Here are the codes that I use to avoid displaying duplicate posts when I use multiple loops to pull in the most recent posts from different categories on a WordPress page.

Here is the code that I use to display the most recent posts from specific categories on a WordPress page:

<h3>Latest posts</h3><br/>
<?php
//for categories 1,5,3, show 1 post
$cat_args=array(
  'include' => '1,5,3',
  'orderby' => 'name',
  'order' => 'ASC'
   );
$categories=get_categories($cat_args);
  foreach($categories as $category) {
    $args=array(
      'showposts' =>3,
      'category__in' => array($category->term_id),
      'caller_get_posts'=>1
    );
    $posts=get_posts($args);
      if ($posts) {
        echo '<p><h4>Latest <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.' Posts</a>: </h4></p> ';
        foreach($posts as $post) {
          setup_postdata($post); ?>
          <p style="margin-left:20px;">
<a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></p>
          <?php
        } // foreach($posts
      } // if ($posts
    } // foreach($categories
?>

And here is the same code with the lines added (highlighted in red) that will prevent your WordPress loops from displaying  duplicate posts:

<h3>Latest posts</h3><br/>
<?php
//for categories 1,5,3, show 1 post
$cat_args=array(
  'include' => '1,5,3',
  'orderby' => 'name',
  'order' => 'ASC'
   );
$do_not_duplicate = array();
$categories=get_categories($cat_args);
  foreach($categories as $category) {
    $args=array(
      'showposts' =>3,
  'post__not_in' => $do_not_duplicate,
      'category__in' => array($category->term_id),
      'caller_get_posts'=>1
    );
    $posts=get_posts($args);
      if ($posts) {
        echo '<p><h4>Latest <a href="' . get_category_link( $category->term_id ) . '" title="' . sprintf( __( "View all posts in %s" ), $category->name ) . '" ' . '>' . $category->name.' Posts</a>: </h4></p> ';
        foreach($posts as $post) {
          setup_postdata($post);
  $do_not_duplicate[] = $post->ID; ?>
          <p style="margin-left:20px;">
<a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></p>
          <?php
        } // foreach($posts
      } // if ($posts
    } // foreach($categories
?>

As always, just let me know if you run into any issues or have any questions.