Functions
AMC 10
1. Let
be a function for which
.
Find the sum of all values of
for which
.
2. Both roots of the quadratic equation
number of possible values of
are prime numbers. The
is
3. What is the sum of the reciprocals of the roots of the equation
?
4. Let ; ;
the first
and ;
;
be two arithmetic progressions. The set
terms of each sequence. How many distinct numbers are in
5. The quadratic equation
of
6. Let
that
is the union of
has roots that are twice those
, and none of
and
,
, and
is zero. What is the value of
be the roots of the equation
and
?
are the roots of the equation
?
. Suppose
. What is ?
7. Let
be a sequence for which
,
What is
, and
?
for each positive integer
.
AMC 12
1. The parabola
where
has vertex
and
-intercept
,
. What is ?
2. The graph of the polynomial
has five distinct
-intercepts, one of which is at
. Which of the following
coefficients cannot be zero?
3. Let
be a function with the following properties:
, and
, for any positive integer
What is the value of
?
4. For all positive integers
, let
.
. Let
.
Which of the following relations is true?
5. For all integers
Let
greater than , define
and
.
. Then
equals
6. The set of all real numbers
is defined is
for which
. What is the value of ?
7. The sum of the zeros, the product of the zeros, and the sum of the coefficients of
the function
are equal. Their common value must also be which
of the following?
8. For some real numbers
and , the equation
has three
distinct positive roots. If the sum of the base- logarithms of the roots is , what is the
value of
9. Points
of
?
and
are on the parabola
. What is the length of
10. The polynomial
, and the origin is the midpoint
?
has integer coefficients and three distinct
positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How
many values of
11. For each
are possible?
in
, define
Let
, and
for each integer
. For how
. For how many polynomials
does there
many values
of
in
is
?
12. Let
exist a polynomial
Since
of degree 3 such that
has degree three, then
degree six, so
?
has degree six. Thus,
must have degree two, since
Hence, we conclude
,
, and
has degree three.
must each be , , or . Since a quadratic is
uniquely determined by three points, there can be
quadratics
after each of the values of
However, we have included
has
,
different
, and
are chosen.
which are not quadratics. Namely,
Clearly, we could not have included any other constant functions. For any linear function,
we have
. Again, it is pretty obvious that we have not included
any other linear functions. Therefore, the desired answer is
.
13. The function
has the property that for each real number
in its domain,
also in its domain and
What is the largest set of real numbers that can be in the domain of
.
?
is