If $x$ and $y$ are positive integers and $x\ne y$, prove that \(xy\left( x+y \right)

Question

If $x$ and $y$ are positive integers and $x\ne y$, prove that $xy\left( x+y \right)<{{x}^{3}}+{{y}^{3}}$. If $a$, $b$, $c$ and $d$ are real numbers such that $a+b+c+d=1$, prove that ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}\ge \frac{1}{4}$.

Accepted Answer

(a) $x^3+y^3-xy(x+y)=(x+y)(x-y)^2>0$. (b) $a^2+b^2+c^2+d^2\ge\frac14$, with equality when $a=b=c=d=\frac14$.

IP3U2 HOT PN1 Q7

IP3U2 HOT PN1 Q7

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