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\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> | ||
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<cmath>\implies a^a b^b c^c d^d \le a^2 +b^2 +c^2 +d^2</cmath> | <cmath>\implies a^a b^b c^c d^d \le a^2 +b^2 +c^2 +d^2</cmath> | ||
− | So, <cmath>(a+2b+3c+4d) a^ab^bc^ | + | So, <cmath>(a+2b+3c+4d) a^ab^bc^cd^d \le (a+2b+3c+4d)(a^2+b^2+c^2+d^2) </cmath> |
Now notice that | Now notice that | ||
− | <cmath>a+2b+3c+4d \text{ will be less then the following | + | <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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So, we get | So, we get | ||
− | <cmath>(a+2b+3c+4d)(a^2+b^2+c^2+d^2) | + | <cmath>\begin{split}&~~~~(a+2b+3c+4d)(a^2+b^2+c^2+d^2) \\ |
− | + | &= a^2(a+2b+3c+4d)+b^2(a+2b+3c+4d)+c^2 (a+2b+3c+4d) +d^2 (a+2b+3c+4d)\\ | |
+ | &\le a^2(a+3b+3c+3d)+b^2(3a+b+3c+3d)+c^2 (3a+3b+c+3d) +d^2 (3a+3b+3c+d)\\ | ||
+ | &<(a+b+c+d)^3 \\&=1\end{split}</cmath> | ||
− | < | + | Now, for equality we must have <math>a=b=c=d=\frac{1}{4}</math> |
− | <cmath> | + | In that case we get <cmath>(a+2b+3c+4d) a^ab^bc^cd^d \le (a+2b+3c+4d)(a^2+b^2+c^2+d^2) =\frac{5}{8} <1</cmath> |
− | |||
− | + | ~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]