2024 AMC 12B Problem 13

Below is the professionally curated solution for Problem 13 of the 2024 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12B solutions, or check the answer key.

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Concepts:completing the square optimization

Difficulty rating: 1640

13.

There are real numbers $x, y, h,$ and $k$ that satisfy the system of equations

$x^2 + y^2 - 6x - 8y = h$

$x^2 + y^2 - 10x + 4y = k.$

What is the minimum possible value of $h + k?$

$-54$

$-46$

$-34$

$-16$

$16$

Solution:

Adding the equations, $h + k = 2x^2 + 2y^2 - 16x - 4y = 2(x - 4)^2 + 2(y - 1)^2 - 34.$ Both squared terms are nonnegative, so the minimum occurs at $x = 4,$ $y = 1,$ giving $h + k = -34.$

Thus, the correct answer is C.

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