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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