# Introduction to Inequalities

Our study starts with the following fundamental result:

Theorem 1.

If $$x$$ is a real number, then $$x^2\geq 0$$. The equality holds if and only if $$x=0$$.

No proofs will be omitted in this text. Except for this one. We will assume that the reader has mature enough to not multiply inequality by a negative number.

We continue our exposition with the first consequence of the previous fact:

Theorem 2.

If $$a,b\in \mathbb R$$ then: \begin{eqnarray*} a^2+b^2\geq 2ab. \quad\quad\quad\quad\quad (1) \end{eqnarray*} The equality holds if and only if $$a=b$$.

Problem 1.

Prove the inequality $$a^2+b^2+c^2\geq ab+bc+ca$$, if $$a,b,c$$ are real numbers.

Problem 2.

Find all real numbers $$a, b, c$$, and $$d$$ such that $a^2+b^2+c^2+d^2=a(b+c+d).$

Problem 3.

If $$a,b,c$$ are positive real numbers that satisfy $$a^2+b^2+c^2=1$$, find the minimal value of $S=\frac{a^2b^2}{c^2}+\frac{b^2c^2}{a^2}+\frac{c^2a^2}{b^2}.$

Problem 4.

If $$x$$ and $$y$$ are two positive numbers less than $$1$$, prove that $\frac1{1-x^2}+\frac1{1-y^2}\geq \frac2{1-xy}.$

Problem 5.

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

Problem 6.

If $$a,b,c$$ are positive real numbers that satisfy $\frac{a^2}{1+a^2}+\frac{b^2}{1+b^2}+\frac{c^2}{1+c^2}=1,$ prove that $$\left|abc\right|\leq \frac1{2\sqrt 2}$$.

Problem 7. (Nesbit’s inequality)

If $$a$$, $$b$$, $$c$$ are positive real numbers prove that $\frac a{b+c}+\frac b{c+a}+\frac c{a+b}\geq \frac32.$

Theorem 3.

If $$a$$, $$b$$, $$c$$ are non-negative real numbers then \begin{eqnarray}\frac{a^3+b^3+c^3}3\geq abc.\end{eqnarray} The equality holds if and only if $$a=b=c$$.

Problem 8.

If $$a$$ and $$b$$ are positive real numbers, prove that $$2a^3+b^3\geq 3a^2b$$.

Problem 9.

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

Problem 10.

If $$a,b,c,d> 0$$, prove that $\frac a{b+c}+\frac b{c+d}+ \frac c{d+a}+ \frac d{a+b}\geq 2.$

Problem 11.

Prove that $\frac{a^3}{a^2+ab+b^2}+ \frac{b^3}{b^2+bc+c^2}+ \frac{c^3}{c^2+ca+a^2} \geq \frac{a+b+c}3,$ for $$a,b,c> 0$$.

Problem 12.

If $$a_1,a_2, \dots, a_n,b_1,b_2,\dots, b_n$$ are two sequences of positive real numbers prove that $\frac{a_1b_1}{a_1+b_1}+\frac{a_2b_2}{a_2+b_2}+\cdots+\frac{a_nb_n}{a_n+b_n}\leq \frac{(a_1+\cdots+a_n)(b_1+\cdots+b_n)}{a_1+\cdots+a_n+b_1+\cdots+b_n}.$

Problem 13.

If $$a$$, $$b$$,$$c$$, $$d$$, $$e$$, $$f$$ are positive real numbers prove that $\sqrt{ab}+\sqrt{cd}+\sqrt{ef}\leq\sqrt{(a+c+e)(b+d+f)}.$

Problem 14.

If $$a$$, $$b$$, and $$c$$ are positive real numbers, prove that $\frac{5a^3-ab^2}{a+b}+\frac{5b^3-bc^2}{b+c}+\frac{5c^3-ca^2}{c+a}\geq 2(a^2+b^2+c^2).$

Problem 15.

Let $$n\geq 3$$ be an integer and let $$x_1, x_2, \dots, x_n$$ be non-negative real numbers such that $$x_1=0$$, $$x_n=1$$. Prove that there exists $$j\in\{1,2,\dots, n-1\}$$ for which: $|x_{j+1}+x_{j-1}-2x_j|\geq \frac4{n^2}.$

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