Difference between revisions of "2021 CIME I Problems/Problem 1"
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== Solution == | == Solution == | ||
| + | |||
| + | <asy> | ||
| + | |||
| + | import geometry; | ||
| + | |||
| + | point B = (0,0); | ||
| + | point C = (16,0); | ||
| + | point A = (0,16); | ||
| + | point D = (16,16); | ||
| + | point P = A * 1/3; | ||
| + | point Q = C * 38/85 + D * 47/85; | ||
| + | |||
| + | // Square ABCD | ||
| + | draw(A--B--C--D--A); | ||
| + | dot(A); | ||
| + | label("A",A,NW); | ||
| + | dot(B); | ||
| + | label("B",B,SW); | ||
| + | dot(C); | ||
| + | label("C",C,SE); | ||
| + | dot(D); | ||
| + | label("D",D,NE); | ||
| + | |||
| + | // Segment PQ | ||
| + | draw(P--Q); | ||
| + | dot(P); | ||
| + | label("P",P,W); | ||
| + | dot(Q); | ||
| + | label("Q",Q,E); | ||
| + | |||
| + | </asy> | ||
| + | |||
<math>\boxed{123}</math> | <math>\boxed{123}</math> | ||
== See also == | == See also == | ||
{{CIME box|year=2021|n=I|before=First Problem|num-a=2}} | {{CIME box|year=2021|n=I|before=First Problem|num-a=2}} | ||
Revision as of 18:12, 24 July 2024
Problem
Let 
be a square. Points 
and 
are on sides 
and 
respectively
such that the areas of quadrilaterals 
and 
are 
and 
respectively. Given that 
then 
where 
and 
are relatively prime positive integers. Find 
.
Solution
![[asy] import geometry; point B = (0,0); point C = (16,0); point A = (0,16); point D = (16,16); point P = A * 1/3; point Q = C * 38/85 + D * 47/85; // Square ABCD draw(A--B--C--D--A); dot(A); label("A",A,NW); dot(B); label("B",B,SW); dot(C); label("C",C,SE); dot(D); label("D",D,NE); // Segment PQ draw(P--Q); dot(P); label("P",P,W); dot(Q); label("Q",Q,E); [/asy]](http://latex.artofproblemsolving.com/4/1/e/41e67c0930c74ea344606d3c8ea9f42444338435.png)
See also
| 2021 CIME I (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 CIME Problems and Solutions | ||
