# Difference between revisions of "2004 IMO Problems/Problem 4"

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== Problem == | == Problem == | ||

− | (''Hojoo Lee'') Let <math>n \geq 3</math> be an integer. Let <math>t_1 | + | (''Hojoo Lee'') Let <math>n \geq 3</math> be an integer. Let <math>t_1, t_2, \dots , t_n</math> be positive real numbers such that |

<cmath>n^2 + 1 > \left( t_1 + t_2 + ... + t_n \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_n} \right).</cmath> | <cmath>n^2 + 1 > \left( t_1 + t_2 + ... + t_n \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_n} \right).</cmath> | ||

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<cmath>\begin{align*}f(n) &:= \left( t_1 + t_2 + ... + t_n \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_n} \right) \\ &= \left( t_1 + t_2 + ... + t_{n-1} \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_{n-1}} \right) + t_n \sum_{i=1}^{n-1} \frac 1{t_i} + \frac{1}{t_n} \sum_{i=1}^{n-1} t_i + 1\end{align*}</cmath> | <cmath>\begin{align*}f(n) &:= \left( t_1 + t_2 + ... + t_n \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_n} \right) \\ &= \left( t_1 + t_2 + ... + t_{n-1} \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_{n-1}} \right) + t_n \sum_{i=1}^{n-1} \frac 1{t_i} + \frac{1}{t_n} \sum_{i=1}^{n-1} t_i + 1\end{align*}</cmath> | ||

− | By [[AM-GM]], <math>\frac{t_n}{t_i} + \frac{t_i}{t_n} \ge 2</math>, so <math>f(n) \ge f(n-1) + 2(n-1) + 1 = f(n-1) + 2n - 1</math>. Then the problem is reduced to proving the statement true for <math>n-1</math> numbers, as desired. <math>\ | + | By [[AM-GM]], <math>\frac{t_n}{t_i} + \frac{t_i}{t_n} \ge 2</math>, so <math>f(n) \ge f(n-1) + 2(n-1) + 1 = f(n-1) + 2n - 1</math>. Then the problem is reduced to proving the statement true for <math>n-1</math> numbers, as desired.<math>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\square</math> |

== See also == | == See also == |

## Latest revision as of 23:50, 20 February 2021

## Problem

(*Hojoo Lee*) Let be an integer. Let be positive real numbers such that

Show that , , are side lengths of a triangle for all , , with .

## Solution

For , suppose (for sake of contradiction) that for ; then (by Cauchy-Schwarz Inequality)

so it is true for . We now claim the result by induction; for , we have

By AM-GM, , so . Then the problem is reduced to proving the statement true for numbers, as desired.

## See also

- <url>viewtopic.php?p=99756#99756 AoPS/MathLinks discussion</url>