2002 AMC 10A — Official Competition Problems (January 2002)
📅 2002 A 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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1
第 1 题
分数与比例
The ratio \frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}} is closest to which of the following numbers?
💡 解题思路
We factor $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ as $\frac{10^{2000}(1+100)}{10^{2001}(1+1)}=\frac{101}{20}$ . As $\frac{101}{20}=5.05$ , our answer is $\boxed{\textbf{(D)}\ 5 }$ .
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第 2 题
综合
Given that a, b, and c are non-zero real numbers, define (a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a} , find (2, 12, 9) .
💡 解题思路
$(2, 12, 9)=\frac{2}{12}+\frac{12}{9}+\frac{9}{2}=\frac{1}{6}+\frac{4}{3}+\frac{9}{2}=\frac{1}{6}+\frac{8}{6}+\frac{27}{6}=\frac{36}{6}=6$ . Our answer is then $\boxed{\textbf{(C) }6}$ .
3
第 3 题
综合
According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\] If the order in which the exponentiations are performed is changed, how many other values are possible?
💡 解题思路
The best way to solve this problem is by simple brute force.
4
第 4 题
整数运算
For how many positive integers m does there exist at least one positive integer n such that m · n \le m + n ?
💡 解题思路
For any $m$ we can pick $n=1$ , we get $m \cdot 1 \le m + 1$ , therefore the answer is $\boxed{\textbf{(E) } \text{infinitely many}}$ .
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第 5 题
几何·面积
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. [图]
💡 解题思路
The outer circle has radius $1+1+1=3$ , and thus area $9\pi$ . The little circles have area $\pi$ each; since there are 7, their total area is $7\pi$ . Thus, our answer is $9\pi-7\pi=\boxed{2\pi\Right
6
第 6 题
工程问题
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?
💡 解题思路
We work backwards; the number that Cindy started with is $3(43)+9=138$ . Now, the correct result is $\frac{138-3}{9}=\frac{135}{9}=15$ . Our answer is $\boxed{\textbf{(A) }15}$ .
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第 7 题
几何·面积
A 45^\circ arc of circle A is equal in length to a 30^\circ arc of circle B. What is the ratio of circle A's area and circle B's area?
💡 解题思路
Let $r_1$ and $r_2$ be the radii of circles $A$ and $B$ , respectively.
8
第 8 题
几何·面积
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let B be the total area of the blue triangles, W the total area of the white squares, and P the area of the red square. Which of the following is correct? [图]
💡 解题思路
The blue that's touching the center red square makes up 8 triangles, or 4 squares. Each of the corners is 1 square and each of the edges is 1, totaling 12 squares. There are 12 white squares, thus we
9
第 9 题
统计
There are 3 numbers A, B, and C, such that 1001C - 2002A = 4004 , and 1001B + 3003A = 5005 . What is the average of A, B, and C?
💡 解题思路
Notice that we don't need to find what $A, B,$ and $C$ actually are, just their average. In other words, if we can find $A+B+C$ , we will be done.
10
第 10 题
规律与数列
Compute the sum of all the roots of (2x+3)(x-4)+(2x+3)(x-6)=0
💡 解题思路
We expand to get $2x^2-8x+3x-12+2x^2-12x+3x-18=0$ which is $4x^2-14x-30=0$ after combining like terms. Using the quadratic part of Vieta's Formulas , we find the sum of the roots is $\frac{14}4 = \box
11
第 11 题
综合
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
💡 解题思路
A $0.8$ MB file can either be on its own disk, or share it with a $0.4$ MB. Clearly it is better to pick the second possibility. Thus we will have $3$ disks, each with one $0.8$ MB file and one $0.4$
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第 12 题
统计
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
💡 解题思路
Let the time he needs to get there in be $t$ and the distance he travels be $d$ . From the given equations, we know that $d=\left(t+\frac{1}{20}\right)40$ and $d=\left(t-\frac{1}{20}\right)60$ . Setti
13
第 13 题
几何·面积
Given a triangle with side lengths 15, 20, and 25, find the triangle's shortest altitude.
💡 解题思路
This is a Pythagorean triple (a $3-4-5$ actually) with legs $15$ and $20$ . The area is then $\frac{(15)(20)}{2}=150$ . Now, consider an altitude drawn to any side. Since the area remains constant, th
14
第 14 题
数论
Both roots of the quadratic equation x^2 - 63x + k = 0 are prime numbers. The number of possible values of k is (A)\ 0 (B)\ 1 (C)\ 2 (D)\ 4 (E) more than 4
💡 解题思路
Consider a general quadratic with the coefficient of $x^2$ being $1$ and the roots being $r$ and $s$ . It can be factored as $(x-r)(x-s)$ which is just $x^2-(r+s)x+rs$ . Thus, the sum of the roots is
15
第 15 题
数论
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers? (A)\ 150 (B)\ 160 (C)\ 170 (D)\ 180 (E)\ 190
💡 解题思路
Since a multiple-digit prime number is not divisible by either 2 or 5, it must end with 1, 3, 7, or 9 in the units place. The remaining digits given must therefore appear in the tens place. Hence our
16
第 16 题
综合
Let a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5 . What is a + b + c + d ? (A)\ -5 (B)\ -10/3 (C)\ -7/3 (D)\ 5/3 (E)\ 5
💡 解题思路
Let $x=a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$ . Since one of the sums involves $a, b, c,$ and $d,$ it makes sense to consider $4x$ . We have $4x=(a+1)+(b+2)+(c+3)+(d+4)=a+b+c+d+10=4(a+b+c+
17
第 17 题
分数与比例
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream? (A) \ \frac{1}{4} (B) \ \frac13 (C) \ \frac38 (D) \ \frac25 (E) \ \frac12
💡 解题思路
We will simulate the process in steps.
18
第 18 题
概率
A 3 x 3 x 3 cube is made of 27 normal dice. Each die's opposite sides sum to 7 . What is the smallest possible sum of all of the values visible on the 6 faces of the large cube? (A)\ 60 (B)\ 72 (C)\ 84 (D)\ 90 (E) 96
💡 解题思路
In a 3x3x3 cube, there are $8$ cubes with three faces showing, $12$ with two faces showing and $6$ with one face showing. The smallest sum with three faces showing is $1+2+3=6$ , with two faces showin
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第 19 题
几何·面积
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? (A)\ 2π/3 (B)\ 2π (C)\ 5π/2 (D)\ 8π/3 (E)\ 3π
💡 解题思路
Part of what Spot can reach is $\frac{240}{360}=\frac{2}{3}$ of a circle with radius 2, which gives him $\frac{8\pi}{3}$ . He can also reach two $\frac{60}{360}$ parts of a unit circle, which combines
20
第 20 题
几何·角度
Points A,B,C,D,E and F lie, in that order, on \overline{AF} , dividing it into five segments, each of length 1. Point G is not on line AF . Point H lies on \overline{GD} , and point J lies on \overline{GF} . The line segments \overline{HC}, \overline{JE}, and \overline{AG} are parallel. Find HC/JE . (A)\ 5/4 (B)\ 4/3 (C)\ 3/2 (D)\ 5/3 (E)\ 2
💡 解题思路
First we can draw an image. [asy] unitsize(0.8 cm); pair A, B, C, D, E, F, G, H, J; A = (0,0); B = (1,0); C = (2,0); D = (3,0); E = (4,0); F = (5,0); G = (-1.5,4); H = extension(D, G, C, C + G - A); J
21
第 21 题
统计
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is (A) 11 (B) 12 (C) 13 (D) 14 (E) 15
💡 解题思路
As the unique mode is $8$ , there are at least two $8$ s.
22
第 22 题
几何·面积
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square , and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one? (A)\ 10 (B)\ 11 (C)\ 18 (D)\ 19 (E)\ 20
💡 解题思路
The pattern is quite simple to see after listing a couple of terms.
23
第 23 题
几何·面积
Points A,B,C and D lie on a line, in that order, with AB = CD and BC = 12 . Point E is not on the line, and BE = CE = 10 . The perimeter of \triangle AED is twice the perimeter of \triangle BEC . Find AB . (A)\ 15/2 (B)\ 8 (C)\ 17/2 (D)\ 9 (E)\ 19/2
💡 解题思路
First, we draw an altitude to $BC$ from $E$ . Let it intersect at $M$ . As $\triangle BEC$ is isosceles, we immediately get $MB=MC=6$ , so the altitude is $8$ . Now, let $AB=CD=x$ . Using the Pythagor
24
第 24 题
概率
Tina randomly selects two distinct numbers from the set \{ 1, 2, 3, 4, 5 \} , and Sergio randomly selects a number from the set \{ 1, 2, ..., 10 \} . What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? (A)\ 2/5 (B)\ 9/20 (C)\ 1/2 (D)\ 11/20 (E)\ 24/25
💡 解题思路
This is not too bad using casework.
25
第 25 题
几何·面积
In trapezoid ABCD with bases AB and CD , we have AB = 52 , BC = 12 , CD = 39 , and DA = 5 . The area of ABCD is [图] (A)\ 182 (B)\ 195 (C)\ 210 (D)\ 234 (E)\ 260
💡 解题思路
It shouldn't be hard to use trigonometry to bash this and find the height, but there is a much easier way. Extend $\overline{AD}$ and $\overline{BC}$ to meet at point $E$ :