2020 AIME II Problems/Problem 14: Difference between revisions
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==Problem== | ==Problem== | ||
For real number <math>x</math> let <math>\lfloor x\rfloor</math> be the greatest integer less than or equal to <math>x</math>, and define <math>\{x\} = x - \lfloor x \rfloor</math> to be the fractional part of <math>x</math>. For example, <math>\{3\} = 0</math> and <math>\{4.56\} = 0.56</math>. Define <math>f(x)=x\{x\}</math>, and let <math>N</math> be the number of real-valued solutions to the equation <math>f(f(f(x)))=17</math> for <math>0\leq x\leq 2020</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>. | For real number <math>x</math> let <math>\lfloor x\rfloor</math> be the greatest integer less than or equal to <math>x</math>, and define <math>\{x\} = x - \lfloor x \rfloor</math> to be the fractional part of <math>x</math>. For example, <math>\{3\} = 0</math> and <math>\{4.56\} = 0.56</math>. Define <math>f(x)=x\{x\}</math>, and let <math>N</math> be the number of real-valued solutions to the equation <math>f(f(f(x)))=17</math> for <math>0\leq x\leq 2020</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
==Solution== | |||
==Video Solution== | |||
https://youtu.be/bz5N-jI2e0U?t=515 | |||
==See Also== | ==See Also== | ||
{{AIME box|year=2020|n=II|num-b=13|num-a=15}} | {{AIME box|year=2020|n=II|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 03:43, 8 June 2020
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
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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
