/

AMC10 2004 B

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

AMC10 · 2004 (B)

Q1
Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club?
Misty Moon Amphitheater 每排有 33 个座位。第 12 到 22 排是为一个青年俱乐部预留的。这个俱乐部预留了多少座位？
Q2
How many two-digit positive integers have at least one 7 as a digit?
有多少个两位正整数至少有一个数字是 7？
Q3
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 罚中的罚球数是前一次的两倍。第五次练习她罚中了 48 个罚球。她第一次练习罚中了多少个罚球？
Q4
A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$?
掷一个标准的六面骰子，$P$ 是可见的五个数字的乘积。什么是最保证能整除 $P$ 的最大数？
Q5
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，虽然不一定按这个顺序。结果的最大可能值是多少？
Q6
Which of the following numbers is a perfect square?
以下哪个数是完全平方数？
Q7
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 $d$?
伊莎贝拉从美国去加拿大旅行时带了 $d$ 美元。在边境她把它们全部兑换，每 7 美元换得 10 加元。花费 60 加元后，她还剩 $d$ 加元。$d$ 的各位数字之和是多少？
Q8
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方 8 英里，明尼阿波利斯市中心东南方 10 英里。圣保罗市中心与明尼阿波利斯市中心之间的英里数最接近下列哪一个？
Q9
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?
一个边长为 10 的正方形，其一顶点为中心半径为 10 的圆。该正方形与圆包围区域的并集面积是多少？
Q10
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?
杂货商摆放罐头，最顶层一行一个罐头，每下一行比上一行多两个罐头。如果展示中共有 100 个罐头，有多少行？
Q11
Two eight-sided dice each have faces numbered 1 through 8. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum?
两个八面骰子，每面分别编号1到8。掷骰子时，每面出现在上面的概率相等。两个上面数字的乘积大于它们的和的概率是多少？
Q12
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?
环形区域是两个同心圆之间的区域。图中的同心圆半径为$b$和$c$，其中$b > c$。设$OX$是大圆的一个半径，$XZ$在$Z$点与小圆相切，$OY$是大圆包含$Z$的半径。设$a = XZ$，$d = YZ$，$e = XY$。环形区域的面积是多少？
stem
Q13
In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these coins is exactly 14 mm high, how many coins are in the stack?
在美国，硬币的厚度如下：美分1.55毫米；镍币1.95毫米；角币1.35毫米；25美分币1.75毫米。如果一摞这些硬币正好高14毫米，这摞硬币有多少个？
Q14
A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only $\frac{1}{3}$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $\frac{1}{5}$ of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?
一个袋子最初只含有红 marbles 和蓝 marbles，蓝 marbles 多于红 marbles。向袋中添加红 marbles 直到袋中蓝 marbles 只占$\frac{1}{3}$。然后添加黄 marbles 直到袋中蓝 marbles 只占$\frac{1}{5}$。最后，袋中蓝 marbles 的数量加倍。现在袋中蓝 marbles 占的比例是多少？
Q15
Patty has 20 coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have 70 cents more. How much are her coins worth?
Patty 有20枚硬币，包括镍币和10美分币。如果她的镍币变成10美分币，而10美分币变成镍币，她将多70美分。她的硬币总值多少？
Q16
Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
三个半径为1的圆相互外切，并与一个更大的圆内切。这个大圆的半径是多少？
stem
Q17
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?
杰克的年龄的两数字与比尔的年龄的两数字相同，但顺序相反。五年后，杰克的年龄将是比尔当时年龄的两倍。他们当前年龄的差是多少？
Q18
In right triangle $\triangle ACE$, we have $AC = 12$, $CE = 16$, and $EA = 20$. Points $B$, $D$, and $F$ are located on $AC$, $CE$, and $EA$, respectively, so that $AB = 3$, $CD = 4$, and $EF = 5$. What is the ratio of the area of $\triangle BDF$ to that of $\triangle ACE$?
在直角三角形$\triangle ACE$中，$AC = 12$，$CE = 16$，$EA = 20$。点$B$、$D$和$F$分别位于$AC$、$CE$和$EA$上，使得$AB = 3$，$CD = 4$，$EF = 5$。$\triangle BDF$的面积与$\triangle ACE$的面积之比是多少？
stem
Q19
In the sequence 2001, 2002, 2003, \dots, 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 2004th term in this sequence?
在数列2001, 2002, 2003, \dots中，从第四项起，每项等于前两项之和减去前一项。例如，第四项为$2001 + 2002 - 2003 = 2000$。这个数列的第2004项是多少？
Q20
In $\triangle ABC$ points $D$ and $E$ lie on $BC$ and $AC$, respectively. If $AD$ and $BE$ intersect at $T$ so that $AT/DT = 3$ and $BT/ET = 4$, what is $CD/BD$?
在$\triangle ABC$中，点$D$和$E$分别在$BC$和$AC$上。若$AD$和$BE$相交于$T$，使得$AT/DT = 3$且$BT/ET = 4$，则$CD/BD$是多少？
stem
Q21
Let 1, 4, \dots and 9, 16, \dots be two arithmetic progressions. The set $S$ is the union of the first 2004 terms of each sequence. How many distinct numbers are in $S$?
让 1, 4, \dots 和 9, 16, \dots 是两个等差数列。集合 $S$ 是每个数列的前 2004 项的并集。$S$ 中有多少个不同的数？
Q22
A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
一个边长为 5、12 和 13 的三角形既有内切圆也有外接圆。那两个圆心的距离是多少？
stem
Q23
Each face of a cube is painted either red or blue, each with probability $\frac{1}{2}$. 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被涂成红色或蓝色，每种颜色概率均为 $\frac{1}{2}$，且每个面的颜色独立确定。涂色后的立方体可以放置在水平面上，使得四个垂直面全为同一颜色的概率是多少？
Q24
In $\triangle ABC$ we have $AB = 7$, $AC = 8$, and $BC = 9$. Point $D$ is on the circumscribed circle of the triangle so that $AD$ bisects $\angle BAC$. What is the value of $AD/CD$?
在 $\triangle ABC$ 中，$AB = 7$，$AC = 8$，$BC = 9$。点 $D$ 在三角形的外接圆上，使得 $AD$ 平分 $\angle BAC$。$AD/CD$ 的值是多少？
Q25
A circle of radius 1 is internally tangent to two circles of radius 2 at points $A$ and $B$, where $AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the figure, that is outside the smaller circle and inside each of the two larger circles?
一个半径为 1 的圆在两个半径为 2 的圆内切于点 $A$ 和 $B$，其中 $AB$ 是小圆的直径。阴影区域（图中所示）是在两个大圆内部但小圆外部的面积是多少？
stem
Time Left 75:00