Difference between revisions of "1996 AIME Problems/Problem 1"
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<center><math>\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*}</math></center> | <center><math>\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*}</math></center> | ||
| + | <math>\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}</math> | ||
| + | |||
| + | <center><math>\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*}</math></center> | ||
<math>\begin{tabular}[t]{|c|c|c|} | <math>\begin{tabular}[t]{|c|c|c|} | ||
| Line 21: | Line 29: | ||
x&19&96\\\hline | x&19&96\\\hline | ||
1&x-95&b\\\hline | 1&x-95&b\\\hline | ||
| − | 114& | + | 114&191&x-190\\\hline |
\end{tabular}</math> | \end{tabular}</math> | ||
| − | < | + | <cmath>3x-285=x+115\Rightarrow 2x=400\Rightarrow x=\boxed{200}</cmath> |
== See also == | == See also == | ||
{{AIME box|year=1996|before=First Problem|num-a=2}} | {{AIME box|year=1996|before=First Problem|num-a=2}} | ||
Revision as of 10:57, 28 February 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 
.
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 | ||
