(x+y)/(a+1) = (y+z)/(a+2) = (z+x)/(a+3) = k
x+y = k(a+1) (1)
y+z = k(a+2) (2)
z+x= k(a+3) (3)
din (3) -(2) ⇒ x - y = k
x + y = k(a+1) ⇒ 2x = k + ka + k = k(a+2) x = k(a+2)/2
y = x - k = k(a+2)/2 - k = ka/2 x/y = (a+2)/a
din (2) - (1) ⇒ z - x = k
z + x = k(a+3) 2z = k(a+4) z = k(a+4)/2
z/x = (a+4)/(a+2)
E = (x/y)² + (z/x)² = (a+2)² /a² + (a+4)² /(a+2)²
a minim = 1 E = 9 + 25/9 = 11,(7) E≤ 11,(7)
x = k(a+2)/2 y = ka/2 z = k(a+4)/2 = k(a+2)/2 + k
x = y + k z = x + k
x/y = 1+ k/y (x/y)² = (1+k/y)²
z/x = 1 + k/x (z/x)² = (1+k/x)²
(x/y)² + (z/x)² = (1+k/y)² + (1+k/x)² = 1 + 2k/y + k²/y² +1+2k/x +k²/x² =
= 2 + 2k(1/x + 1/y) + 2k² (1/x² + 1/y²) > 2
