AMC 12 Mock Exam 5

At a school supply store, $3$ pencils cost the same as $2$ erasers. Mina buys $5$ pencils and $4$ erasers and pays a total of $\$5.50$. How much does one eraser cost?

The lockers in a school hallway are numbered $1$ through $199$. A locker receives a blue sticker if its number is a multiple of $6$ but not a multiple of $4$. How many lockers receive a blue sticker?

The mean score of the $10$ students in a class on a quiz is $72$. Two students who were absent take the quiz later. One of them scores $90$, and after both new scores are included, the mean of all $12$ scores is $74$. What did the other student score?

A rectangle with side lengths $6$ and $8$ is inscribed in a circle. What is the area of the region that is inside the circle but outside the rectangle?

A smoothie bar offers $6$ different fruits. Each smoothie is made from at least one and at most three different fruits, and two smoothies are considered the same if they use the same set of fruits. How many different smoothies can be made?

Real numbers $a$ and $b$ satisfy $2^a=3$ and $3^b=16$. What is the value of $ab$?

An isosceles trapezoid has parallel sides of lengths $8$ and $18$. A circle inside the trapezoid is tangent to all four of its sides. What is the radius of the circle?

In a board game, a player rolls three fair standard six-sided dice and wins the round if the three numbers rolled could be the side lengths of a nondegenerate triangle. (For example, $2,3,4$ wins, but $2,2,4$ does not.) What is the probability that the player wins the round?

Real numbers $b$ and $c$ have the following property: the roots of $x^2+bx+c$ are $r$ and $s$, and the roots of $x^2+cx+b$ are $r+1$ and $s+1$. What is $b+c$?

Two lighthouse beacons flash at regular intervals: the first flashes every $a$ seconds and the second every $b$ seconds, where $a$ and $b$ are positive integers. The beacons flash together at time $0$, and the next time they flash together is exactly $72$ seconds later. How many ordered pairs $(a,b)$ are possible?

As $x$ ranges over all real numbers, what is the least possible value of $\sin^6x+\cos^6x$?

In triangle $ABC$, $\angle BAC=120^\circ$, $AB=3$, and $AC=6$. The bisector of $\angle BAC$ meets side $\overline{BC}$ at $D$, as shown (the figure is not necessarily drawn to scale). What is $AD$?

What is the value of $$\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+\cdots+\frac{1}{20\cdot21\cdot22}\,?$$

A pyramid has a square base, and all eight of its edges have length $6$. A sphere inside the pyramid is tangent to the base and to each of the four triangular faces. What is the radius of the sphere?

In triangle $ABC$, $BC=13$, $CA=15$, and $\angle BAC=60^\circ$. What is the sum of all possible values of $AB$?

For how many positive integers $n\le2026$ is the greatest common divisor of $n^2+1$ and $(n+1)^2+1$ greater than $1$?

A bracelet has six beads equally spaced around a circle. Two of the beads are red, two are green, and two are blue; beads of the same color are indistinguishable. Two bracelets are considered the same if one can be rotated and/or flipped over to match the other. How many different bracelets are possible?

How many ordered triples $(a,b,c)$ of positive integers satisfy both $\operatorname{lcm}(a,b,c)=600$ and $\gcd(a,b,c)=1$?

How many distinct integers appear in the list $$\left\lfloor\frac{1^2}{2026}\right\rfloor,\ \left\lfloor\frac{2^2}{2026}\right\rfloor,\ \left\lfloor\frac{3^2}{2026}\right\rfloor,\ \ldots,\ \left\lfloor\frac{2026^2}{2026}\right\rfloor\,?$$

Triangle $ABC$ has side lengths $AB=5$, $BC=8$, and $CA=7$. What is the distance between the incenter and the circumcenter of triangle $ABC$?

Let $\omega=\cos\dfrac{2\pi}{5}+i\sin\dfrac{2\pi}{5}$. What is the value of $$\frac{1}{2-\omega}+\frac{1}{2-\omega^2}+\frac{1}{2-\omega^3}+\frac{1}{2-\omega^4}\,?$$

A movie theater sells tickets for $\$5$ each. Ten people wait in line: five of them each have a single $\$5$ bill, and the other five each have a single $\$10$ bill. Each person buys one ticket, and whenever someone pays with a $\$10$ bill, the cashier must immediately give back a $\$5$ bill as change. The cashier starts with exactly one $\$5$ bill in the register. If the ten people are lined up in a uniformly random order, what is the probability that the cashier is always able to give change?

In triangle $ABC$, $BC=6$ and $\angle BAC=60^\circ$. What is the maximum possible value of $AB+2\cdot AC$?

How many polynomials of the form $x^3+ax^2+bx+c$, where $a$, $b$, and $c$ are integers, have the property that each of their three roots (counted with multiplicity, possibly nonreal) lies on the unit circle $|z|=1$ in the complex plane?

A positive integer is a palindrome if its base-ten digits read the same forward and backward; for example, $4004$ and $12321$ are palindromes. How many palindromes $n$ with $n<10^7$ are divisible by $1001$?