Difference between revisions of "2001 AMC 12 Problems"
MRENTHUSIASM (talk | contribs) m (→Problem 25) |
|||
(22 intermediate revisions by 13 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AMC12 Problems|year=2001|ab=}} | ||
== Problem 1 == | == Problem 1 == | ||
The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then | The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then | ||
Line 18: | Line 19: | ||
== Problem 3 == | == Problem 3 == | ||
− | The state income tax where Kristin lives is levied at the rate of <math>p%</math> of the first | + | The state income tax where Kristin lives is levied at the rate of <math>p\%</math> of the first |
− | + | <math>\textdollar 28000</math> of annual income plus <math>(p + 2)\%</math> of any amount above <math>\textdollar 28000</math>. Kristin | |
− | noticed that the state income tax she paid amounted to <math>(p + 0.25)%</math> of her | + | noticed that the state income tax she paid amounted to <math>(p + 0.25)\%</math> of her |
annual income. What was her annual income? | annual income. What was her annual income? | ||
− | <math>\text{(A)}\ | + | <math>\text{(A)}\ \$28000\qquad \text{(B)}\ \$32000\qquad \text{(C)}\ \$35000\qquad \text{(D)}\ \$42000\qquad \text{(E)}\ \$56000</math> |
[[2001 AMC 12 Problems/Problem 3|Solution]] | [[2001 AMC 12 Problems/Problem 3|Solution]] | ||
Line 58: | Line 59: | ||
== Problem 7 == | == Problem 7 == | ||
− | A charity sells <math>140</math> benefit tickets for a total of | + | A charity sells <math>140</math> benefit tickets for a total of <math>\textdollar 2001</math>. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? |
− | <math>\text{(A) } | + | <math>\text{(A) }\$782\qquad \text{(B) }\$986\qquad \text{(C) }\$1158\qquad \text{(D) }\$1219\qquad \text{(E) }\$1449</math> |
[[2001 AMC 12 Problems/Problem 7|Solution]] | [[2001 AMC 12 Problems/Problem 7|Solution]] | ||
Line 96: | Line 97: | ||
Let <math>f</math> be a function satisfying <math>f(xy) = \frac{f(x)}y</math> for all positive real numbers <math>x</math> and <math>y</math>. If <math>f(500) =3</math>, what is the value of <math>f(600)</math>? | Let <math>f</math> be a function satisfying <math>f(xy) = \frac{f(x)}y</math> for all positive real numbers <math>x</math> and <math>y</math>. If <math>f(500) =3</math>, what is the value of <math>f(600)</math>? | ||
− | <math> | + | <math>\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ \frac52 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \frac{18}5</math> |
[[2001 AMC 12 Problems/Problem 9|Solution]] | [[2001 AMC 12 Problems/Problem 9|Solution]] | ||
Line 159: | Line 160: | ||
== Problem 12 == | == Problem 12 == | ||
− | How many positive integers not exceeding <math>2001</math> are | + | How many positive integers not exceeding <math>2001</math> are multiples of <math>3</math> or <math>4</math> but not <math>5</math>? |
<math> | <math> | ||
Line 386: | Line 387: | ||
== Problem 23 == | == Problem 23 == | ||
− | A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? | + | A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial? |
<math> | <math> | ||
Line 422: | Line 423: | ||
== Problem 25 == | == Problem 25 == | ||
− | Consider sequences of positive real numbers of the form <math>x, 2000, y, \dots</math> in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of <math>x</math> does the term 2001 appear somewhere in the sequence? | + | Consider sequences of positive real numbers of the form <math>x, 2000, y, \dots</math> in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of <math>x</math> does the term <math>2001</math> appear somewhere in the sequence? |
<math> | <math> | ||
Line 439: | Line 440: | ||
== See also == | == See also == | ||
+ | |||
+ | {{AMC12 box|year=2001|before=[[2000 AMC 12 Problems]]|after=[[2002 AMC 12A Problems]]}} | ||
+ | |||
* [[AMC 12]] | * [[AMC 12]] | ||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
* [[2001 AMC 12]] | * [[2001 AMC 12]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 12:53, 24 January 2024
2001 AMC 12 (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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
The sum of two numbers is . Suppose
is added to each number and then
each of the resulting numbers is doubled. What is the sum of the final two
numbers?
Problem 2
Let and
denote the product and the sum, respectively, of the digits
of the integer
. For example,
and
. Suppose
is a
two-digit number such that
. What is the units digit of
?
Problem 3
The state income tax where Kristin lives is levied at the rate of of the first
of annual income plus
of any amount above
. Kristin
noticed that the state income tax she paid amounted to
of her
annual income. What was her annual income?
Problem 4
The mean of three numbers is more than the least of the numbers and
less than the greatest. The median of the three numbers is
. What is their
sum?
Problem 5
What is the product of all positive odd integers less than 10000?
Problem 6
A telephone number has the form , where each letter represents
a different digit. The digits in each part of the number are in decreasing
order; that is,
,
, and
. Furthermore,
,
, and
are consecutive even digits;
,
,
, and
are consecutive odd
digits; and
. Find
.
Problem 7
A charity sells benefit tickets for a total of
. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
Problem 8
Which of the cones listed below can be formed from a sector of a circle of radius
by aligning the two straight sides?
Problem 9
Let be a function satisfying
for all positive real numbers
and
. If
, what is the value of
?
Problem 10
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
Problem 11
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
Problem 12
How many positive integers not exceeding are multiples of
or
but not
?
Problem 13
The parabola with equation and vertex
is reflected about the line
. This results in the parabola with equation
. Which of the following equals
?
Problem 14
Given the nine-sided regular polygon , how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set
?
Problem 15
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
Problem 16
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
Problem 17
A point is selected at random from the interior of the pentagon with vertices
,
,
,
, and
. What is the probability that
is obtuse?
Problem 18
A circle centered at with a radius of 1 and a circle centered at
with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
Problem 19
The polynomial has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the
-intercept of the graph of
is 2, what is
?
Problem 20
Points ,
,
, and
lie in the first quadrant and are the vertices of quadrilateral
. The quadrilateral formed by joining the midpoints of
,
,
, and
is a square. What is the sum of the coordinates of point
?
Problem 21
Four positive integers ,
,
, and
have a product of
and satisfy:
What is ?
Problem 22
In rectangle , points
and
lie on
so that
and
is the midpoint of
. Also,
intersects
at
and
at
. The area of the rectangle
is
. Find the area of triangle
.
Problem 23
A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?
Problem 24
In ,
. Point
is on
so that
and
. Find
.
Problem 25
Consider sequences of positive real numbers of the form in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of
does the term
appear somewhere in the sequence?
See also
2001 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by 2000 AMC 12 Problems |
Followed by 2002 AMC 12A Problems |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.