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 $(1, 2)?$

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

$19$

$24$

$29$

$34$

$39$

Solution:

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.

2024 AMC 10A Problem 21

Solution:

Problem 21 in Other Years