Question

一个由小正方形组成的矩形网格有 $141$ 行和 $91$ 列。每个小正方形里有放两个数字的空间。Horace 和 Vera 都把从 $1$ 到 $141 \times 91 = 12{,}831$ 的数字填入网格。Horace 按行填写：他把 $1$ 到 $91$ 依次从左到右填入第 $1$ 行，把 $92$ 到 $182$ 依次从左到右填入第 $2$ 行，并如此继续到第 $141$ 行。Vera 按列填写：她把 $1$ 到 $141$ 依次从上到下填入第 $1$ 列，再把 $142$ 到 $282$ 依次从上到下填入第 $2$ 列，并如此继续到第 $91$ 列。有多少个小正方形中两人写下了相同的数字？

A rectangular grid of squares has $141$ rows and $91$ columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from $1$ through $141 \times 91 = 12{,}831$ into the squares. Horace fills the grid horizontally: he puts $1$ through $91$ in order from left to right into row $1,$ puts $92$ through $182$ into row $2$ in order from left to right, and continues similarly through row $141.$ Vera fills the grid vertically: she puts $1$ through $141$ in order from top to bottom into column $1,$ then $142$ through $282$ into column $2$ in order from top to bottom, and continues similarly through column $91.$ How many squares get two copies of the same number?

$7$

$10$

$11$

$12$

$19$

答案：C

知识点：丢番图方程模运算

难度评级：2300

解答：

在第 $i$ 行第 $j$ 列，Horace 写的是 $91(i - 1) + j$ ，Vera 写的是 $141(j - 1) + i$ 。令二者相等并化简，得到 $9i - 14j = -5$ ，所以 $i = \tfrac{14j - 5}{9}$ ，恰好在 $j \equiv 1 \pmod 9$ 时为整数。对于 $j = 1, 10, 19, \ldots, 91$ ，共有 $11$ 个值，且对应的 $i$ 为 $1, 15, 29, \ldots, 141$ ，都在范围内。因此有 $11$ 个小正方形匹配，正确答案是 C。

At row $i,$ column $j,$ Horace writes $91(i - 1) + j$ and Vera writes $141(j - 1) + i.$ Set them equal and simplify to get $9i - 14j = -5,$ so $i = \tfrac{14j - 5}{9},$ an integer exactly when $j \equiv 1 \pmod 9.$ For $j = 1, 10, 19, \ldots, 91,$ that's $11$ values, and $i$ runs $1, 15, 29, \ldots, 141,$ all within range. So $11$ squares match. Thus, C is the correct answer.

2025 AMC 10B 第 23 题

答案：C

难度评级：2300

解答：

其他年份的第 23 题