Difference between revisions of "2020 IMO Problems/Problem 2"
(→Solution) |
|||
Line 15: | Line 15: | ||
Now notice that , | Now notice that , | ||
+ | <cmath>a+2b+3c+4d \text{ will be less then the following expression (and reason is written on right)} </cmath> | ||
+ | <cmath>a+2b+3c+3d ,\text{as} d\le b</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 +3b +c +3d , \text{as} 2c+d \le 2a+b </cmath> | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
So, We get , | So, We get , |
Revision as of 00:31, 27 September 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
On that case we get ,