Difference between revisions of "2020 IMO Problems/Problem 2"
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− | Problem | + | == Problem == |
− | Prove that | + | 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> |
− | < | ||
+ | == Video solution == | ||
+ | |||
+ | https://youtu.be/bDHtM1wijbY [Video covers all day 1 problems] | ||
== Solution == | == Solution == | ||
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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> | ||
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~ftheftics | ~ftheftics | ||
+ | |||
+ | == Video solution == | ||
+ | |||
+ | https://youtu.be/bDHtM1wijbY [Video covers all day 1 problems] | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2020|num-b=1|num-a=3}} | ||
+ | |||
+ | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 11:32, 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]
See Also
2020 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |