Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.

Question

For two positive integer sequences $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_m$ satisfying

• $x_i\neq x_j\quad \text{and}\quad y_i\neq y_j\quad \forall i,j, i \ne j$

• $1<x_1<x_2<\cdots<x_n<y_1<\cdots<y_m.$

• $x_1+x_2+\cdots+x_n>y_1+\cdots+y_m.$

Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.

(from internet)

I don't have an idea for this problem. Thanks for your help.

Answer 1

Here's my approach.

Prove the following version instead.

For two positive integer $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_m$ which satisfy

1. $ x_i \neq x_j\quad \text{and}\quad y_i\neq y_j\quad \forall 1 < i < j $,

2. $ 1<x_2<...<x_n<y_2<...<y_m, $ and $ 1 \leq x_1 $ and $ 1 \leq y_1 $

3. $x_1+x_2+ \ldots +x_n > y_1+ \ldots +y_m.$

Prove that: $ x_1 \times x_2 \times \cdots \times x_n \geq y_1 \times y_2 \times \cdots y_m$.

This version is much easier to work with. We then prove strict inequality by looking at the equality cases.

Hint: Think about what $ x_1, y_1 $ could be made to do.
Hint: How would you minimize the LHS and maximize the RHS?
Hint: Deal with $m=1 $ separately, which results in the equality case. The original version for $m=1$ is straightforward.