Bariz - Problem

Source: St. Petersburg MO 2017 Grade 9 P1

Tags: algebra

ThE-dArK-lOrD

03.05.2018 13:56

Sasha’s computer can do the following two operations: If you load the card with number $a$, it will return that card back and also prints another card with number $a+1$, and if you consecutively load the cards with numbers $a$ and $b$, it will return them back and also prints cards with all the roots of the quadratic trinomial $x^2+ax+b$ (possibly one, two, or none cards.) Initially, Sasha had only one card with number $s$. Is it true that, for any $s> 0$, Sasha can get a card with number $\sqrt{s}$?

Here is my solution. Claim We claim that for any $s$, $\sqrt{s}$ can be achieved. Proof: For any input $\alpha$, we denote by $\gamma(\alpha)$ to be the output. We recommend Sasha to follow the described procedure. 1. Input $s$ and obtain $\gamma(s) = s , s+1$. 2, Consider the equation $x^2+ (s+1)x +s$. The equation has roots $-1$ and $-s$.Thus the next step clearly is $\gamma(s+1,s) \longrightarrow ( s+1, s, -1,-s)$. 3. Input $-1$ and collect $\gamma(-1) = (-1$,$0$$)$ 4. This is the last step: consider $\gamma( 0,-s) \longrightarrow ( 0,-s,\sqrt{s}, -\sqrt{s})$.