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

Revision as of 11:50, 13 August 2008

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.

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

$\begin{eqnarray*}x+19+96=x+1+c\Rightarrow c=19+96-1=114,\\ 114+96+a=x+1+114\Rightarrow a=x-95\end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

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

$\begin{eqnarray*}19+x-95+d=x+d-76=115+x\Rightarrow d=191,\\ 114+191+e=x+115\Rightarrow e=x-190\end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

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

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

@@ Line 1: / Line 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 <math>x</math>.
+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 <math>x</math>.
 <center>[[Image:AIME_1996_Problem_01.png]]</center>
@@ Line 36: / Line 36: @@
 == See also ==
 {{AIME box|year=1996|before=First Problem|num-a=2}}
+[[Category:Introductory Algebra Problems]]

1996 AIME (Problems • Answer Key • Resources)
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Difference between revisions of "1996 AIME Problems/Problem 1"

Revision as of 11:50, 13 August 2008

Problem

Solution

See also