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Difference between revisions of "2020 AIME II Problems/Problem 14"

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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==
+
==Solution 1==
 +
\[
 +
\sum_{x=17}^{2019} \sum_{y=x}^{2019} \sum_{z=y}^{2019} 1
 +
.\]
 +
A combinatorical argument exists which is the official solution, but I will present another here for the sake of variety.
 +
 
 +
Applying algebraic manipulation and the hockey-stick identity <math>3</math> times gives
 +
\begin{align*}
 +
\sum_{x=17}^{2019} \sum_{y=x}^{2019} \sum_{z=y}^{2019} 1
 +
=\sum_{x=17}^{2019} \sum_{y=x}^{2019} \sum_{z=y}^{2019} \binom{z-y}{0}\
 +
=\sum_{x=17}^{2019} \sum_{y=x}^{2019} \binom{2020-y}{1}\
 +
=\sum_{x=17}^{2019} \binom{2021-x}{2}\
 +
=\binom{2005}{3}
 +
\end{align*}
 +
==Solution 2==
  
 
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>.
 
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>.

Revision as of 04:05, 5 August 2020

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 1

\[ \sum_{x=17}^{2019} \sum_{y=x}^{2019} \sum_{z=y}^{2019} 1 .\] A combinatorical argument exists which is the official solution, but I will present another here for the sake of variety.

Applying algebraic manipulation and the hockey-stick identity $3$ times gives ∑x=172019∑y=x2019∑z=y20191=∑x=172019∑y=x2019∑z=y2019(z−y0)=∑x=172019∑y=x2019(2020−y1)=∑x=172019(2021−x2)=(20053)

Solution 2

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, $\boxed{10}$

~aragornmf

Solution 2 (Official MAA)

For any nonnegative integer $n$, the function $f$ increases on the interval $[n,n+1)$, with $f(n)=0$ and $f(x)<n+1$ for every $x$ in this interval. On this interval $f(x)=x(x-n)$, which takes on every real value in the interval $[0,n+1)$ exactly once. Thus for each nonnegative real number $y$, the equation $f(x) = y$ has exactly one solution $x \in [n, n+1)$ for every $n \ge \lfloor y \rfloor$.

For each integer $a\geq 17$ there is exactly one $x$ with $\lfloor x\rfloor=a$ such that $f(x)=17$; likewise for each integer $b\geq a\geq 17$ there is exactly one $x$ with $\lfloor f(x)\rfloor=a$ and $\lfloor x\rfloor=b$ such that $f(f(x))=17$. Finally, for each integer $c \ge b \ge a \ge 17$ there is exactly one $x$ with $\lfloor f(f(x)) \rfloor = a$, $\lfloor f(x)\rfloor=b$, and $\lfloor x\rfloor=c$ such that $f(f(f(x)))=17$.

Thus $f(f(f(x)))=17$ has exactly one solution $x$ with $0\leq x\leq 2020$ for each triple of integers $(a,b,c)$ with $17\leq a\leq b\leq c<2020$, noting that $x=2020$ is not a solution. This nondecreasing ordered triple can be identified with a multiset of three elements of the set of $2003$ integers $\{17,18,19,\ldots,2019\}$, which can be selected in $\binom{2005}3$ ways. Thus \[N=\frac{2005\cdot 2004\cdot 2003}{6}\equiv 10\hskip -.2cm \pmod{1000}.\]

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
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