Difference between revisions of "2019 AMC 12B Problems/Problem 8"
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==Problem== | ==Problem== | ||
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Let <math>f(x) = x^{2}(1-x)^{2}</math>. What is the value of the sum | Let <math>f(x) = x^{2}(1-x)^{2}</math>. What is the value of the sum | ||
<math>f\left(\frac{1}{2019} \right)-f \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots </math> | <math>f\left(\frac{1}{2019} \right)-f \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots </math> | ||
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==Solution== | ==Solution== | ||
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First, note that <math>f(x) = f(1-x)</math>. We can see this since | First, note that <math>f(x) = f(1-x)</math>. We can see this since | ||
<cmath>f(x) = x^2(1-x)^2 = (1-x)^2x^2 = f(1-x)</cmath> | <cmath>f(x) = x^2(1-x)^2 = (1-x)^2x^2 = f(1-x)</cmath> | ||
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+ \left( f \left(\frac{1009}{2019} \right) - f \left(\frac{1009}{2019} \right) \right)</cmath> | + \left( f \left(\frac{1009}{2019} \right) - f \left(\frac{1009}{2019} \right) \right)</cmath> | ||
Now, it is clear that all the terms will cancel out, and so the answer is <math>\boxed{\text{(A) 0}}</math>. | Now, it is clear that all the terms will cancel out, and so the answer is <math>\boxed{\text{(A) 0}}</math>. | ||
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==See Also== | ==See Also== | ||
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{{AMC12 box|year=2019|ab=B|num-b=7|num-a=9}} | {{AMC12 box|year=2019|ab=B|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 22:36, 16 February 2019
Problem
Let 
. What is the value of the sum
?
Solution
First, note that 
. We can see this since
![\[f(x) = x^2(1-x)^2 = (1-x)^2x^2 = f(1-x)\]](http://latex.artofproblemsolving.com/7/7/0/7703ab563430080802dbbf2512e3079d82460ae6.png)
From this, we regroup the terms accordingly:
![\[\left( f \left(\frac{1}{2019} \right) - f \left(\frac{2018}{2019} \right) \right) + \left( f \left(\frac{2}{2019} \right) - f \left(\frac{2017}{2019} \right) \right) + \cdots + \left( f \left(\frac{1009}{2019} \right) - f \left(\frac{1010}{2019} \right) \right)\]](http://latex.artofproblemsolving.com/b/4/9/b4920afe33917c882c3ea0d6b0594effda45801e.png)
![\[= \left( f \left(\frac{1}{2019} \right) - f \left(\frac{1}{2019} \right) \right) + \left( f \left(\frac{2}{2019} \right) - f \left(\frac{2}{2019} \right) \right) + \cdots + \left( f \left(\frac{1009}{2019} \right) - f \left(\frac{1009}{2019} \right) \right)\]](http://latex.artofproblemsolving.com/8/7/3/87361380209081cfc0c5667d1224d7c7ce591457.png)
Now, it is clear that all the terms will cancel out, and so the answer is 
.
See Also
| 2019 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
