Difference between revisions of "2014 AIME I Problems/Problem 13"
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== Solution == | == Solution == | ||
| − | Let <math>w, x, y, z</math> be <math>269k, 275k, 405k</math> and <math>411k</math> respectively. Then the total area of <math>ABCD</math> is <math>1360k</math>. We have <math>w + z = x + y = | + | Let <math>w, x, y, z</math> be <math>269k, 275k, 405k</math> and <math>411k</math> respectively. Then the total area of <math>ABCD</math> is <math>1360k</math>. We have <math>w + z = x + y = 680k</math>, and <math>w + x = 544k = \frac 2 3 (x + y)</math>. |
== See also == | == See also == | ||
{{AIME box|year=2014|n=I|num-b=12|num-a=14}} | {{AIME box|year=2014|n=I|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 23:52, 20 March 2014
Problem 13
On square 
, points 
, and 
lie on sides 
and 
respectively, so that 
and 
. Segments 
and 
intersect at a point 
, and the areas of the quadrilaterals 
and 
are in the ratio 
Find the area of square .
![[asy] pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("$P$",P,dir(60)); label("$w$", intersectionpoint( A--P, E--H )); label("$x$", intersectionpoint( B--P, E--F )); label("$y$", intersectionpoint( C--P, G--F )); label("$z$", intersectionpoint( D--P, G--H ));[/asy]](http://latex.artofproblemsolving.com/e/1/1/e1175984ad8a8e7468972e727cc95e27117b7d92.png)
Solution
Let 
be 
and 
respectively. Then the total area of is 
. We have 
, and 
.
See also
| 2014 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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. 
