Difference between revisions of "2023 AIME II Problems/Problem 3"
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== Diagram == | == Diagram == | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | |||
+ | size(200); | ||
+ | pair A, B, C, P; | ||
+ | |||
+ | A = origin; | ||
+ | B = (0,10*sqrt(5)); | ||
+ | C = (10*sqrt(5),0); | ||
+ | P = intersectionpoints(Circle(A,10),Circle(C,20))[0]; | ||
+ | |||
+ | dot("$A$",A,1.5*SW,linewidth(4)); | ||
+ | dot("$B$",B,1.5*NW,linewidth(4)); | ||
+ | dot("$C$",C,1.5*SE,linewidth(4)); | ||
+ | dot("$P$",P,1.5*NE,linewidth(4)); | ||
+ | |||
+ | markscalefactor=0.125; | ||
+ | draw(rightanglemark(B,A,C,10),red); | ||
+ | draw(anglemark(P,A,B,25),red); | ||
+ | draw(anglemark(P,B,C,25),red); | ||
+ | draw(anglemark(P,C,A,25),red); | ||
+ | add(pathticks(anglemark(P,A,B,25), n = 1, r = 0.1, s = 10, red)); | ||
+ | add(pathticks(anglemark(P,B,C,25), n = 1, r = 0.1, s = 10, red)); | ||
+ | add(pathticks(anglemark(P,C,A,25), n = 1, r = 0.1, s = 10, red)); | ||
+ | |||
+ | draw(A--B--C--cycle^^P--A^^P--B^^P--C); | ||
+ | label("$10$",midpoint(A--P),dir(-30),red); | ||
+ | </asy> | ||
+ | ~MRENTHUSIASM | ||
== Solution 1== | == Solution 1== |
Revision as of 17:34, 16 February 2023
Contents
Problem
Let be an isosceles triangle with
There exists a point
inside
such that
and
Find the area of
Diagram
~MRENTHUSIASM
Solution 1
Since the triangle is a right isosceles triangle, angles B and C are
Let the common angle be
Note that angle PAC is , thus angle APC is
. From there, we know that AC is
Note that ABP is , so from law of sines we have:
Dividing by 10 and multiplying across yields:
From here use the sin subtraction formula, and solve for
Substitute this to find that AC=, thus the area is
~SAHANWIJETUNGA
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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.