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

(Making it prettier)
(Solution)
Line 16: Line 16:
 
<center><math>x+19+96=x+1+cc=19+961=114,114+96+a=x+1+114a=x95</math></center>
 
<center><math>x+19+96=x+1+cc=19+961=114,114+96+a=x+1+114a=x95</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>19+x95+d=x+d76=115+xd=191,114+191+e=x+115e=x190</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&d&e\\hline
+
114&191&x-190\\hline
 
\end{tabular}</math>
 
\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,\ 3x-285=x+115\Rightarrow 2x=400\Rightarrow x=\boxed{200} \end{eqnarray*}</math></center>
+
<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 $x$.

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