HMMT 二月 2015 · 冲刺赛 · 第 12 题
HMMT February 2015 — Guts Round — Problem 12
题目详情
- [ 6 ] For integers a, b, c, d , let f ( a, b, c, d ) denote the number of ordered pairs of integers ( x, y ) ∈
{ 1 , 2 , 3 , 4 , 5 } such that ax + by and cx + dy are both divisible by 5. Find the sum of all possible 2
values of f ( a, b, c, d ).
HMMT FEBRUARY 2015, 21 FEBRUARY 2015 — GUTS ROUND
Organization Team Team ID#
3 2
英文原题
[ 6 ] For integers a, b, c, d , let f ( a, b, c, d ) denote the number of ordered pairs of integers ( x, y ) ∈
{ 1 , 2 , 3 , 4 , 5 } 2 such that ax + by and cx + dy are both divisible by 5. Find the sum of all possible
values of f ( a, b, c, d ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HMMT FEBRUARY 2015, 21 FEBRUARY 2015 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 6 ] For integers a, b, c, d , let f ( a, b, c, d ) denote the number of ordered pairs of integers ( x, y ) ∈
{ 1 , 2 , 3 , 4 , 5 } such that ax + by and cx + dy are both divisible by 5. Find the sum of all possible 2
values of f ( a, b, c, d ).
Answer: 31 Standard linear algebra over the field F (the integers modulo 5). The dimension ofthe solution set is at least 0 and at most 2, and any intermediate value can also be attained. So the 5
answer is 1 + 5 + 5 = 31.2
This also can be easily reformulated in more concrete equation/congruence-solving terms, especiallysince there are few variables/equations.
3 2