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Results tagged with geometric-probability
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user 11919
17
votes
Accepted
Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and th...
The conjecture is true, and it can be verified with fedja's method developed for your earlier question. We present a simplified version of the method. The idea is to scale the triangle so that one of …
- 105k
11
votes
Accepted
If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{...
Both conjectures are true. The proof below uses fedja's method in a simplified form, and also a nice observation by Zacky (see the comments below this post).
Since $ax+by\geq c$ is equivalent to $b\ge …
- 105k
7
votes
Accepted
The vertices of a triangle are three random points on a unit circle. The side lengths are, i...
One can use basic probability theory to prove that
$$P(ab<kc)=\frac{2}{\pi}\arctan k,\qquad k>0.$$
Without loss of generality, the vertices opposite the sides $a,b,c$ are
$$A=e^{2i\beta},\qquad B=e^{- …
- 105k