# 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\ | + | The real numbers <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are such that <math>a \geq b \geq c \geq d > 0</math> and <math>a + b + c + d = 1</math>. Prove that<cmath>(a + 2b + 3c + 4d) a^a b^b c^c d^d < 1.</cmath> |

− | Prove that | ||

− | < | ||

== Video solution == | == Video solution == |

## Revision as of 11:24, 14 May 2021

## Problem

The real numbers , , , are such that and . Prove that

## Video solution

https://youtu.be/bDHtM1wijbY [Video covers all day 1 problems]

## 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

## Video solution

https://youtu.be/bDHtM1wijbY [Video covers all day 1 problems]