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Geometry Difficulty 4.9 AIME Prove it Thailand

Prove that
a2+b2−2ab+b2+c2−2bc≥a2+c2 \sqrt{a^2 + b^2 - \sqrt{2} ab} + \sqrt{b^2 + c^2 - \sqrt{2} bc} \geq \sqrt{a^2 + c^2}
for all positive real numbers aa, bb and cc.

Solution

The inequality results from the triangle inequality,
PQ+PR≥QRPQ + PR \ge QR, as shown in the figure.

Figure 1

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