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If \( x^3 + 2x^2 + 3x = 4 \), what is \[ x^3 \big(x^3 - 15x^2 -43x - 95\big) ? \]

A student notices that the roots of the equation \(x^2+bx+a=0\) are each one less than the roots of the equation \(x^2+ax+b=0\).

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Hi guys!

I know that many of you must be RMO aspirants, and are preparing tough for that. But even all of us know that RMO is not that easy ...

\[\large\displaystyle\dfrac{1}{(1)(2)(3)}+\dfrac{2}{(4)(5)(6)}+\dfrac{3}{(7)(8)(9)}+\dfrac{4}{(10)(11)(12)}+\ldots\]

Let \(\large S\) denote the value of ...

\[ \large1+\frac{2^{3}}{2}+\frac{3^{3}}{2^{2}}+\frac{4^{3}}{2^3}+ \ldots = \ ? \]

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