1996 AIME Problem 1

Question

In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find

x.

Answer

Answer: 200

Concepts:magic square system of equations

Difficulty rating: 1650

Small hint:

Let

e

be the center entry and

S

the common sum

Big hint:

In a

3\times3

magic square,

S=3e

and opposite entries sum to

2e

Solution:

Let the center entry be

e

and the common sum be

S.

In any

3\times3

magic square,

S=3e

and entries opposite across the center sum to

2e.

Thus the bottom-left entry is

2e-96.

The first column and first row give

\begin{aligned}x+1+(2e-96)&=3e,\\x+19+96&=3e.\end{aligned}

The first equation says

e=x-95,

while the second says

3e=x+115.

Hence

3x-285=x+115,

so

x=200.