2001 AMC 10 — Official Competition Problems (January 2001)
📅 2001 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）
1
第 1 题
统计
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?
💡 解题思路
The median of the list is $10$ , and there are $9$ numbers in the list, so the median must be the 5th number from the left, which is $n+6$ .
2
第 2 题
综合
A number x is 2 more than the product of its reciprocal and its additive inverse . In which interval does the number lie?
💡 解题思路
We can write our equation as $x= \left(\frac{1}{x} \right) \cdot (-x) +2 = -1+2 = 1$ . Therefore, $\boxed{\textbf{(C) }0 < x\le 2}$ .
3
第 3 题
规律与数列
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?
💡 解题思路
Suppose the two numbers are $a$ and $b$ , with $a+b=S$ . Then the desired sum is $2(a+3)+2(b+3)=2(a+b)+12=2S +12$ , which is answer $\boxed{\textbf{(E)}}$ .
4
第 4 题
几何·面积
What is the maximum number of possible points of intersection of a circle and a triangle?
💡 解题思路
We can draw a circle and a triangle, such that each side is tangent to the circle. This means that each side would intersect the circle at one point.
5
第 5 题
综合
How many of the twelve pentominoes pictured below have at least one line of symmetry? [图]
💡 解题思路
The ones with lines over the shapes have at least one line of symmetry. Counting the number of shapes that have line(s) on them, we find $\boxed{\textbf{(D)}\ 6}$ pentominoes.
6
第 6 题
规律与数列
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 ?
💡 解题思路
Denote $a$ and $b$ as the tens and units digit of $N$ , respectively. Then $N = 10a+b$ . It follows that $10a+b=ab+a+b$ , which implies that $9a=ab$ . Since $a\neq0$ , $b=9$ . So the units digit of $N
7
第 7 题
分数与比例
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?
💡 解题思路
If $x$ is the number, then moving the decimal point four places to the right is the same as multiplying $x$ by $10000$ . This gives us: \[10000x=4\cdot\frac{1}{x} \implies x^2=\frac{4}{10000}\] Since
8
第 8 题
工程问题
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?
💡 解题思路
By translating the words in the problem into the language of mathematics, the problem is telling us to find the least common multiple of the four numbers given.
9
第 9 题
行程问题
The state income tax where Kristin lives is charged at the rate of p\% of the first \textdollar 28000 of annual income plus (p + 2)\% of any amount above \textdollar 28000 . Kristin noticed that the state income tax she paid amounted to (p + 0.25)\% of her annual income. What was her annual income?
💡 解题思路
Let $A$ , $T$ be Kristin's annual income and the income tax total, respectively. Notice that \begin{align*} T &= p\%\cdot28000 + (p + 2)\%\cdot(A - 28000) \\ &= [p\%\cdot28000 + p\%\cdot(A - 28000)] +
10
第 10 题
逻辑推理
If x , y , and z are positive with xy = 24 , xz = 48 , and yz = 72 , then x + y + z is
💡 解题思路
The first two equations in the problem are $xy=24$ and $xz=48$ . Since $xyz \ne 0$ , we have $\frac{xy}{xz}=\frac{24}{48} \implies 2y=z$ . We can substitute $z = 2y$ into the third equation $yz = 72$
11
第 11 题
几何·面积
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 [图]
💡 解题思路
We can partition the $n^\text{th}$ ring into $4$ rectangles: two containing $2n+1$ unit squares and two containing $2n-1$ unit squares.
12
第 12 题
数论
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 ?
💡 解题思路
Whenever $n$ is the product of three consecutive integers, $n$ is divisible by $3!$ , meaning it is divisible by $6$ .
13
第 13 题
数字运算
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 .
💡 解题思路
We start by noting that there are $10$ letters, meaning there are $10$ digits in total. Listing them all out, we have $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ . Clearly, the most restrictive condition is the co
14
第 14 题
应用题
A charity sells 140 benefit tickets for a total of \2001$ . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
💡 解题思路
Let's multiply ticket costs by $2$ , then the half price becomes an integer, and the charity sold $140$ tickets worth a total of $4002$ dollars.
15
第 15 题
行程问题
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.
💡 解题思路
Drawing the problem out, we see we get a parallelogram with a height of $40$ and a base of $15$ , giving an area of $600$ .
16
第 16 题
统计
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5 . What is their sum?
💡 解题思路
Let $m$ be the mean of the three numbers. Then the least of the numbers is $m-10$ and the greatest is $m + 15$ . The middle of the three numbers is the median, $5$ . So $\dfrac{1}{3}[(m-10) + 5 + (m +
17
第 17 题
几何·面积
Which of the cones listed below can be formed from a 252^\circ sector of a circle of radius 10 by aligning the two straight sides? [图] (A) A cone with slant height of 10 and radius 6(B) A cone with height of 10 and radius 6(C) A cone with slant height of 10 and radius 7(D) A cone with height of 10 and radius 7(E) A cone with slant height of 10 and radius 8
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 [图]
💡 解题思路
Consider any single tile:
19
第 19 题
综合
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?
💡 解题思路
Let's use stars and bars . Let the donuts be represented by $O$ s. We wish to find all possible combinations of glazed, chocolate, and powdered donuts that give us $4$ in all. The four donuts we want
20
第 20 题
几何·面积
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?
💡 解题思路
https://youtu.be/B1OXVB5GDjk
21
第 21 题
概率
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.
💡 解题思路
https://youtu.be/HUM035eNKvU
22
第 22 题
几何·面积
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 , and z . Find y + z . [图]
💡 解题思路
We know that $y+z=2v$ , so we could find one variable rather than two.
23
第 23 题
概率
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?
💡 解题思路
Imagine that we draw all the chips in random order, i.e., we do not stop when the last chip of a color is drawn. To draw out all the white chips first, the last chip left must be red, and all previous
24
第 24 题
几何·角度
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· CD ?
💡 解题思路
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(7cm); real labelscalefactor = 0.5; /* changes label-to-point d
25
第 25 题
数论
How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5 ?
💡 解题思路
Out of the numbers $1$ to $12$ four are divisible by $3$ and three by $4$ , counting $12$ twice. Hence $6$ out of these $12$ numbers are multiples of $3$ or $4$ .