# 2020 INMO Problems/Problem 4

## Problem

Let be an integer and let be real numbers such that . Prove that

## Solution(1)

For , we want to show that where and . This is equivalent to showing that , which is true.

Suppose, now, that the given inequality is true for , where . Now, consider reals with sum . Then, and , so by induction hypothesis,

This means or as desired. ~biomathematics

## Solution(2)

Define,. ,

In general ,.

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Using Tchevbycev inequality we have , .

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.[Applying Induction on successive ].

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.[using GM-AM]

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Since, and , Hence , .

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The RHS inequality is trivial by AM-GM inequality.

For LHS inequality I would like to use induction.

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We have , and .

. Suppose , the statement is true for such that , and .

Now , consider .

Suppose , is median of the sequence, and and .

and .

Our induction step is complete.

This two claim leads and equality for . ~trishan