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`qquad` Решите в целых числах уравнение: `quad x^3-y^3=2xy+8.`

`qquad`1. `quad {(x^3-y^3=2xy+8),(x; quad y in ZZ):} quad iff quad [({(x=y),(x^2=-4),(x; quad y in ZZ):}),({((x-y)(x^2+xy+y^2)=2(xy+4)),(|x-y|=1),(x^2+xy+y^2 equiv 1 (mod 2)),(x; quad y in ZZ):}),({((x-y)(x^2+xy+y^2)=2(xy+4)),(|x-y| ge 2),(x^2+xy+y^2 le |xy+4|), (x; quad y in ZZ):}):} quad iff quad {((x-y)(x^2+xy+y^2)=2(xy+4)),(|x-y| ge 2),(x^2+y^2 le 4), (x; quad y in ZZ):} quad iff quad`

`qquad qquad iff quad {((x-y)(x^2+xy+y^2)=2(xy+4)),(|x-y| ge 2),(|x| le 2),(|y| le 2), (x; quad y in ZZ):} quad iff quad [({(x=2),(y=0):}),({(x=0),(y=-2):}):} quad.`

`qquad` Решите в действительных числах неравенство: `quad (x^9-81x-62)/(x^3) le 18*root(3)(3x+2).`

`qquad`1. `quad f(x)=(x^3-2)/3, quad g(x)=f^(-1)(x)=root(3)(3x+2).`

`qquad`2. `quad (x^9-81x-62)/(x^3) le 18*root(3)(3x+2) quad iff quad ((x^3/3)^3-(2/3)^3-2/3*x^3*root(3)(3x+2)-(3x+2))/(x^3) le 0 quad iff quad`

`qquad iff quad (((x^3-2)/3)^3+3*x^3/3*2/3*((x^3-2)/3)-2/3*x^3*root(3)(3x+2)-(3x+2))/x le 0 quad iff quad (f^3-g^3+2/3*x^3*(f-g))/x le 0 quad iff `

`qquad iff quad ((f-g)(f^2+fg+g^2+2/3*x^3))/x le 0 quad iff quad ((f-g)(f^2+fg+g^2+2f+4/3))/x le 0 quad iff quad ((f-g)(f^2+f(g+2)+g^2+4/3))/x le 0 quad iff `

`qquad iff quad ((f-g)((2f+g+2)^2+3(g-2/3)^2))/(4x) le 0 quad iff quad (f-g)/x le 0 quad iff quad (f(f(x))-f(f^(-1)(x)))/x le 0 quad iff quad (f(f(x))-x)/x le 0 quad iff `

`qquad iff quad (f(x)-x)/x le 0 quad iff quad (x^3-3x-2)/x le 0 quad iff quad ((x+1)^2(x-2))/x le 0 quad iff quad [(0 lt x le 2),(x=-1):} quad . `