1988 AIME Problem 6

Question

It is possible to place positive integers into the vacant twenty-one squares of the

5\times5

square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk

(*).

Answer

Answer: 142

Concepts:arithmetic sequence pattern recognition system of equations

Difficulty rating: 2030

Small hint:

Index the rows and columns from

0

through

4

Big hint:

A grid whose rows and columns are arithmetic has the form

A+Bi+Cj+Dij

Solution:

Index rows and columns from

0

to

4.

The row and column conditions give the general entry

A+Bi+Cj+Dij.

The four shown values yield

\begin{aligned}A+4B&=0,\\A+B+C+D&=74,\\A+2B+4C+8D&=186,\\A+3B+2C+6D&=103.\end{aligned}

Solving gives

A=52,

B=-13,

C=30,

and

D=5.

The asterisk is at

(i,j)=(0,3),

so its value is

A+3C=52+90=142.