# Difference between revisions of "2020 IMO Problems/Problem 2"

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− | Problem | + | ==Problem== |

+ | The real numbers <math>a, b, c, d</math> are such that <math>a\ge b \ge c\ge d > 0</math> and <math>a+b+c+d=1</math>. | ||

Prove that | Prove that | ||

<math>(a+2b+3c+4d)a^a b^bc^cd^d<1</math> | <math>(a+2b+3c+4d)a^a b^bc^cd^d<1</math> |

## Revision as of 10:56, 20 October 2020

## Problem

The real numbers are such that and . Prove that

## Solution

Using Weighted AM-GM we get

So,

Now notice that

So, we get

Now, for equality we must have

In that case we get

~ftheftics