by maxpeed Sun Aug 23, 2009 3:47 pm
hi
as for 1):
let's say x,y, z>2
if x^2+y^2=c, we can choose z^2 = c-1
-this condition complies with 1)-
c^2=x^4 + y^4 +2*x^2*y^2 => x^4 + y^4 = c^2 - 2*x^2*y^2
z^4= (c-1)^2 => z^4 = c^2 -2*c+1
let's check now:
x^4 + y^4 > z^4
c^2 - 2*x^2*y^2 > c^2 -2*c+1
2*(c-x^2*y^2)>1 ----but c=x^2+y^2
x^2+y^2 - x^2*y^2 > 1/2
x^2*y^2*(1/y^2+1/x^2-1) > 1/2
since both x and y are >2, the part in brackets is < 0, therefore this conditions is never true. 1) Not sufficient.
ex x=3, y=4, z=(24)^1/2 : 25>24 but 337<576
ex x=6, y=8, z=(99)^1/2 : 100>99 but 5392<9801
hope it helps.
though i dont know how they pretend students to find an answer within a couple of minutes...
mp