Difference between revisions of "2005 AMC 12B Problems"
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== Problem 22 == | == Problem 22 == | ||
+ | |||
+ | A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule | ||
+ | |||
+ | <cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath> | ||
+ | |||
+ | where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</math>. How many possible values are there for <math>z_{0}</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 1 \qquad | ||
+ | \textbf{(B)}\ 2 \qquad | ||
+ | \textbf{(C)}\ 4 \qquad | ||
+ | \textbf{(D)}\ 2005 \qquad | ||
+ | \textbf{(E)}\ 2^{2005} | ||
+ | </math> | ||
[[2005 AMC 12B Problems/Problem 22|Solution]] | [[2005 AMC 12B Problems/Problem 22|Solution]] |
Revision as of 12:32, 26 March 2010
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
A scout troop buys candy bars at a price of five for
dollars. They sell all the candy bars at the price of two for
dollar. What was their profit, in dollars?
Problem 2
A positive number has the property that
of
is
. What is
?
Problem 3
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
Problem 4
At the beginning of the school year, Lisa's goal was to earn an A on at least of her
quizzes for the year. She earned an A on
of the first
quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?
Problem 5
An -foot by
-foot floor is tiles with square tiles of size
foot by
foot. Each tile has a pattern consisting of four white quarter circles of radius
foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
Problem 6
In , we have
and
. Suppose that
is a point on line
such that
lies between
and
and
. What is
?
Problem 7
What is the area enclosed by the graph of ?
Problem 8
For how many values of is it true that the line
passes through the
vertex of the parabola
?
Problem 9
Problem 10
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?
Problem 11
Problem 12
The quadratic equation has roots twice those of
, and none of
and
is zero. What is the value of
?
Problem 13
Problem 14
Problem 15
The sum of four two-digit numbers is . Non of the eight digits is
and no two of them are the same. Which of the following is not included among the eight digits?
Problem 16
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
Problem 17
How many distinct four-tuples of rational numbers are there with
?
Problem 18
Let and
be points in the plane. Define
as the region in the first quadrant consisting of those points
such that
is an acute triangle. What is the closest integer to the area of the region
?
Problem 19
Let and
be two-digit integers such that
is obtained by reversing the digits of
. The integers
and
satisfy
for some positive integer
. What is
?
Problem 20
Let and
be distinct elements in the set
What is the minimum possible value of
Problem 21
Problem 22
A sequence of complex numbers is defined by the rule
where is the complex conjugate of
and
. Suppose that
and
. How many possible values are there for
?