Putnam preparation (Table of contents)

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