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 x8 +98x4 +1 into two factors with integer (not
necessarily
real) coefficients.
(b) Find the remainder on dividing x 100 - 2x51 + 1 by x2
- 1. (Shelley)
(Hint: (a) 98 = 100 - 2; (b) Bezout’s theorem)
2. If 
and 
are the zeros of the polynomial x2
- 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) = ax2 + 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 (1-x2)(1+x+x2+· · ·+xn)-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 + 4z6 - 10z4 + 4z2 + 1 = 0. (Lei) (Hint: divide it by
z4, and
observe the symmetry)
9. (Putnam 2004-B1) 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 2003-B1) Do there exist polynomials a(x),
b(x), c(y), d(y) such that
1 + xy + x2y2 = a(x)c(y) + b(x)d(y)
holds identically? (Richard)
More Problems:
1. If a and b are two solutions of x4 -x3 -1 = 0, then ab is a solution of x6
+x4 +x3 -
x2 - 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 x4
- x3 +
ax2 - 8x - 8 = 0, find 
and a. (Do not assume that they are real numbers.)
4. (VA 1991) Let f(x) = x5 - 5x3 + 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)(x-d)-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 - bx3 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 2004-A4) 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 2003-A4) Suppose that a, b, c, A,B,C are real
numbers, a ≠ 0 and A ≠ 0,
such that
|ax2 + bx + c| ≤ |Ax2 + Bx + C|
for all real numbers x. Show that
|b2 - 4ac| ≤ |B2 - 4AC|.
12. (Putnam 2003-B1) Do there exist polynomials a(x), b(x), c(y), d(y) such that
1 + xy + x2y2 = a(x)c(y) + b(x)d(y)
holds identically?
13. (Putnam 2003-B4) 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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