Competition · AMC preparation · step 4 of 4

AMC 10 2015B: 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 2015B #1 grade 8+ arithmetic
What is the value of 2-(-2)⁻²?
AMC 10 2015B #2 grade 4+ rate-ratio
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and f…
AMC 10 2015B #3 grade 4+ arithmetic
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and on…
AMC 10 2015B #4 grade 4+ arithmetic
Four siblings ordered an extra large pizza. Alex ate 1/5, Beth 1/3, and Cyril 1/4 of the pizza. Dan got the leftovers. W…
AMC 10 2015B #5 grade 2+ arithmetic
David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of…
AMC 10 2015B #6 grade 2+ arithmetic
Mahdi practices exactly one sport each day of the week. He runs three days a week but never on two consecutive days. On…
AMC 10 2015B #7 grade 6+ arithmetic
Consider the operation "minus the reciprocal of," defined by adiamond b=a-1/b. What is ((1diamond2)diamond3)-(1diamond(2…
AMC 10 2015B #8 grade 8+ arithmetic
The letter F shown below is rotated 90° clockwise around the origin, then reflected in the y-axis, and then rotated a ha…
AMC 10 2015B #9 grade 7+ geometry-2d
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the por…
AMC 10 2015B #10 grade 7+ arithmetic
What are the sign and units digit of the product of all the odd negative integers strictly greater than -2015?
AMC 10 2015B #11 grade 7+ probability
Among the positive integers less than 100, each of whose digits is a prime number, one is selected at random. What is th…
AMC 10 2015B #12 grade 8+ geometry-2d
For how many integers x is the point (x, -x) inside or on the circle of radius 10 centered at (5, 5)?
AMC 10 2015B #13 grade 8+ geometry-2d
The line 12x+5y=60 forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this tr…
AMC 10 2015B #14 grade 8+ algebra
Let a, b, and c be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation (…
AMC 10 2015B #15 grade 6+ arithmetic
The town of Hamlet has 3 people for each horse, 4 sheep for each cow, and 3 ducks for each person. Which of the followin…
AMC 10 2015B #16 grade 7+ probability
Al, Bill, and Cal will each randomly be assigned a whole number from 1 to 10, inclusive, with no two of them getting the…
AMC 10 2015B #17 grade 7+ geometry-3d
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volu…
AMC 10 2015B #18 grade 7+ rate-ratio
Johann has 64 fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails…
AMC 10 2015B #19 grade 8+ geometry-2d
In △ABC, ∠C = 90° and AB = 12. Squares ABXY and ACWZ are constructed outside of the triangle. The points X, Y, Z, and W…
AMC 10 2015B #20 grade 7+ geometry-3d
Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every cor…
AMC 10 2015B #21 grade 6+ rate-ratio
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up th…
AMC 10 2015B #22 grade 8+ arithmetic
In the figure shown below, ABCDE is a regular pentagon and AG=1. What is FG + JH + CD?
AMC 10 2015B #23 grade 6+ arithmetic
Let n be a positive integer greater than 4 such that the decimal representation of n! ends in k zeros and the decimal re…
AMC 10 2015B #24 grade 6+ arithmetic
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin p₀=(0,0) facing to…
AMC 10 2015B #25 grade 7+ geometry-3d
A rectangular box measures a × b × c, where a, b, and c are integers and 1≤ a ≤ b ≤ c. The volume and the surface area o…

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