/

AMC10 2023 B

You are not logged in. After submit, your report may not be available on other devices. Login

AMC10 · 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?
琼斯夫人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，但在第四个玻璃杯只装满 $\frac{1}{3}$ 时汁用完了。琼斯夫人必须从前三个玻璃杯中每个倒出多少杯的量到第四个玻璃杯中，使得四个玻璃杯中的汁量相同？
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纸？
Q5
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard?
麦迪和劳拉看到黑板上写着一列数字。麦迪将列表中每个数字加 $3$，发现新数字的和是 $45$。劳拉将列表中每个数字乘以 $3$，发现新数字的和也是 $45$。黑板上写了多少个数字？
Q6
Let $L_{1}=1, L_{2}=3$, and $L_{n+2}=L_{n+1}+L_{n}$ for $n\geq 1$. How many terms in the sequence $L_{1}, L_{2}, L_{3},...,L_{2023}$ are even?
设 $L_{1}=1, L_{2}=3$，且 $L_{n+2}=L_{n+1}+L_{n}$ 对于 $n\geq 1$。序列 $L_{1}, L_{2}, L_{3},...,L_{2023}$ 中有多少项是偶数？
Q7
Square $ABCD$ is rotated $20^{\circ}$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\angle EAB$? $\text{
正方形 $ABCD$ 围绕其中心顺时针旋转 $20^{\circ}$ 得到正方形 $EFGH$，如下图所示。$\angle EAB$ 的度数是多少？ $\text{
stem
Q8
What is the units digit of $2022^{2023} + 2023^{2022}$?
$2022^{2023} + 2023^{2022}$ 的个位数是多少？
Q9
The numbers $16$ and $25$ are a pair of consecutive positive squares whose difference is $9$. How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$?
数字 $16$ 和 $25$ 是一对相邻正平方数，它们的差是 $9$。有多少对相邻正完全平方数的差小于或等于 $2023$？
Q10
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”包括你猜测一个方格，然后被告知该方格是否被隐藏矩形覆盖。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？
Q11
Suzanne went to the bank and withdrew $\$800$. The teller gave her this amount using $\$20$ bills, $\$50$ bills, and $\$100$ bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
Suzanne去银行取了$\$800$。出纳员使用$\$20$面额、$\$50$面额和$\$100$面额的钞票给她这个金额，至少每种面额各一张。Suzanne可能收到的不同钞票组合有多少种？
Q12
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，剩下的是$11$个不相交的开区间。这些区间中有多少个上$P(x)$为正？
Q13
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？
Q14
How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$?
有多少个整数有序对$(m, n)$满足方程$m^2+mn+n^2 = m^2n^2$？
Q15
What is the least positive integer $m$ such that $m\cdot2!\cdot3!\cdot4!\cdot5!...16!$ is a perfect square?
最小的正整数$m$是多少，使得$m\cdot2!\cdot3!\cdot4!\cdot5!...16!$是一个完全平方数？
Q16
Define an $\textit{upno}$ to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a $\textit{downno}$ to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. For instance, the number $258$ is an upno and $8620$ is a downno. Let $U$ equal the total number of $upnos$ and let $D$ equal the total number of $downnos$. What is $|U-D|$?
将一个正整数定义为\textit{upno}，如果它有$2$位或更多位，且从左到右数字严格递增。类似地，将一个正整数定义为\textit{downno}，如果它有$2$位或更多位，且从左到右数字严格递减。例如，数字$258$是一个upno，$8620$是一个downno。令$U$为所有upno的总数，$D$为所有downno的总数。求$|U-D|$？
Q17
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$\mathcal{P}$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$\frac{1}{2}$。求$\mathcal{P}$连接两个顶点的 longest interior diagonal 的长度？
Q18
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\[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\]以下哪些陈述必然成立？ 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$\gcd(a,14)=\gcd(b,15)=1$。
Q19
Sonya the frog chooses a point uniformly at random lying within the square $[0, 6]$ $\times$ $[0, 6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at random from {north, south, east, west}. All of her choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?
青蛙Sonya在坐标平面上的正方形$[0, 6]$ $\times$ $[0, 6]$内均匀随机选择一个点并跳到该点。然后她均匀随机从$[0, 1]$选择一个距离，并从{north, south, east, west}均匀随机选择一个方向。她的所有选择相互独立。她现在沿选择的方跳跃该距离。求她落在正方形外的概率？
Q20
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is $n$?
在半径为$2$的球体表面上绘制了四个全等的半圆，如图所示，形成一个闭合曲线，将表面分成两个全等区域。该曲线的长度为$\pi\sqrt{n}$。求$n$？
stem
Q21
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，每个球被随机放入 $3$ 个盒子中。以下哪个是最接近于每个盒子中球的个数均为奇数的概率？
Q22
How many distinct values of $x$ satisfy $\lfloor{x}\rfloor^2-3x+2=0$, where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$?
有几个不同的 $x$ 满足 $\lfloor{x}\rfloor^2-3x+2=0$，其中 $\lfloor{x}\rfloor$ 表示小于或等于 $x$ 的最大整数？
Q23
An arithmetic sequence of positive integers has $n \ge 3$ terms, initial term $a$, and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$. The sum of the terms he wrote was $222$. What is $a + d + n$?
一个正整数等差数列有 $n \ge 3$ 个项，首项 $a$，公差 $d > 1$。Carl 正确写下了这个数列的所有项，除了其中一个项，差了 $1$。他写下的项的和是 $222$。求 $a + d + n$？
Q24
What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$, $0\le v\le1,$ and $0\le w\le1$?
由所有可以表示为 $(2u-3w, v+4w)$ 的点的区域的边界的周长是多少，其中 $0\le u\le1$，$0\le v\le1$，且 $0\le w\le1$？
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的中心，形成一个更小的五边形。新五边形的面积是多少？
Time Left 75:00