MathLabs

Problem 4

Let a,b,ca,b,c be the side lengths of a triangle with a+b+c=1a+b+c=1, and let n≥2n\ge2 be an integer. Prove that an+bnn+bn+cnn+cn+ann<1+21/n2\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+\frac{2^{1/n}}2.
Step 1 of 5: Order the sides and bound the largest
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a≥b≥c,a<b+c,a+b+c=1  ⟹  b≤a<12a\ge b\ge c,\qquad a<b+c,\qquad a+b+c=1\implies b\le a<\frac12
Detailed analysis

Relabel the sides so that a≥b≥ca\ge b\ge c. The triangle inequality gives a<b+ca<b+c. Since a+b+c=1a+b+c=1, this implies 2a<12a<1, hence b≤a<1/2b\le a<1/2.