Difference between revisions of "1989 USAMO Problems/Problem 5"
(New page: ==Problem== Let <math>u</math> and <math>v</math> be real numbers such that <math> (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. </math> Det...) |
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Let <math>u</math> and <math>v</math> be real numbers such that | Let <math>u</math> and <math>v</math> be real numbers such that | ||
+ | <cmath> (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. </cmath> | ||
+ | Determine, with proof, which of the two numbers, <math>u</math> or <math>v</math>, is larger. | ||
+ | |||
+ | ==Solution== | ||
+ | |||
+ | The answer is <math>v</math>. | ||
+ | |||
+ | We define real functions <math>U</math> and <math>V</math> as follows: | ||
+ | <cmath>\begin{align*} | ||
+ | U(x) &= (x+x^2 + \dotsb + x^8) + 10x^9 = \frac{x^{10}-x}{x-1} + 9x^9 \\ | ||
+ | V(x) &= (x+x^2 + \dotsb + x^{10}) + 10x^{11} = \frac{x^{12}-x}{x-1} + 9x^{11} . | ||
+ | \end{align*} </cmath> | ||
+ | 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 | ||
+ | <cmath> U(x) = \frac{x^{10}-x}{x-1} + 9x^9 \le 0 < 8 .</cmath> | ||
+ | Similarly, <math>V(x) \le 0 < 8</math>. | ||
− | <math> | + | We also note that if <math>x \ge 9/10 </math>, then |
− | ( | + | <cmath> \begin{align*} |
− | </math> | + | U(x) &= \frac{x-x^{10}}{1-x} + 9x^9 \ge \frac{9/10 - 9^9/10^9}{1/10} + 9 \cdot \frac{9^{9}}{10^9} \\ |
+ | &= 9 - 10 \cdot \frac{9^9}{10^9} + 9 \cdot \frac{9^9}{10^9} = 9 - \frac{9^9}{10^9} > 8. | ||
+ | \end{align*} </cmath> | ||
+ | Similarly <math>V(x) > 8</math>. It then follows that <math>u, v \in (0,9/10)</math>. | ||
− | + | Now, for all <math>x \in (0,9/10)</math>, | |
+ | <cmath> \begin{align*} | ||
+ | V(x) &= U(x) + V(x)-U(x) = U(x) + 10x^{11}+x^{10} -9x^9 \\ | ||
+ | &= U(x) + x^9 (10x -9) (x+1) < U(x) . | ||
+ | \end{align*} </cmath> | ||
+ | Since <math>V</math> and <math>U</math> are both strictly increasing functions over the nonnegative reals, it then follows that | ||
+ | <cmath> V(u) < U(u) = 8 = V(v), </cmath> | ||
+ | so <math>u<v</math>, as desired. <math>\blacksquare</math> | ||
+ | |||
+ | == See Also == | ||
+ | |||
+ | {{USAMO box|year=1989|num-b=4|after=Last Question}} | ||
− | = | + | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356639#356639 Discussion on AoPS/MathLinks] |
+ | {{MAA Notice}} | ||
− | |||
− | + | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 19:16, 18 July 2016
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.
See Also
1989 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.