Giải thích các bước giải:

2.Ta có:

$(a-b)^2+(b-c)^2+(c-a)^2\ge 0$

$\to a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2\ge 0$

$\to 2(a^2+b^2+c^2)\ge 2(ab+bc+ca)$

$\to 3(a^2+b^2+c^2)\ge a^2+b^2+c^2+2(ab+bc+ca)$

$\to 3(a^2+b^2+c^2)\ge (a+b+c)^2$

$\to \dfrac{a^2+b^2+c^2}3\ge(\dfrac{a+b+c}3)^2$

3.Ta có:

$(\dfrac12a-b)^2+(\dfrac12a-c)^2+(\dfrac12a-d)^2+(\dfrac12a-e)^2\ge 0$

$\to \frac{a^2}{4}-ab+b^2+\frac{a^2}{4}-ac+c^2+\frac{a^2}{4}-ad+d^2+\frac{a^2}{4}-ea+e^2\ge 0$

$\to a^2+e^2-ea+b^2-ab+c^2-ac+d^2-ad\ge 0$

$\to a^2+b^2+c^2+d^2+e^2\ge a(b+c+d+e)$

4.Ta có:

$a^4+b^4+c^4+d^4$

$\ge 2a^2b^2+2c^2d^2$

$\ge 2((ab)^2+(cd)^2)$

$\ge 2\cdot 2ab.cd$

$\ge 4abcd$