Difference between revisions of "2018 USAJMO Problems/Problem 2"

(Created page with "== Problem == Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cm...")
 
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==Solution 1==
 
==Solution 1==
WLOG let <math>a \leq b \leq c.</math>
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WLOG let <math>a \leq b \leq c</math>. Add <math>2(ab+bc+ca)</math> to both sides of the inequality and factor to get: <cmath>4(a(a+b+c)+bc) \geq (a+b+c)^2</cmath> <cmath>\frac{4a\sqrt[3]{abc}+bc}{2} \geq 2\sqrt[3]{a^2b^2c^2}</cmath>
Add <math>2(ab+bc+ca)</math> to both sides of the inequality and factor to get: <cmath>4(a(a+b+c)+bc) \geq (a+b+c)^2</cmath> <cmath>\frac{4a\sqrt[3]{abc}+bc}{2} \geq 2\sqrt[3]{a^2b^2c^2}</cmath>
 
  
 
The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.
 
The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.
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==Solution 2==
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WLOG let <math>a \leq b \leq c</math>. Note that the equations are homogeneous, so WLOG let <math>c=1</math>.
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Thus, the inequality now becomes <math>2ab + 2a + 2b + 4 \geq a^2 + b^2 + 1</math>, which simplifies to <math>4ab + 3 \geq (a-b)^2</math>.
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Now we will use the condition. Letting <math>x=a+b</math> and <math>y=a-b</math>, we have
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<math>x+1=\sqrt[3]{16(x^2-y^2)} \implies 16y^2=-x^3+13x^2-3x-1</math>.
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Plugging this into the inequality, we have <math>2x+3 \geq \frac{1}{16}(-x^3+13x^2-3x-1) \implies x^3-13x^2+35x+49 = (x-7)^2(x+1) \geq 0</math> since <math>x \geq 2</math>.
  
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 11:50, 26 April 2018

Problem

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$. Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]

Solution 1

WLOG let $a \leq b \leq c$. Add $2(ab+bc+ca)$ to both sides of the inequality and factor to get: \[4(a(a+b+c)+bc) \geq (a+b+c)^2\] \[\frac{4a\sqrt[3]{abc}+bc}{2} \geq 2\sqrt[3]{a^2b^2c^2}\]

The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.

Solution 2

WLOG let $a \leq b \leq c$. Note that the equations are homogeneous, so WLOG let $c=1$. Thus, the inequality now becomes $2ab + 2a + 2b + 4 \geq a^2 + b^2 + 1$, which simplifies to $4ab + 3 \geq (a-b)^2$.

Now we will use the condition. Letting $x=a+b$ and $y=a-b$, we have $x+1=\sqrt[3]{16(x^2-y^2)} \implies 16y^2=-x^3+13x^2-3x-1$.

Plugging this into the inequality, we have $2x+3 \geq \frac{1}{16}(-x^3+13x^2-3x-1) \implies x^3-13x^2+35x+49 = (x-7)^2(x+1) \geq 0$ since $x \geq 2$.

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See also

2018 USAJMO (ProblemsResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6
All USAJMO Problems and Solutions
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