/

AMC12 2008 A

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

AMC12 · 2008 (A)

Q1
A bakery owner turns on his doughnut machine at $\text{8:30}\ {\small\text{AM}}$. At $\text{11:10}\ {\small\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
一位面包店老板在 $\text{8:30}\ {\small\text{AM}}$ 打开他的甜甜圈机器。到 $\text{11:10}\ {\small\text{AM}}$ 时，机器已完成当天工作量的三分之一。甜甜圈机器将在什么时间完成全部工作？
Q2
What is the reciprocal of $\frac{1}{2}+\frac{2}{3}$?
$\frac{1}{2}+\frac{2}{3}$ 的倒数是多少？
Q3
Suppose that $\tfrac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\tfrac{1}{2}$ of $5$ bananas?
假设 $10$ 根香蕉的 $\tfrac{2}{3}$ 的价值与 $8$ 个橙子相同。多少个橙子的价值与 $5$ 根香蕉的 $\tfrac{1}{2}$ 相同？
Q4
Which of the following is equal to the product \[\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?\]
下列哪一项等于乘积 \[\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?\]
Q5
Suppose that \[\frac{2x}{3}-\frac{x}{6}\] is an integer. Which of the following statements must be true about $x$?
假设 \[\frac{2x}{3}-\frac{x}{6}\] 是一个整数。关于 $x$，下列哪项陈述一定为真？
Q6
Heather compares the price of a new computer at two different stores. Store $A$ offers $15\%$ off the sticker price followed by a $\$90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves $\$15$ by buying the computer at store $A$ instead of store $B$. What is the sticker price of the computer, in dollars?
Heather 在两家不同的商店比较一台新电脑的价格。A 店先在标价基础上打 $15\%$ 折扣，然后再返还 $\$90$；B 店对相同标价打 $25\%$ 折扣，但没有返现。Heather 在 A 店购买比在 B 店购买节省了 $\$15$。这台电脑的标价是多少美元？
Q7
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
当 Steve 和 LeRoy 在离岸 1 英里处钓鱼时，他们的船突然漏水，水以每分钟 10 加仑的恒定速率流入。若进水超过 30 加仑，船将下沉。Steve 以每小时 4 英里的恒定速度开始向岸边划船，同时 LeRoy 往外舀水。为了在不沉没的情况下到达岸边，LeRoy 舀水的最慢速率（加仑/分钟）是多少？
Q8
What is the volume of a cube whose surface area is twice that of a cube with volume 1?
一个立方体的表面积是体积为 1 的立方体表面积的两倍。这个立方体的体积是多少？
Q9
Older television screens have an aspect ratio of $4: 3$. That is, the ratio of the width to the height is $4: 3$. The aspect ratio of many movies is not $4: 3$, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $2: 1$ and is shown on an older television screen with a $27$-inch diagonal. What is the height, in inches, of each darkened strip?
老式电视屏幕的长宽比为 $4: 3$，即宽与高之比为 $4: 3$。许多电影的长宽比不是 $4: 3$，因此有时会用“信箱式画面”(letterboxing) 在电视屏幕上播放——在屏幕顶部和底部各加一条等高的黑边，如图所示。若某电影的长宽比为 $2: 1$，并在一台对角线为 $27$ 英寸的老式电视上播放，则每条黑边的高度（英寸）是多少？
stem
Q10
Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $t$?
Doug 刷完一间房间需要 $5$ 小时。Dave 刷完同一间房间需要 $7$ 小时。Doug 和 Dave 一起刷这间房间，并在中途午休 1 小时。设 $t$ 为他们完成工作所需的总时间（小时），包括午休。下列哪个方程由 $t$ 满足？
Q11
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum?
三个立方体每个都由所示图案制成。然后它们被一个叠一个地堆放在桌子上，使得 $13$ 个可见数字的和尽可能大。这个和是多少？
stem
Q12
A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\{x:a \le x \le b \}$.) What are the domain and range, respectively, of the function $g$ defined by $g(x)=1-f(x+1)$?
函数 $f$ 的定义域是 $[0,2]$，值域是 $[0,1]$。（记号 $[a,b]$ 表示 $\{x:a \le x \le b \}$。）由 $g(x)=1-f(x+1)$ 定义的函数 $g$ 的定义域和值域分别是什么？
Q13
Points $A$ and $B$ lie on a circle centered at $O$, and $\angle AOB = 60^\circ$. A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle?
点 $A$ 和 $B$ 位于以 $O$ 为圆心的圆上，且 $\angle AOB = 60^\circ$。第二个圆内切于第一个圆，并且与 $\overline{OA}$ 和 $\overline{OB}$ 都相切。较小圆的面积与较大圆面积的比是多少？
Q14
What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$?
不等式 $|3x-18|+|2y+7|\le3$ 定义的区域的面积是多少？
Q15
Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
设 $k={2008}^{2}+{2}^{2008}$。$k^2+2^k$ 的个位数是多少？
Q16
The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^\text{th}$ term of the sequence is $\log{b^n}$. What is $n$?
数 $\log(a^3b^7)$，$\log(a^5b^{12})$ 和 $\log(a^8b^{15})$ 是等差数列的前三项，该数列的第 $12^\text{th}$ 项是 $\log{b^n}$。$n$ 是多少？
Q17
Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n=a_{n-1}/2$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$?
设序列 $a_1,a_2,\ldots$ 由如下规则确定：若 $a_{n-1}$ 为偶数，则 $a_n=a_{n-1}/2$；若 $a_{n-1}$ 为奇数，则 $a_n=3a_{n-1}+1$。对于多少个满足 $a_1 \le 2008$ 的正整数 $a_1$，有 $a_1$ 小于 $a_2$、$a_3$ 和 $a_4$ 中的每一个？
Q18
Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$?
三角形 $ABC$ 的边长分别为 $5$、$6$ 和 $7$，其中一个顶点在正 $x$ 轴上，一个在正 $y$ 轴上，一个在正 $z$ 轴上。设 $O$ 为原点。四面体 $OABC$ 的体积是多少？
Q19
In the expansion of \[\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,\] what is the coefficient of $x^{28}$?
在展开式 \[\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,\] 中，$x^{28}$ 的系数是多少？
Q20
Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$?
三角形 $ABC$ 满足 $AC=3$，$BC=4$，$AB=5$。点 $D$ 在 $\overline{AB}$ 上，且 $\overline{CD}$ 平分直角。$\triangle ADC$ 与 $\triangle BCD$ 的内切圆半径分别为 $r_a$ 与 $r_b$。求 $r_a/r_b$。
Q21
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?
$(1,2,3,4,5)$ 的一个排列 $(a_1,a_2,a_3,a_4,a_5)$ 若满足 $a_1 + a_2 < a_4 + a_5$，则称为重尾排列。重尾排列的个数是多少？
Q22
A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?
一个圆桌的半径为 $4$。在桌上放置了六个矩形餐垫。每个餐垫的宽为 $1$，长为 $x$，如图所示。它们的位置使得每个餐垫有两个角在桌边上，这两个角是同一条长度为 $x$ 的边的端点。此外，餐垫的位置使得每个餐垫的内侧角都与相邻餐垫的一个内侧角相接触。求 $x$。
stem
Q23
The solutions of the equation $z^4+4z^3i-6z^2-4zi-i=0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon?
方程 $z^4+4z^3i-6z^2-4zi-i=0$ 的解在复平面中构成一个凸多边形的顶点。该多边形的面积是多少？
Q24
Triangle $ABC$ has $\angle C = 60^{\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\tan{\angle BAD}$?
三角形 $ABC$ 满足 $\angle C = 60^{\circ}$ 且 $BC = 4$。点 $D$ 是 $BC$ 的中点。$\tan{\angle BAD}$ 的最大可能值是多少？
Q25
A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\ldots$ of points in the coordinate plane satisfies $(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)$ for $n = 1,2,3,\ldots$. Suppose that $(a_{100},b_{100}) = (2,4)$. What is $a_1 + b_1$?
坐标平面上的点序列 $(a_1,b_1)$、$(a_2,b_2)$、$(a_3,b_3)$、$\ldots$ 满足 $(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)$，其中 $n = 1,2,3,\ldots$。 已知 $(a_{100},b_{100}) = (2,4)$。求 $a_1 + b_1$。
Time Left 75:00