# 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 (1-x

^{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 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 + 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)(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 - 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 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

|ax^{2} + bx + c| ≤ |Ax^{2} + Bx + C|

for all real numbers x. Show that

|b^{2} - 4ac| ≤ |B^{2} - 4AC|.

12. (Putnam 2003-B1) 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 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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