# Homework 8: Inequalities

Problem 1

If $$a$$ and $$b$$ are positive real number prove that $a^3+b^3\geq ab(a+b).$

Problem 2

If $$a> 0$$ is a real number, prove that $$a^3+11> 9a$$.

Problem 3

If $$\alpha$$, $$\beta$$, and $$\gamma$$ are angles of a triangle, prove that $\sin\alpha+\sin\beta+\sin\gamma\leq \frac{3\sqrt3}2.$

Problem 4

If $$a_1, \dots, a_n$$ are positive real numbers, prove that $\frac{a_1}{a_2+\cdots+ a_n}+\frac{a_2}{a_1+a_3+\cdots + a_{n}}+\cdots + \frac{a_n}{a_1+\cdots+ a_{n-1}}\geq \frac{n}{n-1}.$

Problem 5

If $$a$$, $$b$$, $$c$$ are positive real numbers prove that $(a+b-c)(b+c-a)(c+a-b)\leq abc.$

Problem 6

If $$a$$, $$b$$, $$c$$ are positive real numbers, prove that $2a^3+2b^3+2c^3+a^2b+b^2c+c^2a\geq 3ab^2+3bc^2+3ca^2.$

Problem 7

Assume that $$a$$, $$b$$, $$c$$, and $$d$$ are positive real numbers such that $$abcd\geq 1$$. Prove that $\frac{a}{b+c+2}+\frac{b}{c+d+2}+\frac{c}{d+a+2}+\frac{d}{a+b+2}\geq 1.$

Problem 8

Assume that $$a$$, $$b$$, $$c$$ are positive real numbers such that $$a+b+c=1$$. Prove that $\frac1{bc+a+\frac1a}+\frac1{ca+b+\frac1b}+\frac1{ab+c+\frac1c}\leq \frac{27}{31}.$

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