Competition · AMC preparation · step 4 of 4

AMC 10 2008B: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 10 2008B #1 grade 4+ counting
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers…
AMC 10 2008B #2 grade 2+ arithmetic
A 4× 4 block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed.…
AMC 10 2008B #3 grade 8+ algebra
Assume that x is a positive real number. Which is equivalent to ³√x√x?
AMC 10 2008B #4 grade 5+ arithmetic
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000…
AMC 10 2008B #5 grade 7+ algebra
For real numbers a and b, define a * b=(a-b)². What is (x-y)²*(y-x)²?
AMC 10 2008B #6 grade 6+ rate-ratio
Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD…
AMC 10 2008B #7 grade 6+ geometry-2d
An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side lengt…
AMC 10 2008B #8 grade 6+ counting
A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnat…
AMC 10 2008B #9 grade 7+ algebra
A quadratic equation ax² - 2ax + b = 0 has two real solutions. What is the average of these two solutions?
AMC 10 2008B #10 grade 8+ geometry-2d
Points A and B are on a circle of radius 5 and AB=6. Point C is the midpoint of the minor arc AB. What is the length of…
AMC 10 2008B #11 grade 8+ algebra
Suppose that (u_n) is a sequence of real numbers satifying u_(n+2)=2u_(n+1)+u_n, and that u₃=9 and u₆=128. What is u₅?
AMC 10 2008B #12 grade 6+ arithmetic
Postman Pete has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 00000 on th…
AMC 10 2008B #13 grade 6+ algebra
For each positive integer n, the mean of the first n terms of a sequence is n. What is the 2008^th term of the sequence?
AMC 10 2008B #14 grade 8+ geometry-2d
Triangle OAB has O=(0,0), B=(5,0), and A in the first quadrant. In addition, ∠ ABO=90° and ∠ AOB=30°. Suppose that OA is…
AMC 10 2008B #15 grade 8+ number-theory
How many right triangles have integer leg lengths a and b and a hypotenuse of length b+1, where b<100?
AMC 10 2008B #16 grade 7+ probability
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability…
AMC 10 2008B #17 grade 7+ probability
A poll shows that 70% of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter…
AMC 10 2008B #18 grade 8+ rate-ratio
Bricklayer Brenda takes 9 hours to build a chimney alone, and bricklayer Brandon takes 10 hours to build it alone. When…
AMC 10 2008B #19 grade 8+ geometry-3d
A cylindrical tank with radius 4 feet and height 9 feet is lying on its side. The tank is filled with water to a depth o…
AMC 10 2008B #20 grade 7+ probability
The faces of a cubical die are marked with the numbers 1, 2, 2, 3, 3, and 4. The faces of another die are marked with th…
AMC 10 2008B #21 grade 7+ counting
Ten chairs are evenly spaced around a round table and numbered clockwise from 1 through 10. Five married couples are to…
AMC 10 2008B #22 grade 7+ probability
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no…
AMC 10 2008B #23 grade 7+ geometry-2d, number-theory
A rectangular floor measures a by b feet, where a and b are positive integers and b > a. An artist paints a rectangle on…
AMC 10 2008B #24 grade 8+ geometry-2d
Quadrilateral ABCD has AB = BC = CD, m∠ ABC = 70° and m∠ BCD = 170°. What is the degree measure of ∠ BAD?
AMC 10 2008B #25 grade 7+ rate-ratio
Michael walks at the rate of 5 feet per second on a long straight path. Trash pails are located every 200 feet along the…

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