60 2 Algebra
Solution.An inductive argument based on the recurrence relation shows thatTn(x)is a
polynomial with integer coefficients. And sincean+a−n=Tn(a+a−^1 ), it follows that
this number is an integer.
191.Prove that forn≥1,
Tn+ 1 (x)=xTn(x)−( 1 −x^2 )Un− 1 (x),
Un(x)=xUn− 1 (x)+Tn(x),192.Compute then×ndeterminants
∣
∣∣
∣∣
∣∣
∣∣
∣∣
∣
∣
x 100 ··· 0
12 x 10 ··· 0
012 x 1 ··· 0
..
...
.
..
.
..
.
... ..
.
0000 ··· 1
0000 ··· 2 x∣∣
∣
∣∣
∣∣
∣∣
∣∣
∣∣
and∣∣
∣
∣∣
∣∣
∣∣
∣∣
∣∣
2 x 100 ··· 0
12 x 10 ··· 0
012 x 1 ··· 0
..
...
.
..
.
..
.
... ..
.
0000 ··· 1
0000 ··· 2 x∣∣
∣
∣∣
∣∣
∣∣
∣∣
∣∣
.
193.Prove Chebyshev’s theorem forn=4: namely, show that for any monic fourth-
degree polynomialP(x),
max
− 1 ≤x≤ 1
|P(x)|≥ max
− 1 ≤x≤ 1
∣
∣ 2 −^3 T 4 (x)∣
∣,
with equality if and only ifP(x)= 2 −^3 T 4 (x).194.Letrbe a positive real number such that^6
√
r+√ (^61) r=6. Find the maximum value
of^4
√
r−√ (^41) r.
195.Letα=^2 nπ. Prove that the matrix
⎛
⎜⎜
⎜⎜
⎜
⎝
11 ··· 1
cosα cos 2α ··· cosnα
cos 2α cos 4α ··· cos 2nα
..
...
.
... ..
.
cos(n− 1 )αcos 2(n− 1 )α···cos(n− 1 )nα⎞
⎟⎟
⎟⎟
⎟
⎠
is invertible.196.Find all quintuples(x,y,z,v,w)withx, y, z, v, w∈[− 2 , 2 ]satisfying the system
of equations
x+y+z+v+w= 0 ,
x^3 +y^3 +z^3 +v^3 +w^3 = 0 ,
x^5 +y^5 +z^5 +v^5 +w^5 =− 10.