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 17:45, 1 August 2012
Problem
Let and be real numbers such that 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 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 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 |