2017 AIME I Problems/Problem 8
Problem 8
Two real numbers 
and 
are chosen independently and uniformly at random from the interval 
. Let 
and 
be two points on the plane with 
. Let 
and 
be on the same side of line 
such that the degree measures of 
and 
are and respectively, and 
and 
are both right angles. The probability that 
is equal to 
, where 
and 
are relatively prime positive integers. Find 
.
Solution 1
Noting that and are right angles, we realize that we can draw a semicircle with diameter 
and points and on the semicircle. Since the radius of the semicircle is 
, if 
, then 
must be less than or equal to 
.
This simplifies the problem greatly. Since the degree measure of an angle on a circle is simply half the degree measure of its subtended arc, the problem is simply asking:
Given 
such that 
, what is the probability that 
?
Through simple geometric probability, we get that 
.
The answer is 
~IYN~
Solution 2 (Trig Bash)
Put 
and 
with on the origin and the triangles on the 1st quadrant
the coordinates of and is 
, 
. So 
= 
, which we want to be less then 
.
So
$(200 cos^{2} a - 200cos^{2} a)^{2} +(200 cos a*sin a - 200 cos b * sin b)^{2} ≤ 100$ (Error compiling LaTeX. Unknown error_msg) $(200 cos^{2} a - 200cos^{2} a)^{2} +(200 cos a*sin a - 200 cos b * sin b)^{2} ≤$ (Error compiling LaTeX. Unknown error_msg)
See Also
| 2017 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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