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AMC12 2024 A

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AMC12 · 2024 (A)

Q1
What is the value of $9901\cdot101-99\cdot10101?$
$9901\cdot101-99\cdot10101$ 的值为多少？
Q2
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=aL+bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?
一个用于估计徒步爬到山顶所需时间的模型形式为 $T=aL+bG$，其中 $a$ 和 $b$ 是常数，$T$ 是分钟数，$L$ 是小路长度（英里），$G$ 是海拔上升（英尺）。该模型估计一条长 $1.5$ 英里、上升 $800$ 英尺的小路需要 $69$ 分钟爬到山顶；一条长 $1.2$ 英里、上升 $1100$ 英尺的小路也需要 $69$ 分钟。如果小路长 $4.2$ 英里、上升 $4000$ 英尺，该模型估计需要多少分钟爬到山顶？
Q3
The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
数字 $2024$ 被写成若干（不一定不同）两位数的和。需要最少多少个两位数来表示这个和？
Q4
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
求最小的 $n$，使得 $n!$ 是 $2024$ 的倍数。
Q5
A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set?
一个包含 $20$ 个数字的数据集，其中有些是 $6$，其平均数为 $45$。当所有 $6$ 被移除后，数据集的平均数为 $66$。原来数据集中有多少个 $6$？
Q6
The product of three integers is $60$. What is the least possible positive sum of the three integers?
三个整数的乘积是$60$。这三个整数的最小正和是多少？
Q7
In $\Delta ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum \[\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?\]
在$\Delta ABC$中，$\angle ABC = 90^\circ$且$BA = BC = \sqrt{2}$。点$P_1, P_2, \dots, P_{2024}$位于斜边$\overline{AC}$上，使得$AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$。向量和$\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}$的长度是多少？
Q8
How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$?
多少个角度$\theta$满足$0\le\theta\le2\pi$且$\log(\sin(3\theta))+\log(\cos(2\theta))=0$？
Q9
Let $M$ be the greatest integer such that both $M+1213$ and $M+3773$ are perfect squares. What is the units digit of $M$?
设$M$是最大的整数，使得$M+1213$和$M+3773$都是完全平方数。$M$的个位数是多少？
Q10
Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
设$\alpha$是一个$3{-}4{-}5$直角三角形中最小角的弧度测度。设$\beta$是一个$7{-}24{-}25$直角三角形中最小角的弧度测度。用$\alpha$表示，$\beta$是多少？
Q11
There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$?
存在恰好 $K$ 个正整数 $b$ 满足 $5 \leq b \leq 2024$，使得基数 $b$ 的整数 $2024_b$ 能被 $16$（十进制）整除。$K$ 的各位数字之和是多少？
Q12
The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$
一个几何序列的前三项是整数 $a,\,720,$ 和 $b$，其中 $a<720<b$。最小可能的 $b$ 的各位数字之和是多少？
Q13
The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?
函数 $y=e^{x+1}+e^{-x}-2$ 的图像具有一条对称轴。点 $(-1,\tfrac{1}{2})$ 关于这条对称轴的反射是什么？
Q14
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5$. The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12$, respectively. What number is in position $(1, 2)?$ \[\begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
一个 $5 \times 5$ 的整数数组，每行的数字（按顺序）和每列的数字（按顺序）都形成长度为 $5$ 的等差数列。位置 $(5, 5), \,(2,4),\,(4,3),$ 和 $(3, 1)$ 的数字分别是 $0, 48, 16,$ 和 $12$。位置 $(1, 2)$ 的数字是多少？ \[\begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
Q15
The roots of $x^3 + 2x^2 - x + 3$ are $p, q,$ and $r.$ What is the value of \[(p^2 + 4)(q^2 + 4)(r^2 + 4)?\]
方程 $x^3 + 2x^2 - x + 3$ 的根为 $p, q,$ 和 $r$。$(p^2 + 4)(q^2 + 4)(r^2 + 4)$ 的值是多少？
Q16
A set of $12$ tokens — $3$ red, $2$ white, $1$ blue, and $6$ black — is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
有12个令牌——3个红色的，2个白色的，1个蓝色的，6个黑色的——将随机分给3名游戏玩家，每人4个令牌。某玩家得到所有红色令牌，另一玩家得到所有白色令牌，剩下玩家得到蓝色令牌的概率可以写成 $\frac{m}{n}$，其中 $m$ 和 $n$ 是互质的正整数。求 $m+n$？
Q17
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
整数 $a$、$b$ 和 $c$ 满足 $ab + c = 100$，$bc + a = 87$，$ca + b = 60$。求 $ab + bc + ca$？
Q18
On top of a rectangular card with sides of length $1$ and $2+\sqrt{3}$, an identical card is placed so that two of their diagonals line up, as shown ($\overline{AC}$, in this case). Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled $B$ in the figure?
在一张长宽为 $1$ 和 $2+\sqrt{3}$ 的矩形卡片上方，放置一张相同的卡片，使它们的对角线对齐（如图中的 $\overline{AC}$）。 继续这个过程，在第二张上添加第三张，依此类推，每次顺时针旋转后对齐连续的对角线。总共需要使用多少张卡片，直到一张新卡片的顶点恰好落在图中标记为 $B$ 的顶点上？
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Q19
Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$ and $DA=5$ with $\angle CDA=120^\circ$. What is the length of the shorter diagonal of $ABCD$?
循环四边形 $ABCD$ 有边长 $BC=CD=3$ 和 $DA=5$，且 $\angle CDA=120^\circ$。求 $ABCD$ 的较短对角线的长度？
Q20
Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline {AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\triangle ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$
在等边三角形 $\triangle ABC$ 的边 $\overline{AB}$ 和 $\overline{AC}$ 上，分别均匀独立随机选择点 $P$ 和 $Q$。以下哪个区间包含 $\triangle APQ$ 的面积小于 $\triangle ABC$ 面积一半的概率？
Q21
Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation \[\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}\]for all $n \ge 2.$ What is the greatest integer less than or equal to \[\sum^{100}_{n=1} a_n^2?\]
假设 $a_1 = 2$，序列 $(a_n)$ 满足递推关系 \[\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}\] 对所有 $n \ge 2$。求 $\left\lfloor \sum^{100}_{n=1} a_n^2 \right\rfloor$ 的值。
Q22
The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?
下图显示了一个宽 $8$ 格、高 $3$ 格的虚线网格，由 $1''\times1''$ 正方形组成。Carl 沿一些正方形的边放置 $1$ 英寸牙签，创建一条不自交的闭合回路。单元格中的数字表示该正方形应被牙签覆盖的边的数量，如果未写数字则允许任意数量牙签。Carl 放置牙签的方法有多少种？
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Q23
What is the value of \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]
求 \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}\] 的值。
Q24
A $\textit{disphenoid}$ is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
$\textit{不等四面体}$ 是其三角形面全等四面体。求脸为具有整数边长的不等边三角形的此类不等四面体的最小总表面积。
Q25
A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a,b,c,d)$, where $|a|,|b|,|c|,|d|\le5$ and $c$ and $d$ are not both $0$, is the graph of \[y=\frac{ax+b}{cx+d}\]symmetric about the line $y=x$?
如果图像关于一条直线对称，则该图像在该直线反射后保持不变。对于整数四元组 $(a,b,c,d)$，其中 $|a|,|b|,|c|,|d|\le5$ 且 $c$ 和 $d$ 不全为 $0$，图像 \[y=\frac{ax+b}{cx+d}\] 关于直线 $y=x$ 对称的有多少种？
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