University Maths Solution

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Maths Question
Question 1

$\begin{align}  & \text{if }a,b\text{ are real and unequal show that} \\ & \text{     }{{a}^{2}}+{{b}^{2}}>2ab \\ & \text{Deduce that }a,b\text{ and }c\text{ are real and unequal, then} \\ & {{(a+b+c)}^{2}}>3(ab+bc+ca) \\\end{align}$

Question 2

$\text{If }a,b,c\text{ and }d\text{ are positive, prove that (}a+b+c+d\text{)}\left( \tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}+\tfrac{1}{d} \right)\ge 16$

Question 3

$\text{Show that }{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\ge ab+bc+ca\text{ for all real values of }a,b,c$

Question 4

$\text{Show that }a+\frac{1}{a}>2$

Question 5

$\text{Show that }\left( a+b \right)\left( \frac{1}{a}+\frac{1}{b} \right)>4$

Question 6

$\text{Show that }(a+b)(b+c)(a+c)\ge 8abc$