Advanced book on Mathematics Olympiad

788 Combinatorics and Probability

similarly forr 2. The distribution ofθis uniform on[ 0 ,π]. These three distributions are

independent; hence

E(A(OP Q))=

1

2

(∫ 1

0

2 r^2 dr

) 2 (

1

π

∫π

0

sinθdθ

)

=

4

9 π

,

and

E(A(OP Q)+A(OQR)+A(ORP ))=

4

3 π

.

We now treat the case in whichP′,Q′,R′lie on a semicircle in that order. Set

θ 1 =∠POQandθ 2 =∠QOR; then the distribution ofθ 1 ,θ 2 is uniform on the region

0 ≤θ 1 , 0 ≤θ 2 ,θ 1 +θ 2 ≤π.

In particular, the distribution onθ =θ 1 +θ 2 isπ^2 θ 2 on[ 0 ,π]. SetrP =OP,rQ =

OQ,rR=OR. Again, the distribution onrPis given by 2rPon[ 0 , 1 ], and similarly

forrQ,rR; these are independent of each other and the joint distribution ofθ 1 ,θ 2. Write

E′(X)for the expectation of a random variableXrestricted to this part of the domain.

Letχbe the random variable with value 1 ifQis inside triangleOPRand 0 otherwise.

We now compute

E′(A(OP R))=

1

2

(∫ 1

0

2 r^2 dr

) 2 (∫π

0

2 θ

π^2

sinθdθ

)

=

4

9 π

and

E′(χ A(OP R))=E′

(

2 A(OP R)^2

θ

)

=

1

2

(∫ 1

0

2 r^3 dr

) 2 (∫π

0

2 θ

π^2

θ−^1 sin^2 θdθ

)

=

1

8 π

.

Also, recall that given any triangleXY Z,ifTis chosen uniformly at random insideXY Z,
the expectation ofA(T XY )is the area of triangle bounded byXYand the centroid of
XY Z, namely,^13 A(XY Z).
Letχbe the random variable with value 1 ifQis inside triangleOPRand 0 otherwise.
Then

E′(A(OP Q)+A(OQR)+A(ORP )−A(P QR))

= 2 E′(χ (A(OP Q)+A(OQR))+ 2 E′(( 1 −χ )A(OP R))

= 2 E′(

2

3

χ A(OP R))+ 2 E′(A(OP R))− 2 E′(χ A(OP R))