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 
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Solution
Let's make a table.
![$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table}\\\hline x&19&96\\\hline 1&a&b\\\hline c&d&e\\\hline \end{tabular}$](http://latex.artofproblemsolving.com/f/c/b/fcbcbd888d2a3a3fe0b644846a73317615c7b905.png)
![$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table in progress}\\\hline x&19&96\\\hline 1&x-95&b\\\hline 114&d&e\\\hline \end{tabular}$](http://latex.artofproblemsolving.com/2/5/d/25d8c9b92279741a9f1af09ddde0a35ab424bc67.png)
![$\begin{tabular}[t]{|c|c|c|} \multicolumn{3}{c}{Table in progress}\\\hline x&19&96\\\hline 1&x-95&b\\\hline 114&191&x-190\\\hline \end{tabular}$](http://latex.artofproblemsolving.com/2/3/4/23464c2e73bb0559aa32a1bad044b3f1e8627285.png)
![\[3x-285=x+115\Rightarrow 2x=400\Rightarrow x=\boxed{200}\]](http://latex.artofproblemsolving.com/a/1/1/a117c3a54af4b7782d4681c73bfa6c873212f99f.png)
See also
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
