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

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Now notice that | Now notice that | ||

<cmath>a+2b+3c+4d \text{ will be less then the following expressions (and the reason is written to the right)} </cmath> | <cmath>a+2b+3c+4d \text{ will be less then the following expressions (and the reason is written to the right)} </cmath> | ||

− | <cmath>a+ | + | <cmath>a+3b+3c+3d,\text{as } d\le b</cmath> |

<cmath>3a+3b+3c+d, \text{as } d\le a</cmath> | <cmath>3a+3b+3c+d, \text{as } d\le a</cmath> | ||

<cmath>3a+b+3c+3d, \text{as } b+d\le 2a </cmath> | <cmath>3a+b+3c+3d, \text{as } b+d\le 2a </cmath> |

## Revision as of 10:17, 20 October 2020

Problem 2. 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