Difference between revisions of "2001 AMC 10 Problems/Problem 22"
Pidigits125 (talk | contribs) (Created page with '==Problem== In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by <math> v </math>, <math> …') |
Pidigits125 (talk | contribs) (→Solution) |
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==Solution== | ==Solution== | ||
| + | |||
| + | We know that <math> y+z=2v </math>, so we could find one variable rather than two. | ||
<math> v+24+w=43+v </math> | <math> v+24+w=43+v </math> | ||
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The sum per row is <math> 25+21+20=66 </math>. | The sum per row is <math> 25+21+20=66 </math>. | ||
| − | Thus <math> | + | Thus <math> 66-18-25=66-43=v=23 </math>. |
| + | |||
| − | + | Since we needed <math> 2v </math> and we know <math> v=23 </math>, <math> 23 \times 2 = \boxed{\textbf{(D)}\ 46} </math>. | |
Revision as of 19:47, 16 March 2011
Problem
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by 
, 
, 
, 
, and 
. Find 
.
![[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$w$",(2.5,2.5));[/asy]](http://latex.artofproblemsolving.com/4/e/2/4e24d4a82884d1873aa08b1502e6996d8292e7ed.png)
Solution
We know that 
, so we could find one variable rather than two.
![[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]](http://latex.artofproblemsolving.com/a/c/9/ac93cbfe3fce005ca5ee9103f9f203476e189db6.png)
![[asy] unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$20$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$19$",(2.5,2.5));[/asy]](http://latex.artofproblemsolving.com/1/2/6/1261087702926461e26052453481b55bb6c07a10.png)
The sum per row is 
.
Thus 
.
Since we needed 
and we know 
, 
.
