1960 AMC 12 Problem 35

Question

From point

P

outside a circle, with a circumference of

10

units, a tangent is drawn. Also from

P

a secant is drawn dividing the circle into unequal arcs with lengths

m

and

n.

It is found that

t,

the length of the tangent, is the mean proportional between

m

and

n.

If

m

and

t

are integers, then

t

may have the following number of values:

zero

one

two

three

infinitely many

Use

m+n=10

and

t^2=mn

Test the integer values

m=1,2,\ldots,9,

excluding equal arcs

The arc lengths satisfy

m+n=10,

while the mean-proportional condition gives

t^2=mn=m(10-m).

For integral

m

from

1

through

9,

the possible products, up to symmetry, are

9,

16,

21,

24,

and

25.

The last comes from equal arcs

m=n=5

and is excluded. The remaining squares give

t=3

and

t=4,

two values.

Thus, the correct answer is C.

Problem 35 in other years