Difference between revisions of "1996 AIME Problems/Problem 1"

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Revision as of 18:31, 4 July 2013

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 $x$.

AIME 1996 Problem 01.png

Solution

Let's make a table.

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

$x+19+96=x+1+cc=19+961=114,114+96+a=x+1+114a=x95$ (Error compiling LaTeX. Unknown error_msg)

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

$19+x95+d=x+d76=115+xd=191,114+191+e=x+115e=x190$ (Error compiling LaTeX. Unknown error_msg)

$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table in progress}\\\hline x&19&96\\\hline 1&x-95&b\\\hline 114&191&x-190\\\hline \end{tabular}$

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

See also

1996 AIME (ProblemsAnswer KeyResources)
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

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