Áp dụng bđt Bunhia cho các số dương $(a^4+b^4+c^4+d^4)(a^2+b^2+c^2+d^2) \geq (a^3+b^3+c^3+d^3)^2(1)$
$(a^3+b^3+c^3+d^3)(a+b+c+d) \geq(a^2+b^2+c^2+d^2)^2(2)$
$(a^2+b^2+c^2+d^2).4 \geq (a+b+c+d)^2(3)$
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Từ $(1),(2),(3)$
$\Rightarrow \frac{a^4+b^4+c^4+d^4}{a^3+b^3+c^3+d^3}\geq \frac{a^3+b^3+c^3+d^3}{a^2+b^2+c^2+d^2} \geq \frac{a^2+b^2+c^2+d^2}{a+b+c+d} \geq \frac{a+b+c+d}{4}$
$\Rightarrow P \geq \frac{3}{4}$