Question 133
a) Let $x_{1} \geqslant x_{2}\gt 0$ and $y_{1} \geqslant y_{2}\gt 0$ be real numbers such that
$$
x_{1} \geqslant y_{1} \text{ and } x_{1} x_{2} \geqslant y_{1} y_{2}
$$
Prove that
$$
x_{1}+x_{2} \geqslant y_{1}+y_{2}
$$
b) Let $x_{1} \geqslant x_{2} \geqslant \ldots \geqslant x_{n}\gt 0$ and $y_{1} \geqslant y_{2} \geqslant \ldots \geqslant y_{n}\gt 0$ be real numbers such that
$$
x_{1} x_{2} \cdots x_{i} \geqslant y_{1} y_{2} \cdots y_{i} \text{ for } i=1, \ldots, n
$$
Prove that
$$
x_{1}+x_{2}+\cdots+x_{n} \geqslant y_{1}+y_{2}+\cdots+y_{n}
$$
$$
x_{1} \geqslant y_{1} \text{ and } x_{1} x_{2} \geqslant y_{1} y_{2}
$$
Prove that
$$
x_{1}+x_{2} \geqslant y_{1}+y_{2}
$$
b) Let $x_{1} \geqslant x_{2} \geqslant \ldots \geqslant x_{n}\gt 0$ and $y_{1} \geqslant y_{2} \geqslant \ldots \geqslant y_{n}\gt 0$ be real numbers such that
$$
x_{1} x_{2} \cdots x_{i} \geqslant y_{1} y_{2} \cdots y_{i} \text{ for } i=1, \ldots, n
$$
Prove that
$$
x_{1}+x_{2}+\cdots+x_{n} \geqslant y_{1}+y_{2}+\cdots+y_{n}
$$