2023 AMC 10B Problem 14

Attempt Problem 14 of the 2023 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10B solutions, or check the answer key.

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14.

How many ordered pairs of integers

(m, n)

satisfy the equation

m^2 + mn + n^2 = m^2 n^2?

7

1

3

6

5

m = 0

then

n = 0;

otherwise assume

|m| \le |n|

Then

m^2 n^2 = m^2 + mn + n^2 \le 3n^2

forces

m^2 \le 3

m = 0,

the equation forces

n^2 = 0,

giving

(0, 0).

Otherwise both are nonzero; assume

|m| \le |n|.

Then

m^2 n^2 = m^2 + mn + n^2 \le 3n^2,

m^2 \le 3

and

m = \pm 1.

Take

m = 1:

1 + n + n^2 = n^2

gives

n = -1.

Take

m = -1:

n = 1.

That leaves

(0,0), (1,-1), (-1,1),

three in all. Therefore, the answer is C.

Problem 14 in other years