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HMMT 二月 2016 · 冲刺赛 · 第 17 题

HMMT February 2016 — Guts Round — Problem 17

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 11 ] Compute the sum of all integers 1 ≤ a ≤ 10 with the following property: there exist integers p
    2 2
    and q such that p , q , p + a and q + a are all distinct prime numbers.
解析

英文解析

  1. [ 11 ] Compute the sum of all integers 1 ≤ a ≤ 10 with the following property: there exist integers p
    2 2
    and q such that p , q , p + a and q + a are all distinct prime numbers.
    Proposed by: Evan Chen
    Answer: 20
    Odd a fail for parity reasons and a ≡ 2 (mod 3) fail for mod 3 reasons. This leaves a ∈ { 4 , 6 , 10 } . Itis easy to construct p and q for each of these, take ( p, q ) = (3 , 5) , (5 , 11) , (3 , 7), respectively.