Difference between revisions of "1989 USAMO Problems/Problem 5"
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V(x) &= (x+x^2 + \dotsb + x^{10}) + 10x^{11} = \frac{x^{12}-x}{x-1} + 9x^{11} . | V(x) &= (x+x^2 + \dotsb + x^{10}) + 10x^{11} = \frac{x^{12}-x}{x-1} + 9x^{11} . | ||
\end{align*} </cmath> | \end{align*} </cmath> | ||
| − | We wish to show that if <math>U(u)=V(v)=8</math>, then <math>u | + | We wish to show that if <math>U(u)=V(v)=8</math>, then <math>u <v</math>. |
We first note that when <math>x \le 0</math>, <math>x^{12}-x \ge 0</math>, <math>x-1 < 0</math>, and <math>9x^9 \le 0</math>, so | We first note that when <math>x \le 0</math>, <math>x^{12}-x \ge 0</math>, <math>x-1 < 0</math>, and <math>9x^9 \le 0</math>, so | ||
Revision as of 18:45, 1 August 2012
Problem
Let 
and 
be real numbers such that
![\[(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.\]](http://latex.artofproblemsolving.com/7/4/7/747d227c5c93ab38b45ffe98794d68ac188986dd.png)
Determine, with proof, which of the two numbers, or , is larger.
Solution
The answer is .
We define real functions 
and 
as follows:
We wish to show that if 
, then 
.
We first note that when 
, 
, 
, and 
, so
![\[U(x) = \frac{x^{10}-x}{x-1} + 9x^9 \le 0 < 8 .\]](http://latex.artofproblemsolving.com/a/1/b/a1b8d3aedf346b5aa699546e25156e20d990103d.png)
Similarly, 
.
We also note that if 
, then
Similarly 
. It then follows that 
.
Now, for all 
,
Since and are both strictly increasing functions over the nonnegative reals, it then follows that
![\[V(u) < U(u) = 8 = V(v),\]](http://latex.artofproblemsolving.com/d/d/e/ddeb36defeb65d2a05aa127bdc978a2ec1a4ef25.png)
so 
, as desired. 
Resources
| 1989 USAMO (Problems • Resources) | ||
| Preceded by Problem 4 |
Followed by Final Question | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
