/sci/ - Science & Math

Let <span class="math">a_j,b_j,c_j[/spoiler] be integers for <span class="math">1\leq j\leq N[/spoiler]. Assume for each <span class="math">j[/spoiler], at least one of <span class="math">a_j,b_j,c_j[/spoiler] is odd. Show that there exist integers <span class="math">r[/spoiler], <span class="math">s[/spoiler], <span class="math">t[/spoiler] such that <span class="math">ra_j+sb_j+tc_j[/spoiler] is odd for at least <span class="math">4N/7[/spoiler] values of <span class="math">j[/spoiler], <span class="math">1\leq j\leq N[/spoiler].