Difference between revisions of "2014 AIME I Problems/Problem 13"
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== Solution == | == Solution == | ||
| − | [ | + | [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("<math>A</math>",A,dir(135)); | ||
| + | dot("<math>B</math>",B,dir(215)); | ||
| + | dot("<math>C</math>",C,dir(305)); | ||
| + | dot("<math>D</math>",D,dir(45)); | ||
| + | pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); | ||
| + | pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); | ||
| + | dot("<math>H</math>",H,dir(90)); | ||
| + | dot("<math>F</math>",F,dir(270)); | ||
| + | draw(H--F); | ||
| + | pair E = (0,(sqrt(850)-6)/2); | ||
| + | pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); | ||
| + | dot("<math>E</math>",E,dir(180)); | ||
| + | dot("<math>G</math>",G,dir(0)); | ||
| + | draw(E--G); | ||
| + | pair P = extension(H,F,E,G); | ||
| + | dot("<math>P</math>",P,dir(60)); | ||
| + | label("<math>296k</math>", intersectionpoint( A--P, E--H )); | ||
| + | label("<math>275</math>", intersectionpoint( B--P, E--F )); | ||
| + | label("<math>405k</math>", intersectionpoint( C--P, G--F )); | ||
| + | label("<math>411k</math>", intersectionpoint( D--P, G--H ));[/asy] | ||
== 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 21:55, 15 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
[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,dir(135));
dot("
",B,dir(215));
dot("
",C,dir(305));
dot("
",D,dir(45));
pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850));
pair F = ((2sqrt(850)+sqrt(306)+7)/6,0);
dot("",H,dir(90));
dot("
",F,dir(270));
draw(H--F);
pair E = (0,(sqrt(850)-6)/2);
pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2);
dot("
",E,dir(180));
dot("
",G,dir(0));
draw(E--G);
pair P = extension(H,F,E,G);
dot("",P,dir(60));
label("
", intersectionpoint( A--P, E--H ));
label("
", intersectionpoint( B--P, E--F ));
label("
", intersectionpoint( C--P, G--F ));
label("
", intersectionpoint( D--P, G--H ));[/asy]
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. 
