Unknown

7.5. Other Problems 227

(b)

(u - byyu - c) ’ (b - ai;b - c) ’ (c - uy;c - b)

= (a + b + c)~ - 2(a + b + c)(bc + cu + ub) + abc.

3. Let

i=l

Sk(n) = &l(i) (k 2 2).

i=l

Show that Sk(n) = n(n + 1).. .(n + k)(2n + k)/(k + 2)!.

4. Let m and n be integers with m > n > 0, and let c be any constant.
   Define
   f(m) = e(-1)’ ( y ) (m - k + c)“.
   k=O
   Show that f(m) = 0 if m > n and that f(n) = n!.


5. Let -1 5 u < 1. Determine the smallest number K, which satisfies
   the condition
   IS’WI 5 K”
   whenever g(t) is a polynomial such that degg(t) 2 2 and lg(t)l 5 1
   for -1 5 t 5 1.


6. Let f(x) = (x - xr)...(x - x,) for -1 5 ti 5 1. Prove that there
   cannot exist numbers a, b for which

(9 Iml 2 1

(ii) If(b)1 L 1

(iii) -1 < a < 0 < b < 1.

7. Let ni (0 5 i 5 k) be any k + 1 integers for which no < n1 < n2 <
   ... < nk. Show that

rI (nrni)

o<i<jlk (j - j)

is an integer.

8. Construct, with proof that the construction works, a polynomial p(x)
   over Z such that
   Ip(x) - 0.51 < l/1981
   for each 2 for which 0.19 5 x 5 0.81.