How do you remove the search box on the search results page?
This is a fun one... The Search box is returned as part of the render array returned by search_view().

I suggest that you make a hook_menu_alter() which replaces the default callback for the search page with your own, which is basically a pass-thru to the core function, but that unsets the search form.  Something like this:

function MYMODULE_menu_alter(&amp;$items) {
  $items['search']['page callback'] = 'MYMODULE_search_view';
}

function MYMODULE_search_view($module = NULL, $keys = '') {
  $build = search_view($module, $keys);
  unset($build['search_form']);
  return $build;
}


Don't forget to clear your cache to pick up the new hook and menu changes.

You may need to add additional paths to the hook_menu_alter() as needed (e.g., if you search users, then you may need to add entries for search/user and search/user/%menu_tail).