2015 AMC 12A Problems
Problem 1
What is the value of
Problem 2
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible p
erimeter of the triangle?
Problem 3
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?
Problem 4
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?
Problem 5
Amelia needs to estimate the quantity , where and are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of ?
Problem 6
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be ?
Problem 7
Two right circular cylinders have the same volume. The radius of the second cylinder is more than the radius of the first. What is the relationship between the heights of the two cylinders?
Problem 8
The ratio of the length to the width of a rectangle is : . If the rectangle has diagonal of length , then the area may be expressed as for some constant What is ? .
Problem 9
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?
Problem 10
Integers and with satisfy . What is ?
Problem 11
On a sheet of paper, Isabella draws a circle of radius , a circle of radius , and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly lines. How many different values of are possible?
Problem 12
The parabolas and intersect the coordinate axes in .
exactly four points, and these four points are the vertices of a kite of area What is ?
Problem 13
A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Games either end with one team victorious or else end in a draw. A team scores 2 points for every game it wins and 1 point for every game it draws. Which of the following is NOT a true statement about the list of 12 scores?
Problem 14
What is the value of for which ?
Problem 15
What is the minimum number of digits to the right of the decimal point needed to express the fraction as a decimal?
Problem 16
Tetrahedron has . What is the volume of the tetrahedron? and
Problem 17
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
Problem 18
The zeros of the function the possible values of ? are integers. What is the sum of
Problem 19
For some positive integers , there is a quadrilateral side lengths, perimeter , right angles at and , How many different values of are possible? with positive integer , and .
Problem 20
Isosceles triangles and are not congruent but have the same area and the same perimeter. The sides of have lengths of and , while those of have lengths of and . Which of the following numbers is closest to ?
Problem 21
A circle of radius passes through both foci of, and exactly four points on, the . The set of all possible values of ? is an ellipse with equation interval . What is
Problem 22
For each positive integer , let and be the number of sequences of length , with no more than three s in a row and is divided
consisting solely of the letters no more than three by 12?
s in a row. What is the remainder when
Problem 23
Let be a square of side length 1. Two points are chosen independently at random on the sides of . The probability that the straight-line distance between the points is at least is , where ? and are positive integers and
. What is
Problem 24
Rational numbers the interval with number? and are chosen at random among all rational numbers in where and are integers is a real
that can be written as fractions . What is the probability that
Problem 25
A collection of circles in the upper half-plane, all tangent to the -axis, is constructed in layers as follows. Layer consists of two circles of radii and
that are externally tangent. For , the circles in are ordered according to their points of tangency with the -axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer consists of the circles
constructed in this way. Let
, and for every circle
denote by
its radius. What is
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