**AMC 12 Problem Series online course**.

# 2005 AMC 12A 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

Two is of and of . What is ?

## Problem 2

The equations and have the same solution. What is the value of ?

## Problem 3

A rectangle with diagonal length is twice as long as it is wide. What is the area of the rectangle?

## Problem 4

A store normally sells windows at <dollar/> each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?

## Problem 5

The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers?

## Problem 6

Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

## Problem 7

Square is inside the square so that each side of can be extended to pass through a vertex of . Square has side length and . What is the area of the inner square ?

## Problem 8

Let , and be digits with

What is ?

## Problem 9

There are two values of for which the equation has only one solution for . What is the sum of these values of ?

## Problem 10

A wooden cube units on a side is painted red on all six faces and then cut into unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is ?

## Problem 11

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

## Problem 12

A line passes through and . How many other points with integer coordinates are on the line and strictly between and ?

## Problem 13

The regular 5-point star is drawn and in each vertex, there is a number. Each and are chosen such that all 5 of them came from set . Each letter is a different number (so one possible way is ). Let be the sum of the numbers on and , and so forth. If and form an arithmetic sequence (not necessarily in increasing order), find the value of .

## Problem 14

On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?

## Problem 15

Let be a diameter of a circle and be a point on with . Let and be points on the circle such that and is a second diameter. What is the ratio of the area of to the area of ?

## Problem 16

Three circles of radius are drawn in the first quadrant of the -plane. The first circle is tangent to both axes, the second is tangent to the first circle and the -axis, and the third is tangent to the first circle and the -axis. A circle of radius is tangent to both axes and to the second and third circles. What is ?

## Problem 17

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex ?

## Problem 18

Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000?

## Problem 19

A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?

## Problem 20

For each in , define

Let , and for each integer . For how many values of in is ?

## Problem 21

How many ordered triples of integers , with , , and , satisfy both and ?

## Problem 22

A rectangular box is inscribed in a sphere of radius . The surface area of is 384, and the sum of the lengths of its 12 edges is 112. What is ?

## Problem 23

Two distinct numbers and are chosen randomly from the set . What is the probability that is an integer?

## Problem 24

Let . For how many polynomials does there exist a polynomial of degree 3 such that ?

## Problem 25

Let be the set of all points with coordinates , where and are each chosen from the set . How many equilateral triangles have all their vertices in ?