返回题库

HMMT 十一月 2012 · 冲刺赛 · 第 10 题

HMMT November 2012 — Guts Round — Problem 10

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 8 ] Consider the sequence defined by a = 0 , a = 1, and a = 2013 a + 2012 a + n for n ≥ 2.
    0 1 n n − 1 n − 2
    Find the remainder when a is divided by 2012.
    2012
解析

英文解析

  1. [ 8 ]
    Answer: 1006 Taking both sides modulo 2012, we see that a ≡ a + n (mod 2012). Therefore,
    n n − 1
    (2012)(2013)
    a ≡ a + 2012 ≡ a + 2011 + 2012 ≡ ... ≡ 1 + 2 + ... + 2012 ≡ ≡ (1006)(2013) ≡
    2012 2011 2010
    (1006)(1) ≡ 1006 (mod 2012).2
    Guts Round