Solution:
<?php
//for each category, show 1 post
$cat_args=array(
'orderby' => 'name',
'order' => 'ASC'
);
$categories=get_categories($cat_args);
foreach($categories as $category) {
$args=array(
'showposts' => 5,
'category__in' => array($category->term_id),
'caller_get_posts'=>5
);
$posts=get_posts($args);
if ($posts && $category->category_parent == 0) {
?>
<div style="border:thick solid #0000FF; ">
<?php
echo 'Category: <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.'</a> ';
foreach($posts as $post) {
setup_postdata($post); ?>
<p>
<a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?>
</a>
</p>
<?php
} // foreach($posts
?>
</div>
<?php
} // if ($posts
} // foreach($categories
?>