2

Part of 2007 Turkey Team Selection Test

Problems(2)

Source: turkey TST 2007

Two different points

A

B

\omega

that passes through

A

B

P

is a variable point on

\omega

(different from

A

B

M

is a point such that

MP

is the bisector of the angle

\angle{APB}

M

lies outside of

\omega

MP=AP+BP

. Find the geometrical locus of

M

ratiogeometrysimilar trianglesangle bisectorgeometry proposed

Source: turkey 2007 TST

n

is satisfying the conditions below i)

n

is a positive odd integer; ii) there are some odd integers such that their squares' sum is equal to

n^{4}

. Find all such numbers.

number theory proposednumber theory