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The prism with the regular octagonal base and with all edges of the length equal to

1

is given. The points

M_{1},M_{2},\cdots,M_{10}

are the midpoints of all the faces of the prism. For the point

P

from the inside of the prism denote by

P_{i}

the intersection point (not equal to

M_{i}

) of the line

M_{i}P

with the surface of the prism. Assume that the point

P

is so chosen that all associated with

P

points

P_{i}

do not belong to any edge of the prism and on each face lies exactly one point

P_{i}

. Prove that

\sum_{i=1}^{10}\frac{M_{i}P}{M_{i}P_{i}}=5

Problem Statement