HMMT 二月 2004 · GEN1 赛 · 第 8 题
HMMT February 2004 — GEN1 Round — Problem 8
题目详情
英文原题
- You have a 10 × 10 grid of squares. You write a number in each square as follows: youwrite 1 , 2 , 3 , . . . , 10 from left to right across the top row, then 11 , 12 , . . . , 20 across the second row, and so on, ending with a 100 in the bottom right square. You then writea second number in each square, writing 1 , 2 , . . . , 10 in the first column (from top tobottom), then 11 , 12 , . . . , 20 in the second column, and so forth.
When this process is finished, how many squares will have the property that their twonumbers sum to 101?
解析
英文解析
- You have a 10 × 10 grid of squares. You write a number in each square as follows: youwrite 1 , 2 , 3 , . . . , 10 from left to right across the top row, then 11 , 12 , . . . , 20 across the second row, and so on, ending with a 100 in the bottom right square. You then writea second number in each square, writing 1 , 2 , . . . , 10 in the first column (from top tobottom), then 11 , 12 , . . . , 20 in the second column, and so forth.
When this process is finished, how many squares will have the property that their twonumbers sum to 101?
Solution: 10
The number in the i th row, j th column will receive the numbers 10( i − 1) + j and
10( j − 1) + i , so the question is how many pairs ( i, j ) (1 ≤ i, j ≤ 10) will have
101 = [10( i − 1) + j ] + [10( j − 1) + i ] ⇔ 121 = 11 i + 11 j = 11( i + j ) .
Now it is clear that this is achieved by the ten pairs (1 , 10) , (2 , 9) , (3 , 8) , . . . , (10 , 1) andno others.