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 \[ \frac1{bc+a+\frac1a}+\frac1{ca+b+\frac1b}+\frac1{ab+c+\frac1c}\leq \frac{27}{31}.\]


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us