AMC 10A 2023

Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes toward $B$ 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 $\frac{1}{3}$ of a large pizza together with $3\frac{1}{2}$ cups of orange slices is the same as the weight of $\frac{3}{4}$ of a large pizza together with $\frac{1}{2}$ cup of orange slices. A cup of orange slices weighs $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza?

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

A quadrilateral has all integer side lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?

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

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$. What is the value of the cube?

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 $3$?

Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at $110$ degrees Fahrenheit, which is $0$ degrees on the Breadus scale. Bread is baked at $350$ degrees Fahrenheit, which is $100$ degrees on the Breadus scale. Bread is done when its internal temperature is $200$ degrees Fahrenheit. What is this, in degrees, on the Breadus scale?

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 $2023$ does 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 right triangles, as shown below. What is the ratio of the shorter leg to the longer leg in each shaded right triangle?

The outer square has area $3$ (side $\sqrt{3}$) and the inner tilted square has area $2$ (side $\sqrt{2}$). The four corner triangles are congruent right triangles with legs $a$ (shorter) and $b$ (longer), where $a + b = \sqrt{3}$ and $a^2 + b^2 = 2$.

How many three-digit positive integers $N$ satisfy the following properties?

• The number $N$ is divisible by $7$.
• The number formed by reversing the digits of $N$ is divisible by $5$.

Abdul and Chiang are standing $48$ feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures $60^\circ$. What is the square of the distance (in feet) between Abdul and Bharat?

A number is chosen at random from among the first $100$ positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by $11$?

$\textbf{(A)}~\dfrac{4}{100}\qquad\textbf{(B)}~\dfrac{9}{200}\qquad\textbf{(C)}~\dfrac{1}{20}\qquad\textbf{(D)}~\dfrac{11}{200}\qquad\textbf{(E)}~\dfrac{3}{50}$

An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1$. An example showing $8$ circles is displayed below. What is the least number of circles needed to make the total shaded area at least $2023\pi$?

$\textbf{(A) } 46 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 64$

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?

$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$

Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}$, $\triangle{PCQ}$, and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$?

$\textbf{(A) } 84 \qquad \textbf{(B) } 86 \qquad \textbf{(C) } 88 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 92$

A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?

$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r - s|$?

$\textbf{(A) } \dfrac{1}{4} \qquad \textbf{(B) } \dfrac{1}{2} \qquad \textbf{(C) } \dfrac{3}{4} \qquad \textbf{(D) } \dfrac{2}{3} \qquad \textbf{(E) } 1$

Rows 1, 2, 3, 4, and 5 of Pascal's triangle are shown below.

| Row 1 | 1 | |-------|---| | Row 2 | 1   1 | | Row 3 | 1   2   1 | | Row 4 | 1   3   3   1 | | Row 5 | 1   4   6   4   1 |

In which row of Pascal's triangle do three consecutive entries occur that are in the ratio $3:4:5$?

Let $P(x)$ be the unique polynomial of minimal degree with the following properties: $P(x)$ has a leading coefficient $1$, $1$ is a root of $P(x) - 1$, $2$ is a root of $P(x-2)$, $3$ is a root of $P(3x)$, and $4$ is a root of $4P(x)$. The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime integers. What is $m + n$?

$\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49$

Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\dfrac{1}{2}$. 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 $C_4$?

$\textbf{(A) } \dfrac{1}{14} \qquad \textbf{(B) } \dfrac{1}{12} \qquad \textbf{(C) } \dfrac{1}{10} \qquad \textbf{(D) } \dfrac{3}{28} \qquad \textbf{(E) } \dfrac{1}{9}$

If the positive integer $c$ has positive integer divisors $a$ and $b$ with $c = ab$, then $a$ and $b$ are said to be complementary divisors of $c$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ by $23$. What is the sum of the digits of $N$?

$\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

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 $d(Q, R) > d(R, S)$?

$\textbf{(A) } \dfrac{7}{22} \qquad \textbf{(B) } \dfrac{1}{3} \qquad \textbf{(C) } \dfrac{3}{8} \qquad \textbf{(D) } \dfrac{5}{12} \qquad \textbf{(E) } \dfrac{1}{2}$