Difference between revisions of "2020 IMO Problems/Problem 2"
m (→Solution) |
|||
(2 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | 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> | ||
Line 16: | Line 17: | ||
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> | ||
Line 34: | Line 35: | ||
~ftheftics | ~ftheftics | ||
+ | |||
+ | == Video solution == | ||
+ | |||
+ | https://youtu.be/bDHtM1wijbY [Video covers all day 1 problems] |
Revision as of 02:22, 27 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
Video solution
https://youtu.be/bDHtM1wijbY [Video covers all day 1 problems]