2000 Putnam B1

Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each

$j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers

$r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values

of $j$, $1\leq j\leq N$.

令 $a_j,b_j,c_j$ 为 $1\leq j\leq N$ 的整数。假设对于每个

$j$，$a_j,b_j,c_j$中至少有一个是奇数。证明存在整数

$r$、$s$、$t$ 使得 $ra_j+sb_j+tc_j$ 对于至少 $4N/7$ 值来说是奇数

的$j$，$1\leq j\leq N$。