2023 AMC 12A

Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B.$ Alicia bikes towards $B$ at $18$ miles per hour. Leaving at the same time, Beth bikes toward $A$ at $12$ miles per hour. How many miles from City $A$ will they be when they meet?

The weight of $\tfrac13$ of a large pizza together with $3\tfrac12$ cups of orange slices is the same as the weight of $\tfrac34$ of a large pizza together with $\tfrac12$ cup of orange slices. A cup of orange slices weighs $\tfrac14$ of a pound. What is the weight, in pounds, of a large pizza?

How many positive perfect squares less than $2023$ are divisible by $5?$

How many digits are in the base-ten representation of $8^5\cdot 5^{10}\cdot 15^5?$

Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3?$

Points $A$ and $B$ lie on the graph of $y=\log_2 x.$ The midpoint of $\overline{AB}$ is $(6,2).$ What is the positive difference between the $x$-coordinates of $A$ and $B?$

A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as $20230428.$ For how many dates in $2023$ will each digit appear an even number of times in the $8$-digit display for that date?

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1.$ If she scores an $11$ on each of the next three quizzes, her mean will increase by $2.$ What is the mean of her quiz scores currently?

A square of area $2$ is inscribed in a square of area $3,$ creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?

Positive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2.$ What is $x+y?$

What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac13?$

What is the value of $$\begin{gathered} 2^3-1^3+4^3-3^3+6^3-5^3\\ {}+\cdots+18^3-17^3? \end{gathered}$$

In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?

How many complex numbers satisfy the equation $z^5=\overline{z},$ where $\overline{z}$ is the conjugate of the complex number $z?$

Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}.$ He wants to increase the distance walked by zigzagging as shown in the figure below ($APQRS$). What angle $\theta=\angle PAB$ $=\angle QPC$ $=\angle RQB=\cdots$ will produce a length that is $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)

Consider the set of complex numbers $z$ satisfying $|1+z+z^2|=4.$ The maximum value of the imaginary part of $z$ can be written in the form $\dfrac{\sqrt{m}}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\dfrac{1}{2^m}.$ What is the probability that Flora will eventually land at $10?$

Circle $C_1$ and $C_2$ each have radius $1,$ and the distance between their centers is $\tfrac12.$ Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2.$ Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3.$ What is the radius of $C_4?$

What is the product of all the solutions to the equation $$\begin{gathered} \log_{7x}2023\cdot\log_{289x}2023\\ {}=\log_{2023x}2023? \end{gathered}$$

Rows $1,$ $2,$ $3,$ $4,$ and $5$ of a triangular array of integers are shown below. $$\begin{array}{ccccccccc} &&&&1&&&&\\ &&&1&&1&&&\\ &&1&&3&&1&&\\ &1&&5&&5&&1&\\ 1&&7&&11&&7&&1 \end{array}$$ Each row after the first row is formed by placing a $1$ at each end of the row, and each interior entry is $1$ greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the $2023$ numbers in the $2023$rd row?

If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B.$ For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A,B)=1,$ but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A,B)=2.$ Let $Q,$ $R,$ and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of $20$ equilateral triangles). What is the probability that $d(Q,R)\gt d(R,S)?$

Let $f$ be the unique function defined on the positive integers such that $$\sum_{d\mid n} d\cdot f\left(\frac{n}{d}\right)=1$$ for all positive integers $n,$ where the sum is taken over all positive divisors of $n.$ What is $f(2023)?$

How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation $$\begin{gathered} (1+2a)(2+2b)(2a+b)\\ {}=32ab? \end{gathered}$$

Let $K$ be the number of sequences $A_1,$ $A_2,$ $\ldots,$ $A_n$ such that $n$ is a positive integer less than or equal to $10,$ each $A_i$ is a subset of $\{1,2,3,\ldots,10\},$ and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n,$ inclusive. For example, $\{\},$ $\{5,7\},$ $\{2,5,7\},$ $\{2,5,7\},$ $\{2,5,6,7,9\}$ is one such sequence, with $n=5.$ What is the remainder when $K$ is divided by $10?$

There is a unique sequence of integers $a_1,$ $a_2,$ $\cdots a_{2023}$ such that $$\begin{gathered} \tan 2023x\\ {}=\tiny\dfrac{a_1\tan x+a_3\tan^3 x+a_5\tan^5 x+\cdots+a_{2023}\tan^{2023}x}{1+a_2\tan^2 x+a_4\tan^4 x\cdots+a_{2022}\tan^{2022}x} \end{gathered}$$ whenever $\tan 2023x$ is defined. What is $a_{2023}?$