HMMT 二月 2005 · TEAM2 赛 · 第 15 题

英文解析

15. [40] Suppose that S tiles the set of odd prime numbers. Prove that S has only oneelement.
    Solution: Consider the set S equivalent to S that contains 3. If it contains 5 butnot 7, then the set S equivalent to S containing 7 must contain 9, which is not prime.0
    Likewise, S cannot contain 7 but not 5, because then the set S containing 5 must 1
    0 1
    contain 9. Suppose S contains 3, 5, and 7. Then any other set S of the tiling contains
    0 1
    elements p , p + 2, and p + 4. But not all of these can be prime, because one of them isdivisible by 3. Finally, suppose S contains 3 and has second-smallest element p > 7.
    Then the set S containing 5 does not contain 7 but does contain p + 2, and the set S0
    1 2
    containing 7 contains p + 4. But as before, not all of p , p + 2, and p + 4 can be prime.
    Therefore S has no second-smallest element, so it has only one element. 4