Maths Olympiad Prep

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Algebra Difficulty 6.4 National olympiad Prove it

8. Let a,b,ca, b, c be positive real numbers, prove that:
(b+c−a)2a2+(b+c)2+(c+a−b)2b2+(c+a)2+(a+b−c)2c2+(a+b)2⩾35\frac{(b+c-a)^{2}}{a^{2}+(b+c)^{2}}+\frac{(c+a-b)^{2}}{b^{2}+(c+a)^{2}}+\frac{(a+b-c)^{2}}{c^{2}+(a+b)^{2}} \geqslant \frac{3}{5}
(1997 Japan Mathematical Olympiad)

Solution

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