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AMC10 2023 A

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AMC10 · 2023 (A)

Q1
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?
城市 $A$ 和 $B$ 相距 $45$ 英里。Alicia 住在 $A$，Beth 住在 $B$。Alicia 以每小时 $18$ 英里的速度向 $B$ 骑车。同时出发，Beth 以每小时 $12$ 英里的速度向 $A$ 骑车。他们相遇时距离城市 $A$ 有多少英里？
Q2
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?
一大块披萨的 $\frac{1}{3}$ 加上 $3 \frac{1}{2}$ 杯橙子片的分量，与一大块披萨的 $\frac{3}{4}$ 加上 $\frac{1}{2}$ 杯橙子片的分量相同。一杯橙子片重 $\frac{1}{4}$ 磅。大披萨的重量是多少磅？
Q3
How many positive perfect squares less than $2023$ are divisible by $5$?
小于 $2023$ 的正完全平方数中，有多少个能被 $5$ 整除？
Q4
A quadrilateral has all integer sides lengths, a perimeter of $26$, and one side of length $4$. What is the greatest possible length of one side of this quadrilateral?
一个四边形的所有边长均为整数，周长为 $26$，有一边长为 $4$。该四边形的最长一边可能的最大长度是多少？
Q5
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?
$8^5 \cdot 5^{10} \cdot 15^5$ 的十进制表示中有多少位数字？
Q6
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?
一个立方体的每个顶点都被赋上一个整数。边的值为它连接的两个顶点的值之和，面的值为围绕它的四个边的值之和。立方体的值为其六个面的值之和。假设赋给顶点的整数之和为$21$。立方体的值为多少？
Q7
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 $3$?
Janet掷一个标准的$6$面骰子$4$次，并保持她掷出的数字的累积总和。累积总和在某个时刻等于$3$的概率是多少？
Q8
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?
面包师Barb为她的面包店开发了一种新的温度标度，称为Breadus标度，它是华氏标度的线性函数。面包在$110$华氏度时发酵，这对应Breadus标度的$0$度。面包在$350$华氏度时烘烤，这对应Breadus标度的$100$度。面包内部温度达到$200$华氏度时完成烘烤。这在Breadus标度上是多少度？
Q9
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 $8$-digital display for that date?
一个数字显示屏将当前日期显示为一个$8$位整数，由$4$位年份，后跟$2$位月份，后跟月份内的$2$位日期。例如，今年植树节显示为$20230428$。在$2023$年的多少个日期中，$8$位数字显示中每个数字出现的次数是偶数？
Q10
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?
Maureen正在跟踪她这学期测验成绩的平均分。如果她在下一次测验中得$11$分，她的平均分将增加$1$。如果她在接下来三次测验中各得$11$分，她的平均分将增加$2$。她当前测验成绩的平均分是多少？
Q11
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?
一个面积为$2$的正方形内接在一个面积为$3$的正方形中，形成了四个全等的三角形，如下图所示。阴影直角三角形的短腿与长腿的比率为多少？
stem
Q12
How many three-digit positive integers $N$ satisfy the following properties?
有多少个三位正整数$N$满足以下性质？
Q13
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?
Abdul和Chiang在田野中相距$48$英尺。Bharat站在同一田野中，尽可能远离Abdul，同时他看向Abdul和Chiang的视线形成的夹角为$60^\circ$。Abdul和Bharat之间的距离平方（英尺）是多少？
Q14
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 $11$?
从前$100$个正整数中随机选一个数，然后从该数的正整数因数中随机选一个。选出的因数能被$11$整除的概率是多少？
Q15
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 $2023\pi$?
有偶数个圆嵌套，起始半径为$1$，每次增加$1$，所有圆共有一个公共点。每隔一个圆之间的区域被涂阴影，从半径$2$的圆内但半径$1$的圆外的区域开始。下方显示了$8$个圆的示例。需要最少多少个圆才能使总阴影面积至少为$2023\pi$？
stem
Q16
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?
在一次乒乓球锦标赛中，每位参赛者与其他每位参赛者恰好对战一次。虽然右手球员的数量是左手球员的两倍，但左手球员赢得的比赛数量比右手球员多40%。（没有平局，也没有双手球员。）总共进行了多少场比赛？
Q17
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 $\triangle{APQ}$?
设 $ABCD$ 为矩形，$AB = 30$，$BC = 28$。点 $P$ 和 $Q$ 分别位于 $\overline{BC}$ 和 $\overline{CD}$ 上，使得 $\triangle{ABP}$、$\triangle{PCQ}$ 和 $\triangle{QDA}$ 的所有边长均为整数。求 $\triangle{APQ}$ 的周长。
Q18
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?
菱形十二面体是一个有 $12$ 个全等菱形面的立体图形。在每个顶点，有 $3$ 或 $4$ 条边相交，取决于顶点。有多少个顶点恰好有 $3$ 条边相交？
Q19
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 $|r-s|$?
由 $A(1, 2)$ 和 $B(3, 3)$ 形成的线段绕点 $P(r, s)$ 旋转成由 $A'(3, 1)$ 和 $B'(4, 3)$ 形成的线段。求 $|r-s|$？
Q20
Each square in a $3\times3$ grid of squares is colored red, white, blue, or green so that every $2\times2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
一个 $3\times3$ 的方格网格中的每个小方格涂成红色、白色、蓝色或绿色，使得每个 $2\times2$ 大方格包含每种颜色恰好一个。如下右图所示的一种着色。可能有多少种不同的着色？
stem
Q21
Let $P(x)$ be the unique polynomial of minimal degree with the following properties: The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. What is $m+n$?
设 $P(x)$ 是具有以下性质的最低次数唯一多项式： $P(x)$ 的根都是整数，除了一个例外。那个非整数根可以写成 $\frac{m}{n}$，其中 $m$ 和 $n$ 是互质的整数。求 $m+n$？
Q22
Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. 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 $C_4$?
圆 $C_1$ 和 $C_2$ 半径均为 $1$，两圆心距离为 $\frac{1}{2}$。圆 $C_3$ 是与 $C_1$ 和 $C_2$ 都内切的最大的圆。圆 $C_4$ 与 $C_1$ 和 $C_2$ 都内切，且与 $C_3$ 外切。求 $C_4$ 的半径。
stem
Q23
If the positive integer $c$ has positive integer divisors $a$ and $b$ with $c = ab$, then $a$ and $b$ are said to be $\textit{complementary}$ 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 $N$?
如果正整数 $c$ 有正整数除数 $a$ 和 $b$ 满足 $c = ab$，则称 $a$ 和 $b$ 是 $c$ 的\textit{互补除数}。假设正整数 $N$ 有一对互补除数之差为 $20$，另一对互补除数之差为 $23$。求 $N$ 的各位数字之和。
Q24
Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What is the area of the region inside the frame not occupied by the blocks?
六个边长为 $1$ 单位的正六边形木块排列在一个正六边形框架内。每个木块位于框架内侧边上，并与另外两个木块对齐，如图所示。从框架的任意角到最近的木块顶点的距离为 $\frac{3}{7}$ 单位。求框架内未被木块占据的区域面积。
stem
Q25
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 $d(Q, R) > d(R, S)$?
若 $A$ 和 $B$ 是多面体的顶点，定义距离 $d(A,B)$ 为连接 $A$ 和 $B$ 所需穿越的最少边数。例如，若 $\overline{AB}$ 是多面体的一条边，则 $d(A, B) = 1$；若 $\overline{AC}$ 和 $\overline{CB}$ 是边而 $\overline{AB}$ 不是，则 $d(A, B) = 2$。设正二十面体（由 $20$ 个正三角形组成的正多面体）的三个不同顶点 $Q$、$R$ 和 $S$ 随机选取。求 $d(Q, R) > d(R, S)$ 的概率。
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