AMC Algebra
AMC Combinatorics
AMC Geometry
500
2012 AMC 10A #19
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
(D) 48
500
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
(E) 900
500
2011 AMC 10B #22
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
(A) 5sqr2 - 7
600
2017 AMC 10B #4
Supposed that x and y are nonzero real numbers such that 
. What is the value of 
?
(D) 2
600
2015 AMC 10A #10
How many rearrangements of 
are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either 
or 
.
(C) 2
600
2012 AMC 10B #12
Point B is due east of point A. Point C is due north of point B. The distance between points A and C is 
, and 
. Point D is 20 meters due north of point C. The distance AD is between which two integers?
(B) 31 and 32
700
2018 AMC 10A #12
How many ordered pairs of real numbers 
satisfy the following system of equations?
(C) 3
700
2012 AMC 10A #20
A 3 x 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated 
clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
(A) 49/512
700
2012 AMC 10B #16
Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?
(A) 10pi + 4sqr3
800
2014 AMC 10A #15
David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?
(C) 210
800
The regular octagon ABCDEFGH has its center at J. Each of the vertices and the center are to be associated with one of the digits 1 through 9, with each digit used once, in such a way that the sums of the numbers on the lines AJE, BJF, CJG, and DJH are all equal. In how many ways can this be done?
(C) 1152
800
2015 AMC 10B #19
In 
, 
and 
. Squares ABXY and ACWZ are constructed outside of the triangle. The points X, Y, Z and W lie on a circle. What is the perimeter of the triangle?
(C) 12 + 12sqr2
900
2017 AMC 10B #3
Real numbers x, y, and z satisfy the inequalities 
, 
, and 
. Which of the following numbers is necessarily positive?
(E) y+z
900
2018 AMC 10A #4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
(E) 24
900
2012 AMC 10A #4
Let 
and 
. What is the smallest possible degree measure for 
?
(C) 4