2002A AMC 12 — Official Competition Problems (February 2024)
📅 2024 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
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
2
第 2 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 $\boxed{\textbf{(A) }15}$ .
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
几何·角度
Find the degree measure of an angle whose complement is 25% of its supplement. (A) \ 48 (B) \ 60 (C) \ 75 (D) \ 120 (E) \ 150
💡 解题思路
We can create an equation for the question, $4(90-x)=(180-x)$
5
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
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 $\boxed{\textbf{(E) } \text{infinitely many}}$ .
7
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
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 $0.4$
10
第 10 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.
11
第 11 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
12
第 12 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
13
第 13 题
整数运算
Two different positive numbers a and b each differ from their reciprocals by 1 . What is a+b ? (A) 1 (B) 2 (C) \sqrt 5 (D) \sqrt 6 (E) 3
💡 解题思路
Each of the numbers $a$ and $b$ is a solution to $\left| x - \frac 1x \right| = 1$ .
14
第 14 题
整数运算
For all positive integers n , let f(n)=\log_{2002} n^2 . Let N=f(11)+f(13)+f(14) . Which of the following relations is true? (A) N<1 (B) N=1 (C) 1<N<2 (D) N=2 (E) N>2
💡 解题思路
First, note that $2002 = 11 \cdot 13 \cdot 14$ .
15
第 15 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.
16
第 16 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.
17
第 17 题
数论
Several sets of prime numbers, such as \{7,83,421,659\} use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? (A) 193 (B) 207 (C) 225 (D) 252 (E) 447
💡 解题思路
Neither of the digits $4$ , $6$ , and $8$ can be a units digit of a prime. Therefore the sum of the set is at least $40 + 60 + 80 + 1 + 2 + 3 + 5 + 7 + 9 = 207$ .
18
第 18 题
几何·面积
Let C_1 and C_2 be circles defined by (x-10)^2 + y^2 = 36 and (x+15)^2 + y^2 = 81 respectively. What is the length of the shortest line segment PQ that is tangent to C_1 at P and to C_2 at Q ? (A) 15 (B) 18 (C) 20 (D) 21 (E) 24
💡 解题思路
First examine the formula $(x-10)^2+y^2=36$ , for the circle $C_1$ . Its center, $D_1$ , is located at (10,0) and it has a radius of $\sqrt{36}$ = 6. The next circle, using the same pattern, has its c
19
第 19 题
坐标几何
The graph of the function f is shown below. How many solutions does the equation f(f(x))=6 have? [图] (A) 2 (B) 4 (C) 5 (D) 6 (E) 7
💡 解题思路
First of all, note that the equation $f(t)=6$ has two solutions: $t=-2$ and $t=1$ .
20
第 20 题
分数与比例
Suppose that a and b are digits, not both nine and not both zero, and the repeating decimal 0.\overline{ab} is expressed as a fraction in lowest terms. How many different denominators are possible? (A) 3 (B) 4 (C) 5 (D) 8 (E) 9
💡 解题思路
The repeating decimal $0.\overline{ab}$ is equal to \[\frac{10a+b}{100} + \frac{10a+b}{10000} + \cdots = (10a+b)\cdot\left(\frac 1{10^2} + \frac 1{10^4} + \cdots \right) = (10a+b) \cdot \frac 1{99} =
21
第 21 题
规律与数列
Consider the sequence of numbers: 4,7,1,8,9,7,6,\dots For n>2 , the n -th term of the sequence is the units digit of the sum of the two previous terms. Let S_n denote the sum of the first n terms of this sequence. The smallest value of n for which S_n>10,000 is: (A) 1992 (B) 1999 (C) 2001 (D) 2002 (E) 2004
💡 解题思路
The sequence is infinite. As there are only $100$ pairs of digits, sooner or later a pair of consecutive digits will occur for the second time. As each next digit only depends on the previous two, fro
22
第 22 题
几何·面积
Triangle ABC is a right triangle with \angle ACB as its right angle, m\angle ABC = 60^\circ , and AB = 10 . Let P be randomly chosen inside ABC , and extend \overline{BP} to meet \overline{AC} at D . What is the probability that BD > 5\sqrt2 ? [图]
💡 解题思路
Clearly $BC=5$ and $AC=5\sqrt{3}$ . Choose a $P'$ and get a corresponding $D'$ such that $BD'= 5\sqrt{2}$ and $CD'=5$ . For $BD > 5\sqrt2$ we need $CD>5$ , creating an isosceles right triangle with hy
23
第 23 题
几何·面积
In triangle ABC , side AC and the perpendicular bisector of BC meet in point D , and BD bisects \angle ABC . If AD=9 and DC=7 , what is the area of triangle ABD ? (A)\ 14 (B)\ 21 (C)\ 28 (D)\ 14\sqrt5 (E)\ 28\sqrt5
💡 解题思路
[asy] unitsize(0.25 cm); pair A, B, C, D, M; A = (0,0); B = (88/9, 28*sqrt(5)/9); C = (16,0); D = 9/16*C; M = (B + C)/2; draw(A--B--C--cycle); draw(B--D--M); label("$A$", A, SW); label("$B$", B, N); l
24
第 24 题
综合
Find the number of ordered pairs of real numbers (a,b) such that (a+bi)^{2002} = a-bi . (A) 1001 (B) 1002 (C) 2001 (D) 2002 (E) 2004
💡 解题思路
Let $s=\sqrt{a^2+b^2}$ be the magnitude of $a+bi$ . Then the magnitude of $(a+bi)^{2002}$ is $s^{2002}$ , while the magnitude of $a-bi$ is $s$ . We get that $s^{2002}=s$ , hence either $s=0$ or $s=1$
25
第 25 题
坐标几何
The nonzero coefficients of a polynomial P with real coefficients are all replaced by their mean to form a polynomial Q . Which of the following could be a graph of y = P(x) and y = Q(x) over the interval -4≤ x ≤ 4 ?
💡 解题思路
The sum of the coefficients of $P$ and of $Q$ will be equal, so $P(1) = Q(1)$ . The only answer choice with an intersection between the two graphs at $x = 1$ is (B) . (The polynomials in the graph are