\documentclass[11pt]{article}
\usepackage[letterpaper,margin=1in,headheight=14pt,includeheadfoot]{geometry}

\def\thestudent{Student's Name}
\def\theprob{X}
\def\thepart{Y}

\usepackage{fancyhdr}
\usepackage{lastpage}
\pagestyle{fancy}
\fancyhf{}
\fancyhf[HLEO]{\thestudent}
\fancyhf[HCEO]{6.852 - Problem Set \theprob, Part \thepart}
\fancyhf[HREO]{Problem \theenumi}
\fancyhf[FREO]{Page \thepage\ of \pageref{LastPage}}
\renewcommand\headrulewidth{0.4pt}
\renewcommand\footrulewidth{0.4pt}

\begin{document}
  \begin{enumerate}
    \item Problem 1:

    Here is solution 1.

    \pagebreak
    \item Problem 2:

    Here is solution 2.
    
    \pagebreak
    \item Problem 3:

    Here is solution 3, etc.
  \end{enumerate}
\end{document}
