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HMMT 十一月 2017 · THM 赛 · 第 1 题

HMMT November 2017 — THM Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. T wo ordered pairs ( a, b ) and ( c, d ), where a, b, c, d are real numbers, form a basis of the coordinate plane if ad 6 = bc . Determine the number of ordered quadruples ( a, b, c, d ) of integers between 1 and 3 inclusive for which ( a, b ) and ( c, d ) form a basis for the coordinate plane.
解析
  1. T wo ordered pairs ( a, b ) and ( c, d ), where a, b, c, d are real numbers, form a basis of the coordinate plane if ad 6 = bc . Determine the number of ordered quadruples ( a, b, c, d ) of integers between 1 and 3 inclusive for which ( a, b ) and ( c, d ) form a basis for the coordinate plane. Proposed by: Ashwin Sah Answer: 66 Any pair of distinct points will form a basis except when ( a, b ) and ( c, d ) are both from { (1 , 1) , (2 , 2) , (3 , 3) } , so the answer is 9 · 8 − 3 · 2 = 66.