<p>For a real number <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" /> let <img alt="$\lfloor x\rfloor$" height="19" src="https://latex.artofproblemsolving.com/4/4/1/441416e660d25335f27e55334ac2a06e51541c97.png" width="23" /> be the greatest integer less than or equal to <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" />, and define <img alt="$\{x\} = x - \lfloor x \rfloor$" height="19" src="https://latex.artofproblemsolving.com/0/0/f/00fc6a0d316131539a685aca6deee35129c1dc93.png" width="109" /> to be the fractional part of <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" />. For example, <img alt="$\{3\} = 0$" height="18" src="https://latex.artofproblemsolving.com/f/a/6/fa6411aefd9d2d89cb84de53a3199164e2ee32ed.png" width="60" /> and <img alt="$\{4.56\} = 0.56$" height="20" src="https://latex.artofproblemsolving.com/a/c/3/ac319b6db77546997c49a0bc7cf8340a5a87f89c.png" width="109" />. Define <img alt="$f(x)=x\{x\}$" height="18" src="https://latex.artofproblemsolving.com/9/d/a/9da2ce2595d8b700ce2bda76efb4cde9961df55f.png" width="97" />, and let <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> be the number of real-valued solutions to the equation <img alt="$f(f(f(x)))=17$" height="18" src="https://latex.artofproblemsolving.com/c/1/7/c173b8b75ef125cb1dbc3457f088a7ed5169fc70.png" width="127" /> for <img alt="$0\leq x\leq 2020$" height="15" src="https://latex.artofproblemsolving.com/7/9/f/79ffaea2e5598d534d5c58f6a3efd932898330c3.png" width="103" />. Find the remainder when <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> is divided by <img alt="$1000$" height="13" src="https://latex.artofproblemsolving.com/6/e/e/6ee927e1332358c96c62c277441c907c4f51057f.png" width="35" />.</p>

Question No: 377

For a 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 $1000$ .

Editorial

Tags: Algebra

Question Details