Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC 10A/12A testing, the AoPS Wiki is in read-only mode and no edits can be made.

1998 USAMO Problems/Problem 3: Difference between revisions

Gauss1181 (talk | contribs)
editor
42 edits
Created page with '1998 USAMO Problem #1 Problem: Let <math>a_0,\cdots a_n</math> be real numbers in the interval <math>(0,\frac {\pi}{2})</math> such that <math>\tan{(a_0 - \frac {\pi}{4})} + \t…'
 
Dwrensha (talk | contribs)
editor
4 edits
 
(10 intermediate revisions by 7 users not shown)
Line 1: Line 1:
1998 USAMO Problem #1
== Problem ==
 
Let <math>a_0,\cdots a_n</math> be real numbers in the interval <math>\left(0,\frac {\pi}{2}\right)</math> such that
Problem:
<cmath>\tan{\left(a_0 - \frac {\pi}{4}\right)} + \tan{\left(a_1 - \frac {\pi}{4}\right)} + \cdots + \tan{\left(a_n - \frac {\pi}{4}\right)}\ge n - 1</cmath>
 
Prove that <math>\tan{\left(a_0\right)}\tan{\left(a_1\right)}\cdots \tan{\left(a_n\right)}\ge n^{n + 1}</math>.
Let <math>a_0,\cdots a_n</math> be real numbers in the interval <math>(0,\frac {\pi}{2})</math> such that
<math>\tan{(a_0 - \frac {\pi}{4})} + \tan{(a_1 - \frac {\pi}{4})} + \cdots + \tan{(a_n - \frac {\pi}{4})}\ge n - 1</math>.
Prove that <math>\tan{(a_0)}\tan{(a_1)}\cdots \tan{(a_n)}\ge n^{n + 1}</math>.
 
Solution:


== Solution ==
Let <math>y_i = \tan{(a_i - \frac {\pi}{4})}</math>, where <math>0\le i\le n</math>. Then we have
Let <math>y_i = \tan{(a_i - \frac {\pi}{4})}</math>, where <math>0\le i\le n</math>. Then we have


Line 17: Line 13:
By AM-GM,
By AM-GM,


*<math>\frac {1}{n}\sum_{j\neq i}{(1 - y_j)}\ge \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}</math>
<cmath>\begin{align*}
*<math>\frac {1 + y_i}{n}\ge \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}</math>
\frac {1}{n}\sum_{j\neq i}{(1 - y_j)} &\geq \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}\\
*<math>\prod_{i = 0}^n{\frac {1 + y_i}{n}}\ge \prod_{i = 0}^n{\prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}</math>
\frac {1 + y_i}{n} &\geq \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}\\
*<math>= \prod_{i = 0}^n{(1 - y_i)}</math>
\prod_{i = 0}^n\frac{1 + y_i}{n} &\geq \prod_{i = 0}^{n} {\prod_{j\neq i} {(1 - y_j)}^{\frac {1}{n}}}\\
*<math>\prod_{i = 0}^n{\frac {1 + y_i}{1 - y_i}}\ge \prod_{i = 0}^n{n} = n^{n + 1}</math>
\prod_{i = 0}^n\frac {1 + y_i}{n} &\geq \prod_{i = 0}^n{(1 - y_i)}\\
\prod_{i = 0}^n\frac {1 + y_i}{1 - y_i} &\geq \prod_{i = 0}^n{n}\\
\prod_{i = 0}^n\frac {1 + y_i}{1 - y_i} &\geq n^{n + 1}
\end{align*}</cmath>
 
Note that by the addition formula for tangents, <cmath>\tan{(a_i)} = \tan{[(a_i - \frac {\pi}{4}) + \frac {\pi}{4}]} = \frac {1 + \tan{(a_i - \frac {\pi}{4})}}{1 - \tan{(a_i - \frac {\pi}{4})}} = \frac {1 + y_i}{1 - y_i}</cmath>.


Note that by the addition formula for tangents, <math>\tan{(a_i)} = \tan{[(a_i - \frac {\pi}{4}) + \frac {\pi}{4}]} = \frac {1 + \tan{(a_i - \frac {\pi}{4})}}{1 - \tan{(a_i - \frac {\pi}{4})}} = \frac {1 + y_i}{1 - y_i}</math>.
So <math>\prod_{i = 0}^n{\frac {1 + y_i}{1 - y_i}} = \tan{(a_0)}\tan{(a_1)}\cdots \tan{(a_n)}\ge n^{n + 1}</math>, as desired. <math>\blacksquare</math>


So <math>\prod_{i = 0}^n{\frac {1 + y_i}{1 - y_i}} = \tan{(a_0)}\tan{(a_1)}\cdots \tan{(a_n)}\ge n^{n + 1}</math>, as desired.
==See Also==


<math>\text{QED}</math>
{{USAMO newbox|year=1998|num-b=2|num-a=4}}
[[Category:Olympiad Trigonometry Problems]]
[[Category:Olympiad Algebra Problems]]
[[Category:Olympiad Inequality Problems]]
{{MAA Notice}}

Latest revision as of 12:31, 23 August 2023

Problem

Let $a_0,\cdots a_n$ be real numbers in the interval $\left(0,\frac {\pi}{2}\right)$ such that \[\tan{\left(a_0 - \frac {\pi}{4}\right)} + \tan{\left(a_1 - \frac {\pi}{4}\right)} + \cdots + \tan{\left(a_n - \frac {\pi}{4}\right)}\ge n - 1\] Prove that $\tan{\left(a_0\right)}\tan{\left(a_1\right)}\cdots \tan{\left(a_n\right)}\ge n^{n + 1}$.

Solution

Let $y_i = \tan{(a_i - \frac {\pi}{4})}$, where $0\le i\le n$. Then we have

  • $y_0 + y_1 + \cdots + y_n\ge n - 1$
  • $1 + y_i\ge \sum_{j\neq i}{(1 - y_j)}$
  • $\frac {1 + y_i}{n}\ge \frac {1}{n}\sum_{j\neq i}{(1 - y_j)}$

By AM-GM,

\begin{align*} \frac {1}{n}\sum_{j\neq i}{(1 - y_j)} &\geq \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}\\ \frac {1 + y_i}{n} &\geq \prod_{j\neq i}{(1 - y_j)^{\frac {1}{n}}}\\ \prod_{i = 0}^n\frac{1 + y_i}{n} &\geq \prod_{i = 0}^{n} {\prod_{j\neq i} {(1 - y_j)}^{\frac {1}{n}}}\\ \prod_{i = 0}^n\frac {1 + y_i}{n} &\geq \prod_{i = 0}^n{(1 - y_i)}\\ \prod_{i = 0}^n\frac {1 + y_i}{1 - y_i} &\geq \prod_{i = 0}^n{n}\\ \prod_{i = 0}^n\frac {1 + y_i}{1 - y_i} &\geq n^{n + 1} \end{align*}

Note that by the addition formula for tangents, \[\tan{(a_i)} = \tan{[(a_i - \frac {\pi}{4}) + \frac {\pi}{4}]} = \frac {1 + \tan{(a_i - \frac {\pi}{4})}}{1 - \tan{(a_i - \frac {\pi}{4})}} = \frac {1 + y_i}{1 - y_i}\].

So $\prod_{i = 0}^n{\frac {1 + y_i}{1 - y_i}} = \tan{(a_0)}\tan{(a_1)}\cdots \tan{(a_n)}\ge n^{n + 1}$, as desired. $\blacksquare$

See Also

1998 USAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1998_USAMO_Problems/Problem_3&oldid=197341"