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AMC10 2024 B

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AMC10 · 2024 (B)

Q1
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
在一长队人从左到右排列中，从左数第1013人是同时也是从右数第1010人。队列中共有多少人？
Q2
What is $10! - 7! \cdot 1!$ ?
$10! - 7! \cdot 1!$ 等于多少？
Q3
For how many integer values of $x$ is $|2x| \leq 7 \pi$
有整数 $x$ 满足 $|2x| \leq 7 \pi$ 共有多少个？
Q4
Balls numbered 1, 2, 3, ... are deposited in 5 bins, labeled A, B, C, D, and E, using the following procedure. Ball 1 is deposited in bin A, and balls 2 and 3 are deposited in bin B. The next 3 balls are deposited in bin C, the next 4 in bin D, and so on, cycling back to bin A after balls are deposited in bin E. (For example, balls numbered 22, 23, ..., 28 are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited?
编号为1、2、3、...的小球被放入标记为A、B、C、D、E的5个箱子中，使用以下程序。小球1放入箱子A，小球2和3放入箱子B。接下来的3个小球放入箱子C，接下来的4个放入箱子D，依此类推，在放入箱子E后循环回到箱子A。（例如，第7步将编号22、23、...、28的小球放入箱子B。）小球2024放入哪个箱子？
Q5
In the following expression, Melanie changed some of the plus signs to minus signs: \[1+3+5+7+...+97+99\] When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
在以下表达式中，Melanie将一些加号改为减号： \[1+3+5+7+...+97+99\] 新表达式计算后为负数。Melanie最少改动了多少个加号为减号？
Q6
A rectangle has integer length sides and an area of 2024. What is the least possible perimeter of the rectangle?
一个长宽为整数的矩形，面积为2024。求该矩形的最小可能周长？
Q7
What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19$?
$7^{2024}+7^{2025}+7^{2026}$ 除以 $19$ 的余数是多少？
Q8
Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$?
设 $N$ 是 $42$ 的所有正整数除数的乘积。$N$ 的个位数是多少？
Q9
Real numbers $a, b,$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2,$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac,$ and $bc$?
实数 $a, b,$ 和 $c$ 的算术平均数为 $0$。$a^2, b^2,$ 和 $c^2$ 的算术平均数为 $10$。求 $ab, ac,$ 和 $bc$ 的算术平均数？
Q10
Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of $\triangle CFB$?
四边形 $ABCD$ 是平行四边形，$E$ 是边 $\overline{AD}$ 的中点。$F$ 是直线 $EB$ 与 $AC$ 的交点。求四边形 $CDEF$ 的面积与 $\triangle CFB$ 的面积之比？
Q11
In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$? Note: On certain tests that took place in China, the problem asked for the area of $\triangle MAY$.
下图中 $WXYZ$ 是一个矩形，$WX=4$，$WZ=8$。点 $M$ 在 $\overline{XY}$ 上，点 $A$ 在 $\overline{YZ}$ 上，且 $\angle WMA$ 是直角。$\triangle WXM$ 和 $\triangle WAZ$ 的面积相等。求 $\triangle WMA$ 的面积。 注：某些在中国举行的考试中，该题询问 $\triangle MAY$ 的面积。
stem
Q12
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students?
来自不同国家的 $100$ 名学生参加数学竞赛聚会。 每个学生会说相同数量的语言，而且对于每对学生 $A$ 和 $B$，学生 $A$ 会说某种学生 $B$ 不会说的语言，学生 $B$ 会说某种学生 $A$ 不会说的语言。所有学生所说的语言总数的可能最小值为多少？
Q13
Positive integers $x$ and $y$ satisfy the equation $\sqrt{x} + \sqrt{y} = \sqrt{1183}$. What is the minimum possible value of $x+y$?
正整数 $x$ 和 $y$ 满足方程 $\sqrt{x} + \sqrt{y} = \sqrt{1183}$。$x+y$ 的可能最小值为多少？
Q14
A dartboard is the region $B$ in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target $T$ is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown and lands at a random point in B. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
飞镖盘是坐标平面中区域 $B$，由满足 $|x| + |y| \le 8$ 的点 $(x, y)$ 组成。靶心 $T$ 是区域 $(x^2 + y^2 - 25)^2 \le 49$。飞镖随机落在 $B$ 中的一点。飞镖落在 $T$ 中的概率可表示为 $\frac{m}{n} \cdot \pi$，其中 $m$ 和 $n$ 互质正整数。求 $m + n$？
Q15
A list of $9$ real numbers consists of $1$, $2.2$, $3.2$, $5.2$, $6.2$, and $7$, as well as $x$, $y$ , and $z$ with $x$ $\le$ $y$ $\le$ $z$. The range of the list is $7$, and the mean and the median are both positive integers. How many ordered triples ($x$, $y$, $z$) are possible?
一个包含 $9$ 个实数的列表由 $1$、$2.2$、$3.2$、$5.2$、$6.2$ 和 $7$，以及 $x$、$y$ 和 $z$ 组成，其中 $x \le y \le z$。列表的极差为 $7$，均值和中位数均为正整数。可能的有序三元组 $(x, y, z)$ 有多少个？
Q16
Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase $1$, $2$, $3$, and $5$, and then write either $11$, their sum, or $30$, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?
杰瑞喜欢玩数字。有一天，他在白板上写下了从$1$到$2024$的所有整数。然后他反复选择白板上的四个数字，擦掉它们，并用它们的和或积替换它们。（例如，杰瑞的第一步可能是擦掉$1$、$2$、$3$和$5$，然后写下它们的和$11$或积$30$。）在反复进行这个操作后，杰瑞注意到白板上剩余的所有数字都是奇数。当时白板上可能的最大整数个数是多少？
Q17
In a race among $5$ snails, there is at most one tie, but that tie can involve any number of snails. For example, the result might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?
在5只蜗牛的比赛中，最多有一个平局，但这个平局可以涉及任意数量的蜗牛。例如，结果可能是Dazzler第一；Abby、Cyrus和Elroy并列第二；Bruna第五。比赛可能有多少种不同的结果？
Q18
How many different remainders can result when the $100$th power of an integer is divided by $125$?
整数的$100$次幂除以$125$可能得到多少种不同的余数？
Q19
In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"? \[ \begin{array}{|c|c|c|c|c|} \hline & \text{zero} & \text{exactly one} & \text{exactly two} & \text{more than two} \\ \hline \text{zero slope} & ? & ? & ? & ? \\ \hline \text{nonzero rational slope} & ? & ? & ? & ? \\ \hline \text{irrational slope} & ? & ? & ? & ? \\ \hline \end{array} \]
在下表中，每个问号都要用 “Possible” 或 “Not Possible” 替换，用来表示：具有给定斜率的一条非竖直直线，是否可能包含给定数量的格点（横纵坐标均为整数的点）。12 个表格项中有多少个会是 “Possible”？ \[ \begin{array}{|c|c|c|c|c|} \hline & \text{零个} & \text{恰好一个} & \text{恰好两个} & \text{多于两个} \\ \hline \text{零斜率} & ? & ? & ? & ? \\ \hline \text{非零有理斜率} & ? & ? & ? & ? \\ \hline \text{无理斜率} & ? & ? & ? & ? \\ \hline \end{array} \]
Q20
Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?
三双不同的鞋排成一排，使得没有左鞋与不同双的右鞋相邻。这六只鞋有多少种排列方式？
Q21
Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
两条直管（圆柱体），半径分别为$1$和$\frac{1}{4}$，平行放置并在平坦的地板上接触。下图显示了正面视图。第三条平行管放置在同一地板上，与两者都接触，可能的半径之和是多少？
stem
Q22
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?
$16$人将被分成$4$个不可区分的$4$人委员会。每个委员会有一个主席和一个秘书。进行这些分配的不同方式数可以写成$3^{r}M$，其中$r$和$M$是正整数且$M$不被$3$整除。$r$是多少？
Q23
The Fibonacci numbers are defined by $F_1 = 1, F_2 = 1,$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3.$ What is \[{\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?\]
斐波那契数列定义为$F_1 = 1, F_2 = 1,$且$n \geq 3$时$F_n = F_{n-1} + F_{n-2}$。求\[{\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?\]
Q24
Let \[P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}\] How many of the values $P(2022)$, $P(2023)$, $P(2024)$, and $P(2025)$ are integers?
设 \[P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}\] 其中$P(2022)$、$P(2023)$、$P(2024)$、$P(2025)$中有多少个是整数？
Q25
Each of $27$ bricks (right rectangular prisms) has dimensions $a \times b \times c$, where $a$, $b$, and $c$ are pairwise relatively prime positive integers. These bricks are arranged to form a $3 \times 3 \times 3$ block, as shown on the left below. A $28$th brick with the same dimensions is introduced, and these bricks are reconfigured into a $2 \times 2 \times 7$ block, shown on the right. The new block is $1$ unit taller, $1$ unit wider, and $1$ unit deeper than the old one. What is $a + b + c$?
$27$块砖（直角长方体），尺寸为$a \times b \times c$，其中$a$、$b$、$c$两两互质的正整数。这些砖排列成如下左图所示的$3 \times 3 \times 3$块。引入第$28$块相同尺寸的砖，并重新配置成右图所示的$2 \times 2 \times 7$块。新块比旧块高$1$单位、宽$1$单位、深$1$单位。求$a + b + c$？
stem
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