Practice problems on Triangle Inequality

Problem 1.

Let a, b, c be positive numbers such that

abc \leq \frac{1}{4} and \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}} <9.

Prove that there exists a triangle with side lengths a, b and c.

Problem 2.

Consider the inequality

a^{3}+b^{3}+c^{3} <k(a+b+c)(ab+bc+ca)

where a, b, c are the side lengths of a triangle and k is a real number.

(a) Prove the inequality when k=1.

(b) Find the least value of k such  that the inequality holds true for any triangle.

Problem 3.

Let a, b, c be positive real numbers. Prove that they are side lengths of a triangle if and only if

a^{2}pq+b^{2}qr+c^{2}rp<0

for any real numbers p, q, and r such that p+q+r=0 and pqr \neq 0.

Let me know if you need any help when you attempt it..

Nalin Pithwa

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