2025 AMC 10B Problem 9

How many ordered triples of integers

(x, y, z)

satisfy the following system of inequalities?

-x - y - z \le -2

-x + y + z \le 2

x - y + z \le 2

x + y - z \le 2

Show solution

Solution

Let

p = -x + y + z,

q = x - y + z,

r = x + y - z.

The last three inequalities say

p, q, r \le 2,

the first says

x + y + z \ge 2,

and

p + q + r = x + y + z.

Since

x = \tfrac{q + r}{2}

and so on,

p, q, r

must all share the same parity. Now count triples with each part

\le 2,

equal parity, and sum in

[2, 6].

The even ones are

(2,2,2),

the permutations of

(2,2,0),

(2,0,0),

and of

(2,2,-2),

giving

10.

The only odd one is

(1,1,1).

That’s

11

in all, and each yields a unique

(x, y, z).

Thus, C is the correct answer.