1994 AIME 第 7 题

Question

对于某些实数有序对

(a,b)

，方程组

\begin{aligned}ax+by&=1,\\x^2+y^2&=50\end{aligned}

至少有一个解，并且每个解都是整数有序对

(x,y)

。这样的有序对

(a,b)

有多少个？

For certain ordered pairs

(a,b)

of real numbers, the system of equations

\begin{aligned}ax+by&=1,\\x^2+y^2&=50\end{aligned}

has at least one solution, and each solution is an ordered pair

(x,y)

of integers. How many such ordered pairs

(a,b)

are there?

答案

知识点：格点弦切线

列出圆

x^2+y^2=50

上的所有整点

List all integer points on

x^2+y^2=50

既要计算经过两个非对径整点的弦，也要计算每个整点处的切线

Count both chords through two non-antipodal lattice points and tangents at one lattice point

圆上共有

12

个整点，它们来自

(\pm1,\pm7)

、

(\pm5,\pm5)

和

(\pm7,\pm1)

。一条符合条件的割线由任意两个非对径整点确定，因此共有

\binom{12}{2}-6=60

条；要排除这些对径点对，因为它们所在的直线经过原点，不能写成

ax+by=1

。此外，每个整点处还有一条切线，共

12

条。每条直线都能唯一地规范化为

ax+by=1

，所以总数为

60+12=72

。

The circle has the

12

lattice points obtained from

(\pm1,\pm7),

(\pm5,\pm5),

and

(\pm7,\pm1).

A secant satisfying the condition is determined by any two non-antipodal lattice points. This gives

\binom{12}{2}-6=60

lines; antipodal pairs are excluded because their line passes through the origin and cannot have equation

ax+by=1.

There are also

12

tangents, one at each lattice point. Each line has a unique normalization

ax+by=1,

so the total is

60+12=72.

其他年份的第 7 题