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Difference between revisions of "2009 USAMO Problems/Problem 4"

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== Problem ==
 
== Problem ==
 
For <math>n \ge 2</math> let <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math> be positive real numbers such that  
 
For <math>n \ge 2</math> let <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math> be positive real numbers such that  
−
<cmath> (a_1+a_2+ ... +a_n)\big( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \big) \le \big(n+ {1 \over 2} \big) ^2 </cmath>
+
<center><math> (a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2 </math></center>
−
Prove that max <math>(a_1, a_2, ... ,a_n)</math> <math> \le </math> 4 min <math>(a_1, a_2, ... , a_n)</math>.
+
Prove that <math>\text{max}(a_1, a_2, ... ,a_n) \le 4 \text{min}(a_1, a_2, ... , a_n)</math>.
  
 +
== Solution 1==
 +
Assume without loss of generality that <math>a_1 \geq a_2 \geq \cdots \geq a_n</math>. Now we seek to prove that <math>a_1 \le 4a_n</math>.
  
−
== Solution ==
+
By the [[Cauchy-Schwarz Inequality]], <cmath>\begin{align*}
−
Assume without loss of generality that <math>a_i \ge a_{i+1}</math> for all <math>1\le i \le n-1</math>. Now we seek to prove that <math>a_1 \le 4a_n</math>
+
(a_n+a_2+ a_3 + ... +a_{n-1}+a_1)\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right) &\ge \left( \sqrt{a_n \over a_1} + n-2 + \sqrt{a_1 \over a_n} \right)^2 \\
 +
(n+ {1 \over 2})^2 &\ge \left( \sqrt{a_n \over a_1} + n-2 + \sqrt{a_1 \over a_n} \right)^2 \\
 +
n+ {1 \over 2} &\ge n-2 + \sqrt{a_n \over a_1} + \sqrt{a_1 \over a_n} \\
 +
{5 \over 2} &\ge \sqrt{a_n \over a_1} + \sqrt{a_1 \over a_n} \\
 +
{17 \over 4} &\ge {a_n \over a_1} + {a_1 \over a_n} \\
 +
0 &\ge (a_1 - 4a_n)\left(a_1 - {a_n \over 4}\right) \end{align*}</cmath>
 +
Since <math>a_1 \ge a_n</math>, clearly <math>(a_1 - {a_n \over 4}) > 0</math>, dividing yields:
  
−
Now by Cauchy-Schwartz, <cmath>(a_n+a_2+ a_3 + ... +a_{n-1}+a_1)({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}) \ge \big( \sqrt{a_n \over a_1} + n-2 + \sqrt{a_1 \over a_n} \big)^2</cmath>
+
<cmath>0 \ge (a_1 - 4a_n) \Longrightarrow 4a_n \ge a_1</cmath>
−
<cmath>(n+ {1 \over 2})^2 \ge \big( \sqrt{a_n \over a_1} + n-2 + \sqrt{a_1 \over a_n} \big)^2 </cmath>
+
 
−
<cmath> n+ {1 \over 2} \ge n-2 + \sqrt{a_n \over a_1} + \sqrt{a_1 \over a_n}</cmath>
+
as desired.
−
<cmath>{5 \over 2} \ge \sqrt{a_n \over a_1} + \sqrt{a_1 \over a_n}</cmath>
+
 
−
<cmath>{17 \over 4} \ge {a_n \over a_1} + {a_1 \over a_n}</cmath>
+
== Solution 2==
−
<cmath> 0 \ge (a_1 - 4a_n)(a_1 - {a_n \over 4})</cmath>
+
Assume without loss of generality that <math>a_1 \geq a_2 \geq \cdots \geq a_n</math>.
−
Since <math>a_1 \ge a_n</math>, clearly <math>(a_1 - {a_n \over 4}) > 0</math>,
+
Using the Cauchy–Bunyakovsky–Schwarz inequality and the inequality given, <cmath>\begin{align*}
−
so dividing yields:
+
&\left(a_1+a_2 + ... +a_n + 3a_n -\frac{3a_1}{4}\right)\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right) \\
−
<cmath>0 \ge (a_1 - 4a_n)</cmath>
+
=&\left(\frac{a_1}{4}+a_2 + ... +a_{n-1}+4a_n\right)\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right) \\
−
<math>4a_n \ge a_1</math> as desired.
+
\ge& \left(\frac{1}{2}+n-2+2\right)^2 \\
 +
=&\left(n+\frac{1}{2}\right)^2 \\
 +
\ge& (a_1+a_2 + ... +a_{n})\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right).\end{align*}</cmath>
 +
(Note that <math>n-2 \ge 0</math> since <math>n \ge 2</math> as given!)
 +
This implies that <math>3a_n -\frac{3a_1}{4} \ge 0 \iff 4a_n \ge a_1</math> as desired.
 +
 
 +
~Deng Tianle, username: Leole
 +
 
 +
== See Also ==
 +
{{USAMO newbox|year=2009|num-b=3|num-a=5}}
 +
 
 +
[[Category:Olympiad Algebra Problems]]
 +
[[Category:Olympiad Inequality Problems]]
 +
{{MAA Notice}}

Latest revision as of 23:50, 13 July 2024

Problem

For $n \ge 2$ let $a_1$, $a_2$, ..., $a_n$ be positive real numbers such that

$(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$

Prove that $\text{max}(a_1, a_2, ... ,a_n) \le 4 \text{min}(a_1, a_2, ... , a_n)$.

Solution 1

Assume without loss of generality that $a_1 \geq a_2 \geq \cdots \geq a_n$. Now we seek to prove that $a_1 \le 4a_n$.

By the Cauchy-Schwarz Inequality, \begin{align*} (a_n+a_2+ a_3 + ... +a_{n-1}+a_1)\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right) &\ge \left( \sqrt{a_n \over a_1} + n-2 + \sqrt{a_1 \over a_n} \right)^2 \\ (n+ {1 \over 2})^2 &\ge \left( \sqrt{a_n \over a_1} + n-2 + \sqrt{a_1 \over a_n} \right)^2 \\ n+ {1 \over 2} &\ge n-2 + \sqrt{a_n \over a_1} + \sqrt{a_1 \over a_n} \\ {5 \over 2} &\ge \sqrt{a_n \over a_1} + \sqrt{a_1 \over a_n} \\ {17 \over 4} &\ge {a_n \over a_1} + {a_1 \over a_n} \\ 0 &\ge (a_1 - 4a_n)\left(a_1 - {a_n \over 4}\right) \end{align*} Since $a_1 \ge a_n$, clearly $(a_1 - {a_n \over 4}) > 0$, dividing yields:

\[0 \ge (a_1 - 4a_n) \Longrightarrow 4a_n \ge a_1\]

as desired.

Solution 2

Assume without loss of generality that $a_1 \geq a_2 \geq \cdots \geq a_n$. Using the Cauchy–Bunyakovsky–Schwarz inequality and the inequality given, \begin{align*} &\left(a_1+a_2 + ... +a_n + 3a_n -\frac{3a_1}{4}\right)\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right) \\ =&\left(\frac{a_1}{4}+a_2 + ... +a_{n-1}+4a_n\right)\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right) \\ \ge& \left(\frac{1}{2}+n-2+2\right)^2 \\ =&\left(n+\frac{1}{2}\right)^2 \\ \ge& (a_1+a_2 + ... +a_{n})\left({1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n}\right).\end{align*} (Note that $n-2 \ge 0$ since $n \ge 2$ as given!) This implies that $3a_n -\frac{3a_1}{4} \ge 0 \iff 4a_n \ge a_1$ as desired.

~Deng Tianle, username: Leole

See Also

2009 USAMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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