2025 AMC 12B 第 19 题

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：丢番图方程模运算区间内整数计数

难度评级：2040

小提示：

分别写出第

r

行第

c

列中 Horace 和 Vera 填入的数字，并将结果表示为

r

和

c

的公式

Write the Horace-number and the Vera-number of the square in row

r,

column

c

as formulas in

r

and

c

大提示：

令两式相等可得

9r = 14c - 5

；找出哪些

c

会使

r

是指定范围内的整数

Setting them equal gives

9r = 14c - 5;

find which

c

make

r

an integer in range

解答：

在第

r

行第

c

列，Horace 写的是

(r-1)\cdot 91 + c

，Vera 写的是

(c-1)\cdot 141 + r

。令二者相等，得

90r - 140c = -50

，即

9r = 14c - 5

。这要求

c \equiv 1 \pmod 9

，所以

c = 1, 10, 19, \ldots, 91

，共有

11

个值；每个都给出合法的

r

，且在

1

到

141

之间。因此有

11

个小正方形匹配。

所以正确答案是 C。

In row

r,

column

c,

Horace writes

(r-1)\cdot 91 + c

and Vera writes

(c-1)\cdot 141 + r.

Setting these equal gives

90r - 140c = -50,

i.e.

9r = 14c - 5.

This requires

c \equiv 1 \pmod 9,

so

c = 1, 10, 19, \ldots, 91

— that is

11

values, and each yields a valid

r

between

1

and

141.

So

11

squares match.

Thus, the correct answer is C.

2025 AMC 12B 第 19 题

答案：C

难度评级：2040

小提示：

大提示：

解答：

其他年份的第 19 题