Question 78
Let $n \geqslant 1$ be an integer. Bosphore has written the number $2$ on the board $n$ times. He then performs, $n-1$ times in a row, the following operation: he chooses two numbers written on the board, which he calls $a$ and $b$, then erases them and writes the number $\sqrt{(ab+1)/2}$ in their place. Finally, he calls $x$ the number written on the board after these $n-1$ operations, and $y$ the number $\sqrt{(n+3)/n}$.
a) Prove that $x \geqslant y$.
b) Prove that there are infinitely many integers $n \geqslant 1$ for which we necessarily have $x \gt y$.
c) Prove that there are infinitely many integers $n \geqslant 1$ for which Bosphore can make sure that $x=y$.
a) Prove that $x \geqslant y$.
b) Prove that there are infinitely many integers $n \geqslant 1$ for which we necessarily have $x \gt y$.
c) Prove that there are infinitely many integers $n \geqslant 1$ for which Bosphore can make sure that $x=y$.