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

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Problem 2. 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>.
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== Problem ==
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Prove that
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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>
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<math>(a+2b+3c+4d)a^a b^bc^cd^d<1</math>
 
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== Solution ==
 
== Solution ==
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Using Weighted AM-GM we get
 
Using Weighted AM-GM we get
  
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<cmath>\frac{a. a +b. b +c. c +d. d}{a+b+c+d} \ge (a^a b^b c^c d^d)^{\frac{1}{a+b+c+d}}</cmath>
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<cmath>\frac{a\cdot a +b\cdot b +c\cdot c +d\cdot d}{a+b+c+d} \ge \sqrt[a+b+c+d]{a^a b^b c^c d^d}</cmath>
  
 
<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>
  
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So, <cmath>(a+2b+3c+4d) a^ab^bc^c \le (a+2b+3c+4d)(a^2+b^2+c^2+d^2) </cmath>
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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
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     <cmath>a+2b+3c+4d \text{ will be less then the following expression (and reason is written to the right)} </cmath>
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     <cmath>a+2b+3c+4d \text{ will be less then the following expressions (and the reason is written to the right)} </cmath>
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     <cmath>a+2b+3c+3d,\text{as } d\le b</cmath>
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     <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
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<cmath>(a+2b+3c+4d)(a^2+b^2+c^2+d^2) </cmath>
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<cmath>\begin{split}&~~~~(a+2b+3c+4d)(a^2+b^2+c^2+d^2) \\
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<cmath>= a^2(a+2b+3c+4d)+b^2(a+2b+3c+4d)+c^2 (a+2b+3c+4d) +d^2 (a+2b+3c+4d) </cmath>
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&= a^2(a+2b+3c+4d)+b^2(a+2b+3c+4d)+c^2 (a+2b+3c+4d) +d^2 (a+2b+3c+4d)\\
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&\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)\\
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&<(a+b+c+d)^3 \\&=1\end{split}</cmath>
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Now, for equality we must have <math>a=b=c=d=\frac{1}{4}</math>
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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>
  
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<cmath>\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)</cmath>
 
  
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<cmath>=(a+b+c+d)^3 =1</cmath>
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~Shen Kislay kai
  
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Now, for equality we must have <math>a=b=c=d=\frac{1}{4}</math>
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== Video solution ==
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https://youtu.be/bDHtM1wijbY [Video covers all day 1 problems]
  
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In that case we get <cmath>(a+2b+3c+4d) a^ab^bc^c \le (a+2b+3c+4d)(a^2+b^2+c^2+d^2) =\frac{5}{8} <1</cmath>
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==See Also==
  
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{{IMO box|year=2020|num-b=1|num-a=3}}
  
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~ftheftics
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[[Category:Olympiad Algebra Problems]]

Latest revision as of 12:14, 3 September 2024

Problem

The real numbers $a$, $b$, $c$, $d$ are such that $a \geq b \geq c \geq d > 0$ and $a + b + c + d = 1$. Prove that\[(a + 2b + 3c + 4d) a^a b^b c^c d^d < 1.\]

Solution

Using Weighted AM-GM we get

\[\frac{a\cdot a +b\cdot b +c\cdot c +d\cdot d}{a+b+c+d} \ge \sqrt[a+b+c+d]{a^a b^b c^c d^d}\]

\[\implies a^a b^b c^c d^d \le a^2 +b^2 +c^2 +d^2\]

So, \[(a+2b+3c+4d) a^ab^bc^cd^d \le (a+2b+3c+4d)(a^2+b^2+c^2+d^2)\]

Now notice that 

   \[a+2b+3c+4d \text{ will be less then the following expressions (and the reason is written to the right)}\]
   \[a+3b+3c+3d,\text{as } d\le b\]
   \[3a+3b+3c+d,              \text{as }  d\le a\]
   \[3a+b+3c+3d, \text{as }  b+d\le 2a\]
 \[3a+3b+c+3d, \text{as } 2c+d \le 2a+b\]


So, we get \[\begin{split}&~~~~(a+2b+3c+4d)(a^2+b^2+c^2+d^2) \\ &= a^2(a+2b+3c+4d)+b^2(a+2b+3c+4d)+c^2 (a+2b+3c+4d) +d^2 (a+2b+3c+4d)\\ &\le a^2(a+3b+3c+3d)+b^2(3a+b+3c+3d)+c^2 (3a+3b+c+3d) +d^2 (3a+3b+3c+d)\\ &<(a+b+c+d)^3 \\&=1\end{split}\]

Now, for equality we must have $a=b=c=d=\frac{1}{4}$

In that case we get \[(a+2b+3c+4d) a^ab^bc^cd^d \le (a+2b+3c+4d)(a^2+b^2+c^2+d^2) =\frac{5}{8} <1\]


~Shen Kislay kai

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
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