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2015 AMC 12B Problems

Problem 1

What is the value of ?

Problem 2

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?

Problem 3

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?

Problem 4

David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?

Problem 5

The Tigers beat the Sharks 2 out of the 3 times they played. They then played more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for ?

Problem 6

Back in 1930, Tillie had to memorize her multiplication facts from to . The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?

Problem 7

A regular 15-gon has lines of symmetry, and the smallest positive angle for which it has rotational symmetry is degrees. What is ?

Problem 8

What is the value of ?

Problem 9

Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is , independently of what has happened before. What is the probability that

Larry wins the game?

Problem 10

How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?

Problem 11

The line forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

Problem 12

Let , , and be three distinct one-digit numbers. What is the maximum value of ? the sum of the roots of the equation

Problem 13

Quadrilateral is inscribed in a circle with and . What is ?

Problem 14

A circle of radius 2 is centered at . An equilateral triangle with side 4 has a vertex at . What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?

Problem 15

At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She thinks she has a chance of getting an A in English, and a has a chance of getting an A, and a chance of getting a B. In History, she chance of getting a B, independently of

what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5?

Problem 16

A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

Problem 17

An unfair coin lands on heads with a probability of . When tossed times, the

probability of exactly two heads is the same as the probability of exactly three heads. What is the value of ?

Problem 18

For every composite positive integer the prime factorization of factorization of function , is , define to be the sum of the factors in because the prime . What is the range of the ?

. For example, , and

Problem 19

In , and . Squares constructed outside of the triangle. The points , What is the perimeter of the triangle? , and , and are lie on a circle.

Problem 20

For every positive integer , let be the remainder obtained when is

divided by 5. Define a function recursively as follows:

What is

?

Problem 21

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose that Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of ?

Problem 22

Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?

Problem 23

A rectangular box measures , where , , and are integers and . The volume and the surface area of the box are numerically equal. How many ordered triples are possible?

Problem 24

Four circles, no two of which are congruent, have centers at and points and lie on all four circles. The radius of circle , and the radius of circle and ? is , is , , and times the . . ,

radius of circle Furthermore, What is

times the radius of circle . Let be the midpoint of

Problem 25

A bee starts flying from point once the bee reaches point inches straight to point inches away from and . She flies inch due east to point . For , she turns counterclockwise and then flies . When the bee reaches she is exactly , where , , and are positive integers and ? ,

are not divisible by the square of any prime. What is

Answer key

Cbabb addcc edbdd cddcb ddbdb

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