2024 AMC 10A Problem 21

Question

The numbers, in order, of each row and the numbers, in order, of each column of a

5 \times 5

array of integers form an arithmetic progression of length

5.

The numbers in positions

(5, 5),

(2, 4),

(4, 3),

and

(3, 1)

are

0,

48,

16,

and

12,

respectively. What number is in position

(1, 2)?

\begin{bmatrix} \cdot & ? & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & 48 & \cdot \\ 12 & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & 16 & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 0 \end{bmatrix}

19

24

29

34

39

A grid whose every row and column is arithmetic has entry

f(i, j) = A + Bi + Cj + Dij

at row

i,

column

j

Substitute the four known entries into this form and solve for

A, B, C, D

If every row and every column is an arithmetic progression, the entry at row

i,

column

j

must take the bilinear form

f(i, j) = A + Bi + Cj + Dij.

Plug in

f(5, 5) = 0,

f(2, 4) = 48,

f(4, 3) = 16,

f(3, 1) = 12

and solve:

A = -10,

B = 5,

C = 22,

D = -5.

So position

(1, 2)

is

-10 + 5 + 2 \cdot 22 - 2 \cdot 5 = 29.

Thus, C is the correct answer.

Problem 21 in other years