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glyphomouni

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    Macedonia
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    high school student

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Lepton

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  1. If [math]x^2-2ax+b=0[/math] ,[math]b\neq 0[/math] and has two unequal real roots then show that [math]6a^4-2a^3-2a^2b+a^2-6a^2b+2ab+2b^2-b\geq 0[/math] I know that since the equation has two unequal real roots then [math]4a^2-4b>0[/math] but then how does this figure in the above inequality??
  2. Thank you . But the frame within which we work is high school level. Thus although we know a little bit of vectors we have not done the Cauchy–Schwarz inequality. Anyway thanks for the reference i will look it up
  3. Before i learn your proof ,there is a thing that i am worried about. if we prove [math]\frac{x^4+y^4+z^4}{x^2y^2+x^2z^2+y^2z^2}\geq 1>\frac{3m+2n}{3(m+n)}[/math] that implies [math]\frac{x^4+y^4+z^4}{x^2y^2+x^2z^2+y^2z^2}>\frac{3m+2n}{3(m+n)}[/math] and not [math]\frac{x^4+y^4+z^4}{x^2y^2+x^2z^2+y^2z^2}\geq\frac{3m+2n}{3(m+n)}[/math] . Is that right??
  4. Find the sign of the following polynomial (i.e whether is greater or less than zero) [math]x^4+x^3(a+b)+x^2(a+ab+b)+x(a^2+b^2) +ab[/math],given that [math]|a|<\sqrt{b}[/math] and [math]|b|<\sqrt{a}[/math] Since [math] x^4 ,x^2[/math] are positive if we can show that (a+ab+b)>0,ab>0 and also [math]x^3(a+b)-x(a^2+b^2)>0[/math] are we done??
  5. Thanks . In that case because 2m<3m ,then 2m+3n<3(m+n) or [math]\frac{2m+3n}{3(m+n)}<1\leq\frac{x^4+y^4+z^4}{x^2y^2+x^2z^2+y^2z^2}[/math]. I am worried about the equality sign, because [math]A<1\leq B[/math] =>A<B. Is it easy to prove [math]\frac{x^4+y^4+z^4}{x^2y^2+x^2z^2+y^2z^2}\geq 1[/math]??
  6. Thank you all ,but could you be more specific? R.H.S is a Natural No ,but what manipulations can make the ratio:[math]\frac{x^4+y^4+z^4}{x^2y^2+x^2z^2+y^2z^2}[/math] such a complicated Natural No??. Why not a whole Natural No?? I have to hand in that exercise next Monday and i am still in the dark. Thanx
  7. Thanks i will try in that direction
  8. Here is another problem that made me land on my head: if [math]sec\frac{A}{2}sec\frac{B}{2}sec\frac{C}{2}+sec\frac{A}{2}+sec\frac{B}{2}+sec\frac{C}{2} -2=0[/math] ,then one of the angles (A,B,C) of the triangle ABC is 90 degrees. How do we start??
  9. I was trying to prove the following inequality and really got stuck The inequality is. [math]\frac{x^4+y^4+z^4}{x^2y^2+x^2z^2+y^2z^2}\geq\frac{2m+3n}{3(m+n)}[/math] for all the reals x,y,z different to zero and for all the natural Nos m,n Can we use induction here
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