MathDB

2016-2017 Fall OMO Problem 25

Source:

November 16, 2016

Online Math Open

Problem Statement

Let

X_1X_2X_3

be a triangle with

X_1X_2 = 4, X_2X_3 = 5, X_3X_1 = 7,

and centroid

G

. For all integers

n \ge 3

, define the set

S_n

to be the set of

n^2

ordered pairs

(i,j)

such that

1\le i\le n

and

1\le j\le n

. Then, for each integer

n\ge 3

, when given the points

X_1, X_2, \ldots , X_{n}

, randomly choose an element

(i,j)\in S_n

and define

X_{n+1}

to be the midpoint of

X_i

and

X_j

. The value of

\sum_{i=0}^\infty \left(\mathbb{E}\left[X_{i+4}G^2\right]\left(\dfrac{3}{4}\right)^i\right)

can be expressed in the form

p + q \ln 2 + r \ln 3

for rational numbers

p, q, r

. Let

|p| + |q| + |r| = \dfrac mn

for relatively prime positive integers

m

and

n

. Compute

100m+n

.
Note:

\mathbb{E}(x)

denotes the expected value of

x

.
Proposed by Yang Liu

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