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AMC12 2004 B

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AMC12 · 2004 (B)

Q1
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?
上周每次篮球练习，Jenny 投中的罚球数是前一次练习的两倍。第五次练习她投中了 48 个罚球。她第一次练习投中了多少个罚球？
Q2
In the expression $c\cdot a^b-d$, the values of $a$, $b$, $c$, and $d$ are $0$, $1$, $2$, and $3$, although not necessarily in that order. What is the maximum possible value of the result?
在表达式 $c\cdot a^b-d$ 中，$a$、$b$、$c$ 和 $d$ 的值为 $0$、$1$、$2$ 和 $3$，但不一定按这个顺序。结果的最大可能值是多少？
Q3
If $x$ and $y$ are positive integers for which $2^x3^y=1296$, what is the value of $x+y$?
如果 $x$ 和 $y$ 是正整数，使得 $2^x3^y=1296$，那么 $x+y$ 的值是多少？
Q4
An integer $x$, with $10\leq x\leq 99$, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of $x$ is a 7?
要选择一个整数 $x$，满足 $10\leq x\leq 99$。如果所有选择等可能，至少有一位数字是 7 的概率是多少？
Q5
On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$?
Isabella 从美国去加拿大旅行，带了 $d$ 美元。在边境她把钱全部兑换，按每 $7$ 美元换 $10$ 加元。花了 $60$ 加元后，她还剩 $d$ 加元。$d$ 的各位数字之和是多少？
Q6
Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
明尼阿波利斯-圣保罗国际机场位于圣保罗市中心西南 $8$ 英里处，位于明尼阿波利斯市中心东南 $10$ 英里处。下列哪个选项最接近圣保罗市中心与明尼阿波利斯市中心之间的英里数？
Q7
A square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. What is the area of the union of the regions enclosed by the square and the circle?
一个正方形的边长为 $10$，并且以它的一个顶点为圆心作半径为 $10$ 的圆。由正方形和圆所围成区域的并集面积是多少？
Q8
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $100$ cans, how many rows does it contain?
某杂货商用罐头摆成展示：最上面一行有一个罐头，并且每往下一行比上一行多两个罐头。若该展示共有 $100$ 个罐头，它共有多少行？
Q9
The point $(-3,2)$ is rotated $90^\circ$ clockwise around the origin to point $B$. Point $B$ is then reflected over the line $x=y$ to point $C$. What are the coordinates of $C$?
点 $(-3,2)$ 绕原点顺时针旋转 $90^\circ$ 到点 $B$。然后将点 $B$ 关于直线 $x=y$ 反射到点 $C$。求 $C$ 的坐标。
Q10
An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b>c$. Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$, and let $OY$ be the radius of the larger circle that contains $Z$. Let $a=XZ$, $d=YZ$, and $e=XY$. What is the area of the annulus?
环形区域（annulus）是两个同心圆之间的区域。图中的同心圆半径分别为 $b$ 和 $c$，且 $b>c$。设 $OX$ 为大圆的一条半径，$XZ$ 在 $Z$ 点与小圆相切，$OY$ 为经过 $Z$ 的大圆半径。令 $a=XZ$，$d=YZ$，$e=XY$。该环形区域的面积是多少？
stem
Q11
All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?
代数班的所有学生参加了一场$100$分的测试。有五名学生得了$100$分，每位学生得分至少$60$分，平均分为$76$分。班级中最少可能有多少名学生？
Q12
In the sequence $2001$, $2002$, $2003$, $\ldots$ , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$. What is the $2004^\textrm{th}$ term in this sequence?
在数列$2001$, $2002$, $2003$, $\ldots$中，从第四项开始，每一项由其前两项之和减去前一项得到。例如，第四项是$2001 + 2002 - 2003 = 2000$。该数列的第$2004^\textrm{th}$项是多少？
Q13
If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, what is the value of $a+b$?
若$f(x) = ax+b$且$f^{-1}(x) = bx+a$，其中$a$和$b$为实数，求$a+b$的值。
Q14
In $\triangle ABC$, $AB=13$, $AC=5$, and $BC=12$. Points $M$ and $N$ lie on $AC$ and $BC$, respectively, with $CM=CN=4$. Points $J$ and $K$ are on $AB$ so that $MJ$ and $NK$ are perpendicular to $AB$. What is the area of pentagon $CMJKN$?
在$\triangle ABC$中，$AB=13$，$AC=5$，$BC=12$。点$M$和$N$分别在$AC$和$BC$上，且$CM=CN=4$。点$J$和$K$在$AB$上，使得$MJ$和$NK$都垂直于$AB$。五边形$CMJKN$的面积是多少？
stem
Q15
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
Jack的年龄的两个数字与Bill的年龄的两个数字相同，但顺序相反。五年后，Jack的年龄将是Bill当时年龄的两倍。他们目前年龄的差是多少？
Q16
A function $f$ is defined by $f(z) = i\overline{z}$, where $i=\sqrt{-1}$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?
函数 $f$ 定义为 $f(z) = i\overline{z}$，其中 $i=\sqrt{-1}$ 且 $\overline{z}$ 是 $z$ 的复共轭。满足 $|z| = 5$ 且 $f(z) = z$ 的 $z$ 有多少个值？
Q17
For some real numbers $a$ and $b$, the equation \[8x^3 + 4ax^2 + 2bx + a = 0\] has three distinct positive roots. If the sum of the base-$2$ logarithms of the roots is $5$, what is the value of $a$?
对于某些实数 $a$ 和 $b$，方程 \[8x^3 + 4ax^2 + 2bx + a = 0\] 有三个不同的正根。如果这些根的以 $2$ 为底的对数之和为 $5$，那么 $a$ 的值是多少？
Q18
Points $A$ and $B$ are on the parabola $y=4x^2+7x-1$, and the origin is the midpoint of $AB$. What is the length of $AB$?
点 $A$ 和 $B$ 在抛物线 $y=4x^2+7x-1$ 上，且原点是 $AB$ 的中点。$AB$ 的长度是多少？
Q19
A truncated cone has horizontal bases with radii $18$ and $2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
一个截锥的上下底面为水平圆，半径分别为 $18$ 和 $2$。一个球与该截锥的上底面、下底面以及侧面都相切。该球的半径是多少？
Q20
Each face of a cube is painted either red or blue, each with probability $1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
一个立方体的每个面以概率 $1/2$ 被涂成红色或蓝色，各面颜色独立决定。求涂色后的立方体能放在水平面上，使得四个竖直侧面全为同一种颜色的概率。
Q21
The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\frac yx$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?
方程 $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ 的图像是 $xy$ 平面第一象限内的一个椭圆。设 $a$ 和 $b$ 分别为椭圆上所有点 $(x,y)$ 的 $\frac yx$ 的最大值与最小值。求 $a+b$ 的值。
Q22
The square \[ \begin{array}{|c|c|c|} \hline 50 & b & c\\ \hline d & e & f\\ \hline g & h & 2\\ \hline \end{array} \] is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of $g$?
正方形 \[ \begin{array}{|c|c|c|} \hline 50 & b & c\\ \hline d & e & f\\ \hline g & h & 2\\ \hline \end{array} \] 是一个乘法幻方。也就是说，每一行、每一列以及两条对角线上的数的乘积都相同。如果所有位置上的数都是正整数，$g$ 的所有可能取值之和是多少？
Q23
The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible?
多项式 $x^3 - 2004 x^2 + mx + n$ 的系数为整数，且有三个互不相同的正零点。其中恰好有一个零点是整数，并且它等于另外两个零点之和。问 $n$ 可能有多少个取值？
Q24
In $\triangle ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE = 10$. The values of $\tan \angle CBE$, $\tan \angle DBE$, and $\tan \angle ABE$ form a geometric progression, and the values of $\cot \angle DBE,$ $\cot \angle CBE,$ $\cot \angle DBC$ form an arithmetic progression. What is the area of $\triangle ABC$?
在 $\triangle ABC$ 中，$AB = BC$，且 $\overline{BD}$ 是一条高。点 $E$ 在 $\overline{AC}$ 的延长线上，且 $BE = 10$。$\tan \angle CBE$、$\tan \angle DBE$、$\tan \angle ABE$ 的值成等比数列，而 $\cot \angle DBE,$ $\cot \angle CBE,$ $\cot \angle DBC$ 的值成等差数列。求 $\triangle ABC$ 的面积。
stem
Q25
Given that $2^{2004}$ is a $604$-digit number whose first digit is $1$, how many elements of the set $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ have a first digit of $4$?
已知 $2^{2004}$ 是一个 $604$ 位数，且其首位数字为 $1$。问集合 $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ 中有多少个元素的首位数字为 $4$？
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