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JANGAN DIJAWAB ASALAN dan harus ada cara
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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?
ente berada di halaman yang benar. Kami mempunyai 2 jawaban dariJANGAN DIJAWAB ASALAN dan harus ada cara
jika tak bisa harus diam okay
(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?
. Silakan lihat jawaban selanjutnya di bawah ini: Jawaban: #1: 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.Jawaban: #2: 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.