/

AMC10 2001 A

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

AMC10 · 2001 (A)

Q1
The median of the list $n, n+3, n+4, n+5, n+6, n+8, n+10, n+12, n+15$ is 10. What is the mean?
列表 $n, n+3, n+4, n+5, n+6, n+8, n+10, n+12, n+15$ 的中位数是 10。平均数是多少？
Q2
A number $x$ is 2 more than the product of its reciprocal and its additive inverse. In which interval does the number lie?
一个数 $x$ 比它的倒数与其加法逆元的乘积多 2。它位于哪个区间？
Q3
The sum of two numbers is $S$. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
两个数的和是 $S$。假设每个数加 3，然后将结果每个数乘以 2。最后两个数的和是多少？
Q4
What is the maximum number for the possible points of intersection of a circle and a triangle?
圆与三角形可能相交的最大点数是多少？
Q5
How many of the twelve pentominoes pictured below have at least one line of symmetry?
下面图片中的十二个五连方中有多少个至少有一条对称轴？
stem
Q6
Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N) + S(N)$. What is the units digit of $N$?
设 $P(n)$ 和 $S(n)$ 分别表示整数 $n$ 的各位数字的乘积和之和。例如，$P(23) = 6$ 和 $S(23) = 5$。假设 $N$ 是一个两位数，使得 $N = P(N) + S(N)$。$N$ 的个位数字是多少？
Q7
When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?
将某个正小数的十进制点向右移动四位后，新数字是原数字倒数的四倍。原数字是多少？
Q8
Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?
学校的数学实验室有四位助教：Wanda、Darren、Beatrice 和 Chi。他们的工作安排如下：Darren 每三个学校日工作一次，Wanda 每四个学校日工作一次，Beatrice 每六个学校日工作一次，Chi 每七个学校日工作一次。今天他们都在数学实验室工作。从今天起多少学校日后他们将再次一起在实验室辅导？
Q9
The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $28000$ of annual income plus $(p + 2)\%$ of any amount above $28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?
Kristin 所在州的所得税税率为年收入前 $28000$ 美元的部分按 $p\%$ 征收，超过 $28000$ 美元的部分按 $(p + 2)\%$ 征收。Kristin 注意到她缴纳的州所得税相当于她年收入的 $(p + 0.25)\%$。她的年收入是多少？
Q10
If $x, y, z$ are positive with $xy = 24, xz = 48, yz = 72$, then $x + y + z$ is
若 $x, y, z$ 为正数，且 $xy = 24$，$xz = 48$，$yz = 72$，则 $x + y + z$ 是
Q11
Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, the number of unit squares in the $100^{th}$ ring is
考虑一个单位正方形阵列中的深色正方形，部分显示如下。围绕这个中心正方形的第一个环包含 8 个单位正方形。第二个环包含 16 个单位正方形。如果我们继续这个过程，第 $100^{th}$ 环中的单位正方形数量是
stem
Q12
Suppose that n is the product of three consecutive integers and that n is divisible by 7. Which of the following is not necessarily a divisor of n?
假设 $n$ 是三个连续整数的乘积，并且 $n$ 能被 7 整除。以下哪一个不一定是 $n$ 的因数？
Q13
A telephone number has the form ABC −DEF −GHIJ, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, A > B > C, D > E > F, and G > H > I > J. Furthermore, D, E, and F are consecutive even digits; G, H, I, and J are consecutive odd digits; and A + B + C = 9. Find A.
一个电话号码的形式是 ABC −DEF −GHIJ，其中每个字母代表不同的数字。号码每个部分的数字是递减的，即 A > B > C, D > E > F, 和 G > H > I > J。此外，D, E, 和 F 是连续的偶数数字；G, H, I, 和 J 是连续的奇数数字；并且 A + B + C = 9。求 A。
Q14
A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?
一个慈善机构售出了 140 张总价为 $2001 的福利票。有些票按全价（整美元金额）出售，其余按半价出售。全价票筹集了多少钱？
Q15
A street has parallel curbs 40 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 15 feet and each stripe is 50 feet long. Find the distance, in feet, between the stripes.
一条街道有相距 40 英尺的平行路缘。一个由两条平行条纹界定的横道以一定角度穿过街道。条纹之间的路缘长度为 15 英尺，每条条纹长 50 英尺。求条纹之间的距离，单位英尺。
Q16
The mean of three numbers is 10 more than the least of the numbers and less than the greatest. The median of the three numbers is 5. What is their sum?
三个数的平均数比最小的数大10，并且小于最大的数。这三个数的中间数是5。它们的和是多少？
Q17
Which of the cones below can be formed from a 252° sector of a circle of radius 10 by aligning the two straight sides?
下面哪个圆锥可以由半径为10的圆的252°扇形通过将两条直边对齐而成？
stem
Q18
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
平面被全等的正方形和全等的五边形如图所示铺满。五边形包围的平面百分比最接近
stem
Q19
Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?
Pat想从充足的三种甜甜圈（釉面、巧克力和糖粉）中买四个甜甜圈。有多少种不同的选择？
Q20
A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length 2000. What is the length of each side of the octagon?
通过从边长为2000的正方形的每个角切掉一个等腰直角三角形，形成一个正八边形。正八边形的每边长是多少？
Q21
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder.
一个直圆柱，其直径等于其高度，内接于一个直圆锥中。圆锥的直径为10，高为12，圆柱与圆锥的轴线重合。求圆柱的半径。
Q22
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v, w, x, y, z$. Find $y + z$.
在所示的幻方中，每行、每列和对角线上的数字之和相同。其中五个数字用 $v, w, x, y, z$ 表示。求 $y + z$。
stem
Q23
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
一个盒子中恰好有五张筹码，三张红色，两张白色。随机依次无放回取出筹码，直到取出所有红色筹码或所有白色筹码。最后一張取出的筹码为白色的概率是多少？
Q24
In trapezoid $ABCD$, $\overline{AB}$ and $\overline{CD}$ are perpendicular to $\overline{AD}$, with $AB + CD = BC$, $AB < CD$, and $AD = 7$. What is $AB \cdot CD$?
在梯形 $ABCD$ 中， $\overline{AB}$ 和 $\overline{CD}$ 都垂直于 $\overline{AD}$，有 $AB + CD = BC$，$AB < CD$，且 $AD = 7$。求 $AB \cdot CD$。
Q25
How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5?
不超过 2001 的正整数中，能被 3 或 4 整除但不被 5 整除的有多少个？
Time Left 75:00