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Number Theory Level 5

10 y 2 − 2 b 2 − 8 b y = x \sqrt{10y^{2}-2b^{2}-8by} = x

y y is the repeated root(at least two roots are same) of the function y 3 − 12 y + c y^{3}-12y+c and b b is any number and c is positive

Let p p be total number of different integral values of x x ; (q) be sum of all different integral values of x x ; r r be number of integral values that b b can take to get an integral x x .

Find p + q + r + c p + q + r + c


The answer is 30.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

John Gilling
Sep 18, 2015

The statement of this problem isn't entirely clear. The polynomial y 3 − 12 y + c y^{3}-12y+c has a repeated root at y = 2 y=2 if c = 16 c=16 , but can also have a repeated root at y = − 2 y=-2 if c = − 16 c=-16 . I recommend that you insist upon c being positive in order to ensure a unique solution here.

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Thanks I have edited it.

A Former Brilliant Member - 5 years, 8 months ago

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