Algebra 407190.Letnbe the degree ofP(x). Suppose that we can find polynomials with integer
coefficientsR 1 (x)andR 2 (x)of degree at most 2n−1 such thatQ(x) =P(x^2 ) =
R 1 (x)R 2 (x). Then we also haveQ(x) =Q(−x)=R 1 (−x)R 2 (−x). LetF(x)be
the greatest common divisor ofR 1 (x)andR 1 (−x). SinceF(x)=F(−x), we can write
F(x)=G(x^2 )with the degree ofG(x)at mostn−1. SinceG(x^2 )dividesQ(x)=P(x^2 ),
we see thatG(x)dividesP(x)and has lower degree; hence by the irreducibility of
P(x),G(x)is constant. Similarly, the greatest common divisor ofR 2 (x)andR 2 (−x)is
constant. HenceR 1 (−x)dividesR 2 (x), whileR 2 (x)dividesR 1 (−x). HenceR 1 (x)and
R 2 (x)both have degreen,R 2 (x)=cR 1 (−x), andQ(x)=cR 1 (x)R 1 (−x). Because
P(x)is monic, we computec=(− 1 )nandP( 0 )=(− 1 )nR 1 ( 0 )^2. Hence|P( 0 )|is a
square, contradicting the hypothesis.
(Romanian Team Selection Test for the International Mathematical Olympiad, 2003,
proposed by M. Piticari)
191.These are just direct consequences of the trigonometric identities
cos(n+ 1 )θ=cosθcosnθ−sinθsinnθand
sin(n+ 1 )θ
sinθ=cosθ
sinnθ
sinθ+cosnθ.192.Denote the second determinant byDn. Expanding by the first row, we obtain
Dn= 2 xDn− 1 −∣
∣∣
∣∣
∣∣
∣
∣∣
∣
110 ··· 0
02 x 1 ··· 0
012 x··· 0
..
...
.
..
.
... ..
.
000 ··· 2 x∣
∣∣
∣∣
∣∣
∣
∣∣
∣
= 2 xDn− 1 −Dn− 2.SinceD 1 = 2 xandD 2 = 4 x^2 −1, we obtain inductivelyDn=Un(x),n≥1. The same
idea works for the first determinant, except that we expand it by the last row. With the
same recurrence relation and with the valuesxforn=1 and 2x^2 −1 forn=2, the
determinant is equal toTn(x)for alln.
193.LetP(x)=x^4 +ax^3 +bx^2 +cx+dand denote byMthe maximum of|P(x)|on
[− 1 , 1 ]. From−M≤P(x)≤M, we obtain the necessary condition−M≤^12 (P (x)+
P(−x))≤Mforx∈[− 1 , 1 ]. With the substitutiony=x^2 , this translates into
−M≤y^2 +by+d≤M, fory∈[ 0 , 1 ].For a monic quadratic function to have the smallest variation away from 0 on[ 0 , 1 ],it
needs to have the vertex (minimum) at^12. The variation is minimized by(y−^12 )^2 −^18 ,
and so we obtainM≥^18. Equality is attained for^18 T 4 (x).