AMC 12 Mock Exam 3

A community theater charges $\$12$ for an adult ticket and $\$7$ for a child ticket. For Saturday's matinee it sold $50$ more child tickets than adult tickets, and its ticket revenue for the matinee was $\$1490$. How many tickets in all did the theater sell for the matinee?

A craftsman has $n$ identical boxes, each containing $2025$ unit cubes. He wants to use all of the unit cubes in the boxes, with none left over, to build a single solid cube. What is the smallest positive integer $n$ for which this is possible?

An empty tank can be filled by pipe $A$ alone in $6$ hours or by pipe $B$ alone in $9$ hours, and a drain at the bottom of the tank can empty a full tank in $18$ hours. A worker opens both pipes and, by mistake, also leaves the drain open, all starting at the same moment. How many hours does it take to fill the empty tank?

A carpenter builds a frame in the shape of a right triangle from three straight boards with a total length of $70$ inches. The longest board, which forms the hypotenuse, is $29$ inches long. What is the area, in square inches, of the region enclosed by the frame?

A nonzero real number $x$ satisfies $x+\dfrac1x=5$. What is $x^3+\dfrac1{x^3}$?

A club has $6$ juniors, two of whom are Alex and Bree, and $5$ seniors. A $4$-person committee is to be chosen that includes at least one junior and at least one senior. Alex and Bree have a scheduling conflict, so the committee may not include both of them. How many different committees are possible?

What is the value of $(\log_2 9)(\log_3 16)(\log_4 25)(\log_5 8)$?

Let $A=(1,2)$ and $B=(7,10)$. The perpendicular bisector of $\overline{AB}$ meets the $x$-axis at a point $P$. What is the area of $\triangle ABP$?

What is the greatest common divisor of $2^{30}-1$ and $2^{42}-1$?

A complex number $z$ satisfies $z+2\overline{z}=6-4i$, where $\overline z$ is the complex conjugate of $z$. What is $|z|$?

The sum of the first $10$ terms of an arithmetic sequence is $155$, and the sum of the first $20$ terms is $610$. What is the sum of the $21$st through $30$th terms of the sequence?

A right circular cone has base radius $6$ and height $8$. A sphere is inscribed in the cone: it is tangent to the base and tangent to the lateral surface along a circle. A second, smaller sphere, whose center also lies on the axis of the cone, lies above the first sphere; it is tangent to the first sphere and tangent to the lateral surface along a circle. What is the radius of the smaller sphere?

A fair $6$-sided die with faces numbered $1$ through $6$ and a fair $8$-sided die with faces numbered $1$ through $8$ are rolled. What is the probability that the sum of the two numbers rolled is a prime number?

The polynomial $x^3-14x^2+kx-64$ has three real roots that form a geometric progression. What is $k$?

The incircle of $\triangle ABC$ is tangent to side $\overline{BC}$ at $D$, where $BD=3$ and $DC=5$, as shown (the figure is not necessarily drawn to scale). If $\angle BAC=60^\circ$, what is the area of $\triangle ABC$?

A string of letters, each $A$ or $B$, is called odd-blocked if every maximal block of consecutive identical letters has odd length. For example, $ABBBA$ and $BAAAAAB$ are odd-blocked, but $AABAB$ is not. How many odd-blocked strings of length $12$ are there?

Kai computes each of the $100$ numbers $\dbinom21,\ \dbinom42,\ \dbinom63,\ \ldots,\ \dbinom{200}{100}$, that is, $\dbinom{2n}{n}$ for $n=1,2,\ldots,100$. How many of these $100$ numbers are not divisible by $4$?

Quadrilateral $ABCD$ is inscribed in a circle, and its side lengths are $AB=2$, $BC=3$, $CD=5$, and $DA=6$. What is the radius of the circle?

Real numbers $a$, $b$, and $c$, each greater than $1$, satisfy $\log_a(bc)=2$ and $\log_b(ca)=3$. What is $\log_c(ab)$?

Real numbers $a$, $b$, and $c$ satisfy $a+b+c=6$ and $ab+bc+ca=9$. What is the greatest possible value of $abc$?

Three points are chosen independently and uniformly at random on a circle of radius $1$. What is the probability that the triangle with these three points as vertices is acute and each of its sides has length greater than $1$?

What is the smallest positive integer $n$ such that $3^n-1$ is divisible by $7\cdot2^{20}$?

A right triangle has legs of lengths $3$ and $4$. On each of its three sides, an equilateral triangle is constructed outside the right triangle, as shown (the figure is not necessarily drawn to scale). What is the area of the triangle whose vertices are the centers of these three equilateral triangles?

How many real numbers $x$ in the interval $[0,2\pi]$ satisfy $$\cos x\cdot\cos2x\cdot\cos4x\cdot\cos8x=\frac1{16}\,?$$

A $2\times10$ board of unit squares is to be tiled, with no gaps or overlaps, using two kinds of tiles: dominoes ($1\times2$ rectangles, placed horizontally or vertically) and L-trominoes (three unit squares forming an L, in any of its four orientations), as shown. Any number of each kind of tile may be used. In how many ways can the board be tiled?