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
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
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
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 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
?