Smoothing An Inequality

I've come across the following question on the internet:

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

$x_i < x_j$ and $y_i < y_j, \forall i < j$ ,
$1<x_1<x_2<...<x_n<y_1<...<y_m,$
$x_1+x_2+ \ldots +x_n > y_1+ \ldots +y_m.$

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

#Algebra #Smoothing

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

can we proove this by taking log in the proove part and as log is an increasing function the following can be true ?

Ashish Nagpal - 7 years ago

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

$x_i < x_j$ and $y_i < y_j, \forall 1 < i < j$ ,
$1<x_2<...<x_n<y_2<...<y_m,$ and $1 \leq x_1$ and $1 \leq y_1$
$x_1+x_2+ \ldots +x_n > y_1+ \ldots +y_m.$

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. Why would I want to make them special?
Hint: How would you minimize the LHS and maximize the RHS? Try smoothing.
Hint: Deal with $m=1$ separately. In particular, it leads to the only equality cases. Proving $m=1$ in the original question is straightforward.

Calvin Lin Staff - 7 years ago

You can use Abel summation techniques. :)

A Brilliant Member - 7 years ago

Interesting. Can you elaborate?

Calvin Lin Staff - 7 years ago

i think the condition n>m must be mentioned if all are positive integers greater than 1 .. !

Ramesh Goenka - 7 years ago

That can be deduced from the conditions. In fact, I believe we have $m < n < 2m$ .

Calvin Lin Staff - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$