AMC 12 Mock Exam 1

A courier rides from a depot to a warehouse at a steady $12$ miles per hour. After waiting $20$ minutes at the warehouse while a package is loaded, she rides back to the depot along the same route at a steady $18$ miles per hour. From the moment she leaves the depot until the moment she returns, exactly $5$ hours and $20$ minutes elapse. How many miles is the warehouse from the depot?

A list of five positive integers has mean $14$, median $12$, and a unique mode, which equals $9$. What is the greatest possible value of the largest integer in the list?

Lena's bicycle lock opens with a three-digit code, and the first digit of the code is not $0$. Lena has forgotten the code, but she remembers that the product of its three digits is $36$. How many codes are consistent with what Lena remembers?

What is the units digit of $3^{2026}+7^{2026}$?

A rhombus has side length $10$ and one diagonal of length $12$. What is the radius of the circle inscribed in the rhombus?

A linear function $f(x)=ax+b$, where $a$ and $b$ are real numbers, satisfies $f(f(x))=4x+9$ for all real numbers $x$. What is the sum of all possible values of $f(1)$?

A jar contains $4$ red marbles, $3$ blue marbles, and $2$ green marbles. Three marbles are drawn at random from the jar without replacement. What is the probability that the three marbles drawn are all different colors?

How many positive divisors of $2^4\cdot3^3\cdot5^2$ are perfect squares or perfect cubes (or both)?

Positive real numbers $a$ and $b$ satisfy $\log_8 a+\log_4 b=5$ and $\log_4 a+\log_8 b=5$. What is $a+b$?

Circles $\omega_1$ and $\omega_2$ have radii $3$ and $12$, respectively, and are externally tangent to each other. Both circles are tangent to a line $\ell$ and lie on the same side of $\ell$. Circle $\omega_3$ is tangent to $\ell$ and externally tangent to both $\omega_1$ and $\omega_2$, and it lies in the region bounded by $\ell$, $\omega_1$, and $\omega_2$, as shown (the figure is not necessarily drawn to scale). What is the radius of $\omega_3$?

For how many positive integers $n\le100$ is $n^2-1$ divisible by $40$?

In triangle $ABC$, $AB=10$, $BC=12$, and $CA=16$. A circle passes through $A$ and $B$ and is tangent to line $BC$ at $B$. The circle meets side $\overline{AC}$ again at a point $D$ other than $A$. What is $BD$?

What is the sum of all real numbers $x$ in the interval $[0,2\pi)$ that satisfy $\sin 2x=\cos 3x$?

A regular tetrahedron has edge length $6$. A sphere centered at the center of the tetrahedron is tangent to each of the six edges of the tetrahedron. What is the surface area of the sphere?

Real numbers $x$, $y$, and $z$ satisfy $x+y+z=6$, $x^2+y^2+z^2=14$, and $x^3+y^3+z^3=36$. What is $x^4+y^4+z^4$?

A florist has $12$ flower pots in a row, numbered $1$ through $12$. She chooses some of the pots (possibly none) to receive a ribbon, subject to the rule that no two chosen pots have numbers that differ by exactly $3$. In how many ways can she choose the pots?

For a real number $t$, let $\{t\}=t-\lfloor t\rfloor$ denote the fractional part of $t$. How many real numbers $x$ with $1\le x<10$ satisfy $\{x^2\}=\{x\}^2$?

Square $ABCD$ has side length $4$. Circle $\omega_1$ passes through $A$ and $B$ and is tangent to side $\overline{CD}$. Circle $\omega_2$ lies inside the square, is tangent to sides $\overline{BC}$ and $\overline{CD}$, and is externally tangent to $\omega_1$, as shown. What is the radius of $\omega_2$?

For a positive integer $m$, let $S(m)$ denote the sum of the digits of $m$ in base ten. For how many positive integers $n<1000$ is $S(2n)=S(n)$?

What is the value of the product $$\left(3-2\cos\frac{2\pi}{11}\right)\left(3-2\cos\frac{4\pi}{11}\right)\left(3-2\cos\frac{6\pi}{11}\right)\left(3-2\cos\frac{8\pi}{11}\right)\left(3-2\cos\frac{10\pi}{11}\right)?$$

A function $\sigma:\{1,2,\dots,10\}\to\{1,2,\dots,10\}$ is chosen uniformly at random from among the $10!$ bijections of $\{1,2,\dots,10\}$ to itself. Let $N$ be the number of integers $x$ in $\{1,2,\dots,10\}$ for which $\sigma(\sigma(\sigma(x)))=x$. What is the expected value of $N$?

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. A circle $\omega$ is tangent to both rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and is internally tangent to the circumcircle of triangle $ABC$ (so $\omega$ lies inside the circumcircle, as shown; the figure is not necessarily drawn to scale). What is the radius of $\omega$?

Let $f$ be a function from the positive integers to the positive integers such that $f$ is strictly increasing and $f(f(n))=3n$ for every positive integer $n$. What is $f(2026)$?

What is the value of $$\sum_{k=1}^{9}\frac{1}{\sin^2\left(\dfrac{k\pi}{10}\right)}=\frac{1}{\sin^2\frac{\pi}{10}}+\frac{1}{\sin^2\frac{2\pi}{10}}+\cdots+\frac{1}{\sin^2\frac{9\pi}{10}}\,?$$

A cashier has $12$ envelopes on a table, containing $\$1, \$2, \$3, \ldots, \$12$, respectively. She will pick up some of the envelopes, possibly none, so that the total amount of money she picks up is a multiple of $\$12$. (Picking up no envelopes gives a total of $\$0$, which counts.) How many different sets of envelopes can she pick up?