1

Part of 1986 Iran MO (2nd round)

Problems(2)

Find the limit of the function - [Iran Second Round 1986]

Source:

f

be a function such that

f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.

Find the limit of

f

x_0=1.

functiontrigonometrylimitalgebra unsolvedalgebra

Midpoint is a Fixed Point - [Iran Second Round 1986]

Source:

O

is a point in the plane. Let

O'

be an arbitrary point on the axis

Ox

of the plane and let

M

be an arbitrary point. Rotate

M

90^\circ

clockwise around

O

to get the point

M'

M

90^\circ

anticlockwise around

O'

to get the point

M''.

Prove that the midpoint of the segment

MM''

is a fixed point.

rotationgeometrygeometric transformationreflectionsymmetrygeometry proposed