Difference between revisions of "2020 AIME II Problems/Problem 14"
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==Solution== | ==Solution== | ||
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+ | To solve <math>f(f(f(x)))=17</math>, we need to solve <math>f(x) = y</math> where <math>f(f(y))=17</math>, and to solve that we need to solve <math>f(y) = z</math> where <math>f(z) = 17</math>. | ||
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+ | It is clear to see for some integer <math>a \geq 17</math> there is exactly one value of <math>z</math> in the interval <math>[a, a+1)</math> where <math>f(z) = 17</math> To understand this, imagine the graph of <math>f(z)</math> on the interval <math>[a, a+1)</math> The graph starts at <math>0</math>, is continuous and increasing, and approaches <math>a+1</math>. So as long as <math>a+1 > 17</math>, there will be a solution for <math>z</math> in the interval. | ||
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+ | Using this logic, we can find the number of solutions to <math>f(x) = y</math>. For every interval <math>[a, a+1)</math> where <math>a \geq \left \lfloor{y}\right \rfloor </math> there will be one solution for x in that interval. However, the question states <math>0 \leq x \leq 2020</math>, but because <math>x=2020</math> doesn't work we can change it to <math>0 \leq x < 2020</math>. Therefore, <math>\left \lfloor{y}\right \rfloor \leq a \leq 2019</math>, and there are <math>2020 - \left \lfloor{y}\right \rfloor</math> solutions to <math>f(x) = y</math>. | ||
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+ | We can solve <math>f(y) = z</math> similarly. <math>0 \leq y < 2020</math> to satisfy the bounds of <math>x</math>, so there are <math>2020 - \left \lfloor{z}\right \rfloor</math> solutions to <math>f(y) = z</math>, and <math>0 \leq z < 2020</math> to satisfy the bounds of <math>y</math>. | ||
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+ | Going back to <math>f(z) = 17</math>, there is a single solution for z in the interval <math>[a, a+1)</math>, where <math>17 \leq a \leq 2019</math>. (We now have an upper bound for <math>a</math> because we know <math>z < 2020</math>.) There are <math>2003</math> solutions for <math>z</math>, and the floors of these solutions create the sequence <math>17, 18, 19, ..., 2018, 2019</math> | ||
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+ | Lets first look at the solution of <math>z</math> where <math>\left \lfloor{z}\right \rfloor = 17</math>. Then <math>f(y) = z</math> would have <math>2003</math> solutions, and the floors of these solutions would also create the sequence <math>17, 18, 19, ..., 2018, 2019</math>. | ||
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+ | If we used the solution of <math>y</math> where <math>\left \lfloor{y}\right \rfloor = 17</math>, there would be <math>2003</math> solutions for <math>f(x) = y</math>. If we used the solution of <math>y</math> where <math>\left \lfloor{y}\right \rfloor = 18</math>, there would be <math>2002</math> solutions for <math>x</math>, and so on. So for the solution of <math>z</math> where <math>\left \lfloor{z}\right \rfloor = 17</math>, there will be <math>2003 + 2002 + 2001 + ... + 2 + 1 = \binom{2004}{2}</math> solutions for <math>x</math> | ||
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+ | If we now look at the solution of <math>z</math> where <math>\left \lfloor{z}\right \rfloor = 18</math>, there would be <math>\binom{2003}{2}</math> solutions for <math>x</math>. If we looked at the solution of <math>z</math> where <math>\left \lfloor{z}\right \rfloor = 19</math>, there would be <math>\binom{2002}{2}</math> solutions for <math>x</math>, and so on. | ||
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+ | The total number of solutions to <math>x</math> is <math>\binom{2004}{2} + \binom{2003}{2} + \binom{2002}{2} + ... + \binom{3}{2} + \binom{2}{2}</math>. Using the hockey stick theorem, we see this equals <math>\binom{2005}{3}</math>, and when we take the remainder of that number when divided by <math>1000</math>, we get the answer, <math>\boxed{10}</math> | ||
==Video Solution== | ==Video Solution== |
Revision as of 19:29, 12 June 2020
Contents
Problem
For real number let be the greatest integer less than or equal to , and define to be the fractional part of . For example, and . Define , and let be the number of real-valued solutions to the equation for . Find the remainder when is divided by .
Solution
To solve , we need to solve where , and to solve that we need to solve where .
It is clear to see for some integer there is exactly one value of in the interval where To understand this, imagine the graph of on the interval The graph starts at , is continuous and increasing, and approaches . So as long as , there will be a solution for in the interval.
Using this logic, we can find the number of solutions to . For every interval where there will be one solution for x in that interval. However, the question states , but because doesn't work we can change it to . Therefore, , and there are solutions to .
We can solve similarly. to satisfy the bounds of , so there are solutions to , and to satisfy the bounds of .
Going back to , there is a single solution for z in the interval , where . (We now have an upper bound for because we know .) There are solutions for , and the floors of these solutions create the sequence
Lets first look at the solution of where . Then would have solutions, and the floors of these solutions would also create the sequence .
If we used the solution of where , there would be solutions for . If we used the solution of where , there would be solutions for , and so on. So for the solution of where , there will be solutions for
If we now look at the solution of where , there would be solutions for . If we looked at the solution of where , there would be solutions for , and so on.
The total number of solutions to is . Using the hockey stick theorem, we see this equals , and when we take the remainder of that number when divided by , we get the answer,
Video Solution
https://youtu.be/bz5N-jI2e0U?t=515
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.