# 2001 AMC 12 Problems

## 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 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 2001 appear somewhere in the sequence?

## See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.