Is it distinct 3? (Are the Solutions to this Problem Distinct?)

Algebra Level 4

Let $⋆$ be the number of positive integral solutions $(x,y,z)$ for this equation:

$xyz+xy+yz+zx+x+y+z=1000 \ \cdot$

What is the value of:

$25 \cdot \left( \lim_{ \alpha \rightarrow 0} \dfrac{ \sin( \alpha ⋆) + \sin( \alpha) + \sin \left( \dfrac{\alpha}{⋆}\right) + \sin \left( \dfrac{\alpha}{⋆ ^2}\right) + \cdots }{ \alpha } \right) \ ?$

Assume that $0$ is not a positive integer.

For more problems like this, try answering this set .

The answer is 180.

1 solution

Christian Daang
Jul 23, 2017

$xyz + xy + yz + zx + x + y + z + 1 = (x+1)(y+1)(z+1) = 1001 = 7(11)(13) \implies ⋆ = 3! = 6$

$\begin{aligned} \therefore \displaystyle 25 \cdot \lim_{ \alpha \rightarrow 0} \dfrac{ \sin( \alpha ⋆) + \sin( \alpha) + \sin \left( \dfrac{\alpha}{⋆}\right) + \sin \left( \dfrac{\alpha}{⋆ ^2}\right) + \cdots }{ \alpha } \\ &= 25 \cdot \lim_{ \alpha \rightarrow 0} \left[ ⋆\cos( \alpha ⋆) + \cos( \alpha) + \dfrac{1}{⋆}\cos \left( \dfrac{\alpha}{⋆}\right) + \dfrac{1}{⋆ ^2}\cos \left( \dfrac{\alpha}{⋆ ^2}\right) + \cdots \right] \\ &= 25 \cdot \left( ⋆ + 1 + \dfrac{1}{⋆} + \dfrac{1}{⋆ ^2} + \cdots\right) = 25 \cdot \dfrac{⋆}{1 - \dfrac{1}{⋆}} \\ &= 25 \cdot \dfrac{⋆ ^2}{⋆ - 1} \\ &= 25 \cdot \dfrac{36}{5} = \boxed{180} \end{aligned}$

FYI It is generally better to focus the problem. IE ask directly for the number of solutions, instead of using this number in a subsequent problem.

Calvin Lin Staff - 3 years, 10 months ago

Sorry sir, I just want to put some segways in the problem.

Christian Daang - 3 years, 10 months ago

Is it distinct 3? (Are the Solutions to this Problem Distinct?)

For more problems like this, try answering this set .

The answer is 180.

1 solution

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