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I am trying to add a custom header to a pdf via markdown. This page has been very helpfull. My working solution is below.

\addtolength{\headheight}{1cm} 
\fancypagestyle{plain}{} 
\rhead{\center\includegraphics[height=3cm]{path}} 
\renewcommand{\headrulewidth}{0pt} 

While this provides the desired solution on the title page with the correctly sized logo and space between the header and the text, there is extra blank space on the following pages. How can I remove (or control) the header spacing on pages > 1?

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In your current working solution, you are using some LaTeX commands to build a custom header. If you want a custom header to exist only on the title page, then you could use the LaTeX commands that control the title page. Specifically, \begin{titlepage} \end{titlepage} and nest your header within. For more information on title pages, see this overleaf post.

For directly solving issues with spacing on all pages, you can use the geometry package in LaTeX. Here are some resources describing how it works: rmarkdown - set margins, overleaf geometry package. This will allow you more direct control over the spacing on every page.

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