Using Vieta's Formula

#MathOlympiadProblem #UniversalLanguage #Math

[universallanguage]

The correct answer is 753, instead of 751..Sorry guys it's typo error. 😬

[BRUBRUETNONO]

Thanks for your interesting problem.
Let me show below how I solved it.
Of course, I didn't look at your solution.
Tell me if you like my solution.
Keep up the good work and greetings !

Let (i) 8x^3+1001x+2008=0 with its three roots r, s, t
and a=(r+s)^3+(s+t)^3+(t+r)^3

From the Vieta's formula or the symetric functions of the roots,
(ii) r+s+t=-0/8=0
(iii) rs+st+tr=1001/8
(iv) rst=-2008/8=-251

From (ii), we have
(v) (s+t)=-r, (t+r)=-s and (r+s)=-t
Then we can write

Recall
Then again from (v) and (ii) we can write

(0)^3=(r^3+s^3+t^3)-3rst
So we have -(r^3+s^3+t^3)=-3rst

Then as written above


To conclude
If (r, s, t) are the roots of 8x^3+1001x+2008=0,
then (r+s)^3+(s+t)^3+(t+r)^3 take the value 753.
END

[mathmurthy]

last step must be 753. It is minor calculation eeror