1996 AIME Problems/Problem 1

Problem

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 $x$ .

Solution

Let's make a table.

$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table}\\\hline x&19&96\\\hline 1&a&b\\\hline c&d&e\\\hline \end{tabular}$

$x+19+96=x+1+c\Rightarrow c=19+96-1=114$ .

$114+96+a=x+1+114\Rightarrow a=x-95$

$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table in progress}\\\hline x&19&96\\\hline 1&x-95&b\\\hline 114&d&e\\\hline \end{tabular}$

$19+x-95+d=x+d-76=115+x\Rightarrow d=191$ .

$114+191+e=x+115\Rightarrow e=x-190$

$3x-285=x+115\Rightarrow 2x=400\Rightarrow x=\boxed{200}$

1996 AIME (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search

1996 AIME Problems/Problem 1

Problem

Solution

See also