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 \( a^2+b^2+c^2=1 \), prove that \[ \frac{a^2b^2}{c^2}+\frac{b^2c^2}{a^2}+\frac{c^2a^2}{b^2}\geq 2.\]

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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