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

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<cmath>3x-285=x+115\Rightarrow 2x=400\Rightarrow x=\boxed{200}</cmath>
 
<cmath>3x-285=x+115\Rightarrow 2x=400\Rightarrow x=\boxed{200}</cmath>
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==Solution 2==
 +
Use the table from above. Obviously <math>c = 114</math>. Hence <math>a+e = 115</math>. Similarly, <math>1+a = 96 + e \Rightarrow a = 95+e</math>.
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Substitute that into the first to get <math>2e = 20 \Rightarrow e=10</math>, so <math>a=105</math>, and so the value of <math>x</math> is just <math>115+x = 210 + 105 \Rightarrow x = \boxed{200}</math>
  
 
== See also ==
 
== See also ==

Revision as of 14:35, 25 July 2019

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.

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

\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*}

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

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

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

Solution 2

Use the table from above. Obviously $c = 114$. Hence $a+e = 115$. Similarly, $1+a = 96 + e \Rightarrow a = 95+e$.

Substitute that into the first to get $2e = 20 \Rightarrow e=10$, so $a=105$, and so the value of $x$ is just $115+x = 210 + 105 \Rightarrow x = \boxed{200}$

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

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