1

Part of 1997 Putnam

Problems(2)

Source:

HOMF

HO=11

OM=5

\Delta ABC

H

as orthocentre,

O

as circumcentre,

M

be the midpoint of

BC

F

is the feet of altitude from

A

. What is the length of

BC

?
[asy] unitsize(0.3 cm); pair F, H, M, O; F = (0,0); H = (0,5); O = (11,5); M = (11,0); draw(H--O--M--F--cycle); label("

F

", F, SW); label("

H

", H, NW); label("

M

", M, SE); label("

O

", O, NE); [/asy]

Putnamgeometryrectanglegeometric transformationreflectioncircumcircleanalytic geometry

Source:

x

\{x\}

to be the difference between

x

and the closest integer to

x

. For each positive integer

n

S_n=\sum_{m=1}^{6n-1}\min \left(\left\{\frac{m}{6n}\right\},\left\{\frac{m}{3n}\right\}\right)

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