(a, b)$ amongst numbers on the board (if any), crossing them out and replacing them by the number
$$
\frac{(a+b)(s+1) - a^2 - ab - b^2}{s-a-b+1}.
$$
a) Prove that there is a sequence of $n-1$ steps such that at each stage the numbers on the board form a suitable collection; and
b) Determine the final number on the board in terms of the initial numbers.
Fix an integer $n \ge 2$. A number of $n$ pairwise distinct real numbers are written on a board. A step consists in choosing an eligible pair $(a, b)$ amongst numbers on the board (if any), crossing them out and replacing them by the number
$$
\frac{(a+b)(s+1) - a^2 - ab - b^2}{s-a-b+1}.
$$
a) Prove that there is a sequence of $n-1$ steps such that at each stage the numbers on the board form a suitable collection; and
b) Determine the final number on the board in terms of the initial numbers.