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ALGEBRA PUTNAM PROBLEMS

2007-A-1. Find all values of α for which the curves y = αx2 + αx+1/24 and x = αy2 + 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 x3 + 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/(xk−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 xn in the expansion of (1 + x + x2)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 xk in the expansion
of (1 + x + x2 + x3)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.)

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