There exists a unique positive integer <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" /> for which the sum <img alt="\[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\]" height="50" src="https://latex.artofproblemsolving.com/6/b/0/6b010978a55642b7f7aabb2df2f7cf45c9db937a.png" width="148" /> is an integer strictly between <img alt="$-1000$" height="13" src="https://latex.artofproblemsolving.com/d/5/1/d51ea422092ef90627083ec6fe2db733d1038957.png" width="49" /> and <img alt="$1000$" height="13" src="https://latex.artofproblemsolving.com/6/e/e/6ee927e1332358c96c62c277441c907c4f51057f.png" width="35" />. For that unique <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" />, find <img alt="$a+U$" height="14" src="https://latex.artofproblemsolving.com/e/8/f/e8f03b93d0a768b4232dd7cb3afcfd626a67a935.png" width="46" />. (Note that <img alt="$\lfloor x\rfloor$" height="19" src="https://latex.artofproblemsolving.com/4/4/1/441416e660d25335f27e55334ac2a06e51541c97.png" width="23" /> denotes the greatest integer that is less than or equal to <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" />.)

Question No: 341

There exists a unique positive integer $a$ for which the sum

$U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor$

is an integer strictly between $-1000$ and $1000$ . For that unique $a$ , find $a+U$ .

(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$ .)

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Tags: Algebra

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