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Math Problems Set 8
Discussion: Oct. 25, Oct. 27 (on polynomials and floor
functions) The name after
the problem is the designated writer of the solution of that problem . (Beth,
Nicholas, and
Frank are exempted this week)
Discussion Problems
1. (a) Factor the polynomial x^{8} +98x^{4} +1 into two factors with integer (not
necessarily
real) coefficients.
(b) Find the remainder on dividing x ^{100}  2x^{51} + 1 by x^{2}
 1. (Shelley)
(Hint: (a) 98 = 100  2; (b) Bezout’s theorem)
2. If and are the zeros of the polynomial x^{2}
 6x + 1, then for every
nonnegative
integer n,
is an integer and not divisible by 5. (Derek) (Hint: how about
induction?)
3. (VA 1982) Let p(x) be a polynomial of the form p(x) = ax^{2} + bx + c, where a,
b and
c are integers, with the property that 1 < p(1) < p(p(1)) < p(p(p(1))). Show
that
a ≥ 0. (Brett) (Hint: by contradiction)
4. (VA 1987) A sequence of polynomials is given by
,
for
n ≥ 0, where and, for n ≥ 0,
. Denote by
and
the roots of , with
. Find
and . (Ben) (Hint:
think about and
.)
5. (VA 1991) Prove that if α is a real root of (1x^{2})(1+x+x^{2}+· · ·+x^{n})x = 0 which
lies
in (0, 1), with n = 1, 2, · · · , then is also a root of
.
(Lei) (Hint: use
6. (VA 1996) Let , i = 1, 2, 3, 4, be real numbers such that
.
Show that for arbitrary real numbers , i = 1,
2, 3, the equation
has at least one real root which is on the
interval
1 ≤ x ≤1. (Tina) (Hint: think integral)
7. (VA 1995) Let
. Show that
for every positive integer
n. Here [r] denotes the largest integer that is not larger than r. (David Rose)
(Hint:
prove ≥ and ≤ both hold.)
8. Solve the equation z ^{8} + 4z^{6}  10z^{4} + 4z^{2} + 1 = 0. (Lei) (Hint: divide it by
z^{4}, and
observe the symmetry)
9. (Putnam 2004B1) Let
be a polynomial with
integer
coefficients. Suppose that r is a rational number such that P(r) = 0. Show that
the
n numbers
are integers. (Davis Edmonson)
10. (Putnam 2003B1) Do there exist polynomials a(x),
b(x), c(y), d(y) such that
1 + xy + x^{2}y^{2} = a(x)c(y) + b(x)d(y)
holds identically? (Richard)
More Problems:
1. If a and b are two solutions of x^{4} x^{3} 1 = 0, then ab is a solution of x^{6}
+x^{4} +x^{3} 
x^{2}  1 = 0.
2. Suppose that a, b, c are distinctive integers. Prove
for any x ∈ R.
3. (VA 1997) Suppose that and
. If and
are roots of x^{4}
 x^{3} +
ax^{2}  8x  8 = 0, find and a. (Do not assume that they are real numbers.)
4. (VA 1991) Let f(x) = x^{5}  5x^{3} + 4x. In each part (i)–(iv), prove or disprove
that
there exists a real number c for which f(x)  c = 0 has a root of multiplicity
(i) one,
(ii) two, (iii) three, (iv) four.
5. (VA 1985) Let , where the coefficients
are real.
Prove that
p(x) = 0 has at least one root in the interval 0 ≤ x ≤1 if
.
6. (VA 1989) Let a,b, c,d be distinct integers such that the equation (x  a)(x
 b)(x 
c)(xd)9 = 0 has an integer root r. Show that 4r = a+b+c+d. (This is
essentially
a problem from the 1947 Putnam examination.)
7. (VA 1988) Find positive real numbers a and b such that f(x) = ax  bx^{3} has
four
extrema on [1, 1], at each of which f(x) = 1.
8. (VA 1987) Let p(x) be given by and let p(x)
≤ x
on [1, 1]. (a) Evaluate . (b) Prove that
.
9. (VA 1990) Suppose that P(x) is a polynomial of degree 3 with integer
coefficients and
that P (1) = 0, P(2) = 0. Prove that at least one of its four coefficients is
equal to or
less than 2.
10. (Putnam 2004A4) Show that for any positive integer n, there is an integer N
such
that the product can be expressed identically in the form
where the are rational numbers and each is one of the numbers 1, 0, 1.
11. (Putnam 2003A4) Suppose that a, b, c, A,B,C are real
numbers, a ≠ 0 and A ≠ 0,
such that
ax^{2} + bx + c ≤ Ax^{2} + Bx + C
for all real numbers x. Show that
b^{2}  4ac ≤ B^{2}  4AC.
12. (Putnam 2003B1) Do there exist polynomials a(x), b(x), c(y), d(y) such that
1 + xy + x^{2}y^{2} = a(x)c(y) + b(x)d(y)
holds identically?
13. (Putnam 2003B4) Let
where a, b, c, d, e are integers, a ≠ 0. Show that if
is a rational
number and
, then is a rational number.
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