2019 AMC 12A Problem 25

Question

Let

\triangle A_0 B_0 C_0

be a triangle whose angle measures are exactly

59.999^\circ,

60^\circ,

and

60.001^\circ.

For each positive integer

n

define

A_n

to be the foot of the altitude from

A_{n-1}

to line

B_{n-1}C_{n-1}.

Likewise, define

B_n

to be the foot of the altitude from

B_{n-1}

to line

A_{n-1}C_{n-1},

and

C_n

to be the foot of the altitude from

C_{n-1}

to line

A_{n-1}B_{n-1}.

What is the least positive integer

n

for which

\triangle A_n B_n C_n

is obtuse?

10

11

13

14

15

Answer: E

Concepts:altitude recursion angle chasing

Difficulty rating: 2520

Small hint:

For an acute triangle, the orthic triangle has angles

180^\circ - 2\alpha

Big hint:

Track the deviation from

60^\circ:

each step multiplies it by

-2,

and the triangle turns obtuse when a deviation exceeds

30^\circ

Solution:

For an acute triangle, the orthic triangle (feet of the altitudes) has angles

180^\circ - 2\alpha

for each original angle

\alpha.

Writing an angle as

60^\circ + x,

the new angle is

60^\circ - 2x,

so each deviation from

60^\circ

is multiplied by

-2.

The initial deviations are

\pm 0.001^\circ.

After

n

steps a deviation has magnitude

0.001 \cdot 2^n

degrees. The triangle first becomes obtuse when this exceeds

30^\circ,

i.e.

2^n \gt 30000.

Since

2^{14} = 16384

and

2^{15} = 32768,

the least such

n

is

15.

Thus, the correct answer is E.

2019 AMC 12A Problem 25

Answer: E

Difficulty rating: 2520

Small hint:

Big hint:

Solution:

Problem 25 in other years