2020 AIME II Problems/Problem 14

Problem

For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ , and define $\{x\} = x - \lfloor x \rfloor$ to be the fractional part of $x$ . For example, $\{3\} = 0$ and $\{4.56\} = 0.56$ . Define $f(x)=x\{x\}$ , and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $0\leq x\leq 2020$ . Find the remainder when $N$ is divided by $1000$ .

Solution

To solve $f(f(f(x)))=17$ , we need to solve $f(x) = y$ where $f(f(y))=17$ , and to solve that we need to solve $f(y) = z$ where $f(z) = 17$ .

It is clear to see for some integer $a \geq 17$ there is exactly one value of $z$ in the interval $[a, a+1)$ where $f(z) = 17$ To understand this, imagine the graph of $f(z)$ on the interval $[a, a+1)$ The graph starts at $0$ , is continuous and increasing, and approaches $a+1$ . So as long as $a+1 > 17$ , there will be a solution for $z$ in the interval.

Using this logic, we can find the number of solutions to $f(x) = y$ . For every interval $[a, a+1)$ where $a \geq \left \lfloor{y}\right \rfloor$ there will be one solution for x in that interval. However, the question states $0 \leq x \leq 2020$ , but because $x=2020$ doesn't work we can change it to $0 \leq x < 2020$ . Therefore, $\left \lfloor{y}\right \rfloor \leq a \leq 2019$ , and there are $2020 - \left \lfloor{y}\right \rfloor$ solutions to $f(x) = y$ .

We can solve $f(y) = z$ similarly. $0 \leq y < 2020$ to satisfy the bounds of $x$ , so there are $2020 - \left \lfloor{z}\right \rfloor$ solutions to $f(y) = z$ , and $0 \leq z < 2020$ to satisfy the bounds of $y$ .

Going back to $f(z) = 17$ , there is a single solution for z in the interval $[a, a+1)$ , where $17 \leq a \leq 2019$ . (We now have an upper bound for $a$ because we know $z < 2020$ .) There are $2003$ solutions for $z$ , and the floors of these solutions create the sequence $17, 18, 19, ..., 2018, 2019$

Lets first look at the solution of $z$ where $\left \lfloor{z}\right \rfloor = 17$ . Then $f(y) = z$ would have $2003$ solutions, and the floors of these solutions would also create the sequence $17, 18, 19, ..., 2018, 2019$ .

If we used the solution of $y$ where $\left \lfloor{y}\right \rfloor = 17$ , there would be $2003$ solutions for $f(x) = y$ . If we used the solution of $y$ where $\left \lfloor{y}\right \rfloor = 18$ , there would be $2002$ solutions for $x$ , and so on. So for the solution of $z$ where $\left \lfloor{z}\right \rfloor = 17$ , there will be $2003 + 2002 + 2001 + ... + 2 + 1 = \binom{2004}{2}$ solutions for $x$

If we now look at the solution of $z$ where $\left \lfloor{z}\right \rfloor = 18$ , there would be $\binom{2003}{2}$ solutions for $x$ . If we looked at the solution of $z$ where $\left \lfloor{z}\right \rfloor = 19$ , there would be $\binom{2002}{2}$ solutions for $x$ , and so on.

The total number of solutions to $x$ is $\binom{2004}{2} + \binom{2003}{2} + \binom{2002}{2} + ... + \binom{3}{2} + \binom{2}{2}$ . Using the hockey stick theorem, we see this equals $\binom{2005}{3}$ , and when we take the remainder of that number when divided by $1000$ , we get the answer, $\boxed{10}$

Video Solution

https://youtu.be/bz5N-jI2e0U?t=515

2020 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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2020 AIME II Problems/Problem 14

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