AMC 12 Mock Exam 2

A rectangular garden is $5$ meters longer than it is wide. A gravel path $1$ meter wide is built around the outside of the garden, so that the garden and the path together form a larger rectangle. The path alone covers $62$ square meters. What is the area of the garden, in square meters?

Call a three-digit positive integer elevenish if it is a multiple of $11$ and its three digits add up to $11$. How many elevenish integers are there?

A store raises the price of a jacket by $25\%$. Later, during a sale, it lowers the new price by $x\%$, and the sale price turns out to be $10\%$ below the original price. What is $x$?

Square $ABCD$ has side length $8$. Point $E$ lies on side $\overline{BC}$ with $BE=6$, and point $F$ lies on side $\overline{CD}$ with $CF=2$. What is the area of triangle $AEF$?

Raffle tickets are numbered $1$ through $200$. A ticket wins a prize if its number is divisible by $6$ or by $15$; however, any ticket whose number is divisible by $10$ is void and cannot win. How many tickets win a prize?

The graph of $y=x^2+bx+c$ crosses the $x$-axis at two points that are $6$ units apart, and the vertex of the graph lies on the line $y=3x+6$. What is $c$?

An angle $\theta$ with $0<\theta<\frac{\pi}{2}$ satisfies $\sin\theta+2\cos\theta=2$. What is $\tan\theta$?

How many different arrangements of the seven letters of the word $\text{BANANAS}$ have no two $\text{A}$'s next to each other?

What is the least possible value of $$\sqrt{x^2+9}+\sqrt{x^2-16x+89}$$ over all real numbers $x$?

What is the sum of all positive integers $n$ such that $n+12$ is a divisor of $n^2+12$?

In triangle $ABC$, angles $A$ and $B$ are acute, $\sin A=\dfrac35$, and $\sin B=\dfrac{5}{13}$. What is $\cos C$?

Rectangle $ABCD$ has $AB=12$ and $BC=5$. Point $E$ lies on side $\overline{CD}$ with $DE=4$. Segment $\overline{BE}$ meets diagonal $\overline{AC}$ at $F$, as shown (the figure is not necessarily drawn to scale). What is the area of the shaded triangle $BCF$?

Maya rolls three fair standard six-sided dice. What is the probability that one of the three numbers she rolls is equal to the sum of the other two?

How many real numbers $x$ in the interval $[0,\pi]$ satisfy $$\cos x+\cos3x+\cos5x=0?$$

In regular hexagon $ABCDEF$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $P$. What fraction of the area of the hexagon is the area of triangle $BCP$?

A rectangular box with an open top (no lid) is to have a volume of exactly $32$ cubic feet. Its bottom and four sides are all made from the same sheet material. What is the least possible total area, in square feet, of material needed to make the box?

Each of the six faces of a cube is to be painted with one of four colors so that no two faces that share an edge are painted the same color. Not every color needs to be used. Colorings that differ by a rotation of the cube are counted as different. How many colorings are possible?

A regular octahedron has edge length $6$. A plane passes through the center of the octahedron and is parallel to one of its faces. What is the area of the cross-section of the octahedron cut by this plane?

How many integers $n$ with $0\le n<1024$ have exactly five $1$'s in their base-$2$ representation and are divisible by $3$?

For how many ordered pairs $(a,b)$ of integers with $2\le a\le100$ and $2\le b\le100$ is $\log_a b$ a rational number?

A panel has four light switches, all of which are initially off. Each minute, one of the four switches is chosen uniformly at random, independently of all earlier choices, and flipped (from off to on, or from on to off). What is the expected number of minutes until all four switches are on for the first time?

The polynomial $P(x)=x^3+ax^2+bx+c$ has integer coefficients and three integer roots, not necessarily distinct, and $a+b+c=23$. How many such polynomials are there?

In triangle $ABC$, $AB=7$, $AC=8$, and $BC=9$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. A circle passes through $A$ and is tangent to line $BC$ at $D$. This circle meets $\overline{AB}$ again at $E$ and $\overline{AC}$ again at $F$, as shown (the figure is not necessarily drawn to scale). What is $EF$?

Eight people sit in equally spaced seats around a round table. They are to be split into four pairs for a game, and no pair may consist of two people who are sitting in adjacent seats. In how many ways can the eight people be split into pairs?

Real numbers $x$, $y$, and $z$ satisfy $x+y+z=0$ and $x^2+y^2+z^2=6$. What is the greatest possible value of $x^2y+y^2z+z^2x$?