AMC 12 Mock Exam 4

Identical plastic cups are stacked one inside another. A stack of $12$ cups is $40$ centimeters tall, and a stack of $20$ cups is $56$ centimeters tall. How many centimeters tall is a stack of $30$ of these cups?

What is the sum of the digits of the product $2026\cdot9999$?

A right triangle has legs of lengths $20$ and $21$. What is the radius of the circle inscribed in the triangle?

A bakery sells muffins individually for $\$3$ each or in boxes of $4$ muffins for $\$10$ per box. Maya spent exactly $\$47$ on muffins at the bakery. How many muffins did she buy?

Five friends, including Ana, Ben, and Cal, line up in a row for a photo. Ana and Ben insist on standing next to each other, and Cal refuses to stand at either end of the row. In how many ways can the five friends line up?

How many ordered pairs $(a,b)$ of positive integers satisfy $ab=2592$ and $\gcd(a,b)=6$?

The lines $y=2x$ and $y=-\tfrac12x+10$ and the $x$-axis bound a triangle. What is the area of that triangle?

Let $f(x)=x^2-6x+c$, where $c$ is a constant. The smallest value that $f(x)$ takes as $x$ ranges over all real numbers is $4$. What is $f(-1)$?

A teacher writes the numbers $1$ through $10$ on ten cards, one number per card, shuffles the cards, and deals three of them to Ana. Ana wins a prize if she can arrange her three numbers in some order so that they form an arithmetic progression (for example, $2$, $5$, $8$). What is the probability that Ana wins a prize?

What is the value of $\dfrac{(1+i)^{2026}}{(1-i)^{2022}}$, where $i=\sqrt{-1}$?

A sequence is defined by $a_1=2$, $a_2=3$, and $$a_{n+1}=\frac{a_n+1}{a_{n-1}}\qquad\text{for } n\ge2.$$ What is $a_1+a_2+a_3+\cdots+a_{2026}$?

A pyramid has a square base with side length $8$, and each of its four lateral edges (the edges from the apex to the vertices of the base) has length $9$, as shown. What is the volume of the pyramid?

What is the sum of all positive integers $n$ for which $n^2+30n$ is a perfect square?

Let $\theta$ be an acute angle with $\tan\theta=2$. What is $\sin3\theta$?

A gardener's triangular plot has corners at $(0,0)$, $(30,0)$, and $(0,20)$ in the coordinate plane. She plants a tree at every point with integer coordinates that lies strictly inside the triangle. Points on the boundary of the triangle are reserved for the fence, so no trees are planted there. How many trees does she plant?

An orchard has a tree at every point $(x,y)$ with integer coordinates satisfying $1\le x\le10$ and $1\le y\le10$. Lena stands at the origin $(0,0)$. A tree is visible to Lena if the line segment from $(0,0)$ to that tree contains no point with integer coordinates other than its two endpoints. (Treat each tree as a single point.) How many trees are visible to Lena?

Square $ABCD$ has side length $1$. Point $E$ lies on side $\overline{BC}$ with $BE=\tfrac13$, and point $F$ lies on side $\overline{CD}$ so that $\angle EAF=45^\circ$, as shown (the figure is not necessarily drawn to scale). What is the area of $\triangle AEF$?

What is the product of all positive real numbers $x$ that satisfy $$\log_x2+\log_{2x}2+\log_{4x}2=0\,?$$

Regular hexagon $ABCDEF$ has side length $2$. A point $P$ inside the hexagon is chosen so that the area of $\triangle PAB$ is $1$ and the area of $\triangle PCD$ is $2$, as shown (the figure is not necessarily drawn to scale). What is the area of $\triangle PEF$?

Priya is designing a weekly exercise routine, Monday through Sunday, that she will repeat every week indefinitely. On each day of the week she either trains or rests. To avoid injury, she must never train on three consecutive days, and this includes stretches that cross from one week into the next (for example, Saturday, Sunday, and the following Monday). How many different weekly routines are possible?

Real numbers $x$, $y$, and $z$ satisfy $x^2+y^2+z^2=1$. What is the greatest possible value of $xy+2yz$?

A function $f$ is defined for all real numbers except $0$ and $1$, and satisfies $$f(x)+f\left(\frac{1}{1-x}\right)=x$$ for every real number $x$ other than $0$ and $1$. What is $f(3)$?

Consider all ordered pairs $(x,y)$ of positive real numbers satisfying both $$x^{x+y}=y^{12}\qquad\text{and}\qquad y^{x+y}=x^{3}.$$ What is the sum of the values of $x+y$ over all such ordered pairs?

Each of the six members of a club chooses one member of the club to be her leader; a member may choose herself. The club has one rule: any member who is chosen as a leader by at least one member must have chosen herself as her own leader. In how many ways can the six choices be made?

Jun rolls a fair six-sided die repeatedly. He stops immediately after the first roll that shows a number strictly less than the number shown on the roll just before it. For example, if his rolls are $2$, $2$, $5$, $3$, he stops after the fourth roll. What is the expected number of times Jun rolls the die?