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(1995 Russian Math Olympiad) Is it p ossible to nd three quadratic

p olynomials f (x ) ; g (x ) ; h (x ) such that the equation f (g (h (x ))) = 0 has

the eight ro ots 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8?

2

Jawabanmu

2014-03-19T16:56:42+07:00
Solution   . Supp  ose there  are such  f ; g ; h: Then h (1); h (2); : : : ; h (8) will      b e the   ro ots of the  4-th  degree   p olynomial   f (g (x )) : Since h (a ) =      h (b) ; a =6 b if and only  if a; b are symmetric    with  resp ect to the  axis      of  the  parab  ola, it follows  that   h (1) =   h (8); h (2) =  h (7); h (3) =      h (6); h (4) =  h (5) and  the  parab  ola y  =  h (x ) is symmetric   with   re-      sp ect to  x =  9=2 : Also, we  have   either h (1) <  h (2) <  h (3) <  h (4) or      h (1) >  h (2) > h (3) >  h (4):            Now  g (h (1)); g (h (2)); g (h (3)); g (h (4)) are the ro ots of the quadratic      p olynomial   f (x ) ; so g (h (1)) = g (h (4)) and g (h (2)) = g (h (3)); which      implies  h (1) + h (4) = h (2) + h (3): For h (x ) = Ax2 + B x + C ; this would      force  A =  0 ; a contradiction.


2014-03-19T17:20:10+07:00
Solution   . Supp  ose there  are such  f ; g ; h: Then h (1); h (2); : : : ; h (8) will      b e the   ro ots of the  4-th  degree   p olynomial   f (g (x )) : Since h (a ) =      h (b) ; a =6 b if and only  if a; b are symmetric    with  resp ect to the  axis      of  the  parab  ola, it follows  that   h (1) =   h (8); h (2) =  h (7); h (3) =      h (6); h (4) =  h (5) and  the  parab  ola y  =  h (x ) is symmetric   with   re-      sp ect to  x =  9=2 : Also, we  have   either h (1) <  h (2) <  h (3) <  h (4) or      h (1) >  h (2) > h (3) >  h (4):            Now  g (h (1)); g (h (2)); g (h (3)); g (h (4)) are the ro ots of the quadratic      p olynomial   f (x ) ; so g (h (1)) = g (h (4)) and g (h (2)) = g (h (3)); which      implies  h (1) + h (4) = h (2) + h (3): For h (x ) = Ax2 + B x + C ; this would      force  A =  0 ; a contradiction.