Question bank AMC12 2015

What is the value of $$
(2^0-1+5^2+0)^{-1}\times 5?
$$

Two of the three sides of a triangle are $20$ and $15.$ Which of the following numbers is not a possible perimeter of the triangle?

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?

The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?

Amelia needs to estimate the quantity $\dfrac{a}{b}-c,$ where $a,$ $b,$ and $c$ 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 $\dfrac{a}{b}-c?$

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 $2 : 1?$

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?

The ratio of the length to the width of a rectangle is $4 : 3.$ If the rectangle has diagonal of length $d,$ then the area may be expressed as $kd^2$ for some constant $k.$ What is $k?$

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?

Integers $x$ and $y$ with $x \gt y \gt 0$ satisfy $x+y+xy = 80.$ What is $x?$

On a sheet of paper, Isabella draws a circle of radius $2,$ a circle of radius $3,$ and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \ge 0$ lines. How many different values of $k$ are possible?

The parabolas $y = ax^2 - 2$ and $y = 4 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12.$ What is $a + b?$

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?

What is the value of $a$ for which $$
\dfrac{1}{\log_2 a} + \dfrac{1}{\log_3 a} + \dfrac{1}{\log_4 a} = 1?
$$

What is the minimum number of digits to the right of the decimal point needed to express the fraction $$
\dfrac{123456789}{2^{26}\cdot 5^4}
$$ as a decimal?

Tetrahedron $ABCD$ has $AB = 5,$ $AC = 3,$ $BC = 4,$ $BD = 4,$ $AD = 3,$ and $CD = \dfrac{12}{5}\sqrt{2}.$ What is the volume of the tetrahedron?

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?

The zeros of the function $f(x) = x^2 - ax + 2a$ are integers. What is the sum of the possible values of $a?$

For some positive integers $p,$ there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p,$ right angles at $B$ and $C,$ $AB = 2,$ and $CD = AD.$ How many different values of $p \lt 2015$ are possible?

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5,$ $5,$ and $8,$ while those of $T'$ have lengths $a,$ $a,$ and $b.$ Which of the following numbers is closest to $b?$