**2007-A-1. **Find all values of α for which the curves y
= αx^{2} + αx+1/24 and x = αy^{2} + αy +1/24 are

tangent to each other.

**2007-A-2. **A repunit is a positive integer whose digits in base 10 are all
ones. Find all polynomials f

with real coefficients such that if n is a repunit, then so is f(n).

**2007-B-1.** Let f be a polynomial with positive integer coefficients. Prove
that if n is a positive integer,

then f(n) divides f (f(n) + 1) if and only if n = 1.

**2007-B-4. **Let n be a positive integer. Find the number of pairs P,Q of
polynomials with real

coefficients such that

and deg P > deg Q.

**2007-B-5.** Let k be a positive integer. Prove that there exist polynomials

(which may depend on k) such that, for any integer n,

**2006-B-1.** Show that the curve x^{3} + 3xy + y3 = 1 contains
only one set of three distinct points A,B,

and C, which are the vertices of an equilateral triangle, and find its area.

**2005-A-3.** Let p(z) be a polynomial of degree n, all of whose zeros have absolute
value 1 in the complex

plane. Put . Show that all zeros of g'(z) = 0 have absolute
value 1.

**2005-B-1.** Find a nonzero polynomial P(x, y) such that
for all
real numbers a. (Note:

is the greatest integer less than or equal to
.)

2005-B-5. Let denote a polynomial with
real coefficients in the variables , and

suppose that

(identically)

and that

divides

Show that P = 0 identically.

**2004-A-4.** Show that for any positive integer n there is an integer N such
that the product

can be ex pressed identically in the form

where the are rational numbers and each
is one of
the numbers, −1, 0, 1.

**2004-B-1. **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.

**2003-A-4.** Suppose that a, b, c, A,B,C are real numbers, a ≠ 0 and A ≠ 0, such
that

for all real numbers x. Show that

**2003-B-1.** Do there exist polynomials a(x), b(x), c(y),
d(y) such that

holds identically?

2003-B-4. Let

where a, b, c, d, e are integers, a ≠ 0. Show that if
is a rational number, and if

then is a rational number.

**2002-A-1. **Let k be a positive integer. The nth derivative of 1/(x^{k}−1) has the
form

where is a polynomial. Find
.

2002-B-6. Let p be a prime number. Prove that the de terminant of the matrix

is congruent modulo p to a product of polynomials in the
form ax + by + cz, where a, b, c are integers. (We

say two integer polynomials are congruent modulo p if corresponding coefficients
are congruent modulo p.)

**2001-A-3.** For each integer m, consider the polynomial

For what values of m is the product of two nonconstant polynomials with integer coefficients?

**2001-B-2. **Find all pairs of real numbers (x, y) satisfying the system of
equations

**2000-A-6.** Let f(x) be a polynomial with integer
coefficients. Define a sequence of integers

such that and
for n ≥ 0. Prove that if there exists a positive
integer m for which

, then either
or .

**1999-A-1.** Find polynomials f(x), g(x) and h(x), if they
exist, such that, for all x,

**1999-A-2.** Let p(x) be a polynomial that is non- negative
for all x . Prove that, for some k, there are

polynomials such that

**1999-B-2. **Let P(x) be a polynomial of degree n such that
P(x) = Q(x)P''(x), where Q(x) is a

quadratic polynomial and P''(x) is the second derivative of P(x). Show that if
P(x) has at least two distinct

roots then it must have n distinct roots. [The roots may be either real or
complex.]

1997-B-4. Let denote the coefficient of x^{n} in the expansion of (1 + x +
x^{2})^{m}. Prove that for all

k ≥ 0,

**1995-B-4. **Evaluate

Express your answer in the form
, where a, b, c, d are integers.

**1993-B-2.** For nonnegative integers n and k, define Q(n, k) to be the coefficient
of x^{k} in the expansion

of (1 + x + x^{2} + x^{3})^{n}. Prove that

where is the standard
binomial coefficient . (Reminder: For integers a and b with a ≥ 0,

for 0 ≤ b ≤ a and = 0 otherwise.)