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AMC12 2023 B

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AMC12 · 2023 (B)

Q1
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
琼斯太太正在为她的四个儿子往四个相同的玻璃杯里倒橙汁。她把前三个玻璃杯完全装满，但当第四个玻璃杯只装满 $\frac{1}{3}$ 时汁用完了。琼斯太太必须从前三个玻璃杯的每个杯子里倒出多少杯的量到第四个玻璃杯里，使得四个玻璃杯都有相同量的汁？
Q2
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $\$43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
卡洛斯去体育用品店买跑鞋。跑鞋打折，每双鞋的价格降低 $20\%$ 。卡洛斯还知道他需要支付折扣价的 $7.5\%$ 销售税。他有 $\$$43 。他能买得起的最贵的鞋的原价（打折前）是多少？
Q3
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
一个 $3-4-5$ 直角三角形内接于圆 $A$ ，一个 $5-12-13$ 直角三角形内接于圆 $B$ 。圆 $A$ 的面积与圆 $B$ 的面积之比是多少？
Q4
Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint?
杰克逊的画笔画出一条宽度为 $6.5$ 毫米的窄条。杰克逊有足够的颜料画一条 $25$ 米长的条。那么杰克逊能涂多少平方厘米的面纸？
Q5
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
你正在玩一个游戏。一个 $2 \times 1$ 矩形覆盖 $3 \times 3$ 方格网格中两个相邻的方格（可以水平或垂直放置），但你不知道覆盖了哪两个方格。你的目标是找到至少一个被矩形覆盖的方格。一“回合”是你猜测一个方格，然后被告知该方格是否被隐藏的矩形覆盖。为了确保至少有一个猜测的方格被矩形覆盖，你需要的最少回合数是多少？
Q6
When the roots of the polynomial \[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot \cdot (x-10)^{10}\] are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?
当多项式 \[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot \cdot (x-10)^{10}\] 的根从数轴上移除后，剩余的是11个不相交的开区间。这些区间中有多少个上$P(x)$为正？
Q7
For how many integers $n$ does the expression\[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}\]represent a real number, where log denotes the base $10$ logarithm?
有且仅有有多少个整数$n$，使得表达式\[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}\]表示一个实数，其中log表示以10为底的对数？
Q8
How many nonempty subsets $B$ of $\{0, 1, 2, 3, \cdots, 12\}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = \{4, 6, 8, 11\}$ satisfies the condition.
集合$\{0, 1, 2, 3, \cdots, 12\}$有多少个非空子集$B$，使得$B$的元素个数等于$B$的最小元素？例如，$B = \{4, 6, 8, 11\}$满足条件。
Q9
What is the area of the region in the coordinate plane defined by $| | x | - 1 | + | | y | - 1 | \le 1$?
坐标平面中由 $| | x | - 1 | + | | y | - 1 | \le 1$定义的区域的面积是多少？
Q10
In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
在$xy$平面中，一个半径为4、圆心在正$x$轴上的圆与$y$轴在原点相切，一个半径为10、圆心在正$y$轴上的圆与$x$轴在原点相切。这两个圆相交的两点的连线斜率是多少？
Q11
What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other?
具有腿长为$1$且一条底边是另一条底边两倍长的等腰梯形的最大面积是多少？
Q12
For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation $u \otimes v = ac + bdi$ Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$. What is $|z|$?
对于复数$u = a+bi$和$v = c+di$（其中$i=\sqrt{-1}$），定义二元运算 $u \otimes v = ac + bdi$ 假设$z$是一个复数使得$z\otimes z = z^{2}+40$。$|z|$是多少？
Q13
A rectangular box $\mathcal{P}$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$?
一个长方体$\mathcal{P}$具有不同的边长$a$、$b$和$c$。$\mathcal{P}$所有$12$条边的长度和为$13$，所有$6$个面的面积和为$\frac{11}{2}$，体积为$\frac{1}{2}$。连接$\mathcal{P}$两个顶点的 longest interior diagonal 的长度是多少？
Q14
For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?
多项式$x^3+ax^2+bx+6$具有$3$个不同的整数根，有多少个整数有序对$(a,b)$？
Q15
Suppose $a$, $b$, and $c$ are positive integers such that\[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\]Which of the following statements are necessarily true? I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$. II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both. III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.
假设$a$、$b$和$c$是正整数，使得\[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\]以下哪些陈述一定是真的？ I. 如果$\gcd(a,14)=1$或$\gcd(b,15)=1$或两者皆然，则$\gcd(c,210)=1$。 II. 如果$\gcd(c,210)=1$，则$\gcd(a,14)=1$或$\gcd(b,15)=1$或两者皆然。 III. $\gcd(c,210)=1$当且仅当$\gcd(a,14)=\gcd(b,15)=1$。
Q16
In the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of $x?$
在Coinland国，硬币的面值为$6$、$10$和$15$美分。设$x$是用这些硬币精确支付Coinland国最贵的无法购买的物品的价值（美分）。$x$的各位数字之和是多少？
Q17
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$?
三角形$ABC$的三边长成等差数列，最小边长为$6$。若三角形有一个$120^\circ$的角，求$ABC$的面积。
Q18
Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was $3$ points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was $18$ points higher than her average for the first semester and was again $3$ points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true?
上学年，Yolanda和Zelda选修了不同的课程，每学期测验数量不一定相同。第一学期，Yolanda所有测验的平均分比Zelda第一学期所有测验平均分高3分。第二学期，Yolanda所有测验的平均分比她第一学期平均分高18分，并且再次比Zelda第二学期所有测验平均分高3分。以下哪个陈述不可能为真？
Q19
Each of $2023$ balls is randomly placed into one of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
有$2023$个球随机放入$3$个盒子。以下哪个选项最接近每个盒子含有奇数个球的概率？
Q20
Cyrus the frog jumps $2$ units in a direction, then $2$ more in another direction. What is the probability that he lands less than $1$ unit away from his starting position?
青蛙Cyrus先向一个方向跳$2$个单位，再向另一个方向跳$2$个单位。他落在距起点不到$1$个单位处的概率是多少？
Q21
A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is $3\sqrt3$ inches, its top diameter is $6$ inches, and its bottom diameter is $12$ inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?
一个灯罩由右圆锥台的侧面构成。圆锥台的高度为 $3\sqrt3$ 英寸，上底直径为 $6$ 英寸，下底直径为 $12$ 英寸。一只虫子在灯罩底部，有一团蜂蜜在灯罩上边缘的最远离虫子的位置。虫子想爬到蜂蜜处，但必须留在灯罩表面上。它的最短路径长度是多少英寸？
stem
Q22
A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ \[f(a + b) + f(a - b) = 2f(a) f(b).\] Which one of the following cannot be the value of $f(1)?$
一个实值函数 $f$ 具有如下性质：对所有实数 $a$ 和 $b$，\[f(a + b) + f(a - b) = 2f(a) f(b).\] 以下哪一个不可能是 $f(1)$ 的值？
Q23
When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$?
当掷 $n$ 个标准六面骰子时，所掷数字的乘积可以是 $936$ 个可能的值。$n$ 是多少？
Q24
Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations. \[abcd=2^6\cdot 3^9\cdot 5^7\] \[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\] \[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\] \[\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2\] \[\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2\] What is $\text{gcd}(a,b,c,d)$?
假设 $a$，$b$，$c$ 和 $d$ 是满足以下所有关系的正整数。 \[abcd=2^6\cdot 3^9\cdot 5^7\] \[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\] \[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\] \[\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2\] \[\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2\] $\text{gcd}(a,b,c,d)$ 是多少？
Q25
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
一个面积为 $\sqrt{5}+1$ 的正五边形印在纸上并剪下。五边形的五个顶点被折向五边形中心，形成一个更小的五边形。新五边形的面积是多少？
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