2008A 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
📋 答题说明
共 25 道题，每题从 A、B、C、D、E 五个选项中选一个答案，点击选项即可选择
答题过程中可随时更改选项，选完后点击底部「提交答案」统一批改
提交后显示对错、正确答案和简短解题思路
点击题目右侧 ⭐ 可收藏难题，方便后续复习
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
行程问题
A bakery owner turns on his doughnut machine at 8:30\ {\smallAM} . At 11:10\ {\smallAM} the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? (A)\ 1:50\ {\smallPM} (B)\ 3:00\ {\smallPM} (C)\ 3:30\ {\smallPM} (D)\ 4:30\ {\smallPM} (E)\ 5:50\ {\smallPM}
💡 解题思路
The machine completes one-third of the job in $\text{11:10}-\text{8:30}=\text{2:40}$ hours. Thus, the entire job is completed in $3\cdot(\text{2:40})=\text{8:00}$ hours.
2
第 2 题
综合
What is the reciprocal of \frac{1}{2}+\frac{2}{3} ? (A)\ \frac{6}{7} (B)\ \frac{7}{6} (C)\ \frac{5}{3} (D)\ 3 (E)\ \frac{7}{2}
💡 解题思路
Here's a cheapshot: Obviously, $\frac{1}{2}+\frac{2}{3}$ is greater than $1$ . Therefore, its reciprocal is less than $1$ , and the answer must be $\boxed{\frac{6}{7}}$ .
3
第 3 题
统计
Suppose that \tfrac{2}{3} of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as \tfrac{1}{2} of 5 bananas? (A)\ 2 (B)\ \frac{5}{2} (C)\ 3 (D)\ \frac{7}{2} (E)\ 4
💡 解题思路
If $\frac{2}{3}\cdot10\ \text{bananas}=8\ \text{oranges}$ , then $\frac{1}{2}\cdot5\ \text{bananas}=\left(\frac{1}{2}\cdot 5\ \text{bananas}\right)\cdot\left(\frac{8\ \text{oranges}}{\frac{2}{3}\cdot1
4
第 4 题
综合
Which of the following is equal to the product \[\frac{8}{4}·\frac{12}{8}·\frac{16}{12}··s·\frac{4n+4}{4n}··s·\frac{2008}{2004}?\]
Suppose that \[\frac{2x}{3}-\frac{x}{6}\] is an integer. Which of the following statements must be true about x ? (A)\ It is negative. ; (B)\ It is even, but not necessarily a multiple of 3. ; (C)\ It is a multiple of 3, but not necessarily even. ; (D)\ It is a multiple of 6, but not necessarily a multiple of 12. ; (E)\ It is a multiple of 12.
💡 解题思路
\[\frac{2x}{3}-\frac{x}{6}\quad\Longrightarrow\quad\frac{4x}{6}-\frac{x}{6}\quad\Longrightarrow\quad\frac{3x}{6}\quad\Longrightarrow\quad\frac{x}{2}\] For $\frac{x}{2}$ to be an integer, $x$ must be e
6
第 6 题
应用题
Heather compares the price of a new computer at two different stores. Store A offers 15\% off the sticker price followed by a \90 rebate, and store B offers 25\% off the same sticker price with no rebate. Heather saves \15 by buying the computer at store A instead of store B . What is the sticker price of the computer, in dollars? (A)\ 750 (B)\ 900 (C)\ 1000 (D)\ 1050 (E)\ 1500
💡 解题思路
Let the sticker price be $x$ .
7
第 7 题
行程问题
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? (A)\ 2 (B)\ 4 (C)\ 6 (D)\ 8 (E)\ 10
💡 解题思路
It will take $\frac{1}{4}$ of an hour or $15$ minutes to get to shore.
8
第 8 题
几何·面积
What is the volume of a cube whose surface area is twice that of a cube with volume 1? (A)\ √(2) (B)\ 2 (C)\ 2√(2) (D)\ 4 (E)\ 8
💡 解题思路
A cube with volume $1$ has a side of length $\sqrt[3]{1}=1$ and thus a surface area of $6 \cdot 1^2=6$ .
9
第 9 题
分数与比例
Older television screens have an aspect ratio of 4: 3 . That is, the ratio of the width to the height is 4: 3 . The aspect ratio of many movies is not 4: 3 , so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of 2: 1 and is shown on an older television screen with a 27 -inch diagonal. What is the height, in inches, of each darkened strip? [图] (A)\ 2 (B)\ 2.25 (C)\ 2.5 (D)\ 2.7 (E)\ 3
💡 解题思路
Let the width and height of the screen be $4x$ and $3x$ respectively, and let the width and height of the movie be $2y$ and $y$ respectively.
10
第 10 题
方程
Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let t be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by t ? (A)\ (\frac{1}{5}+\frac{1}{7})(t+1)=1 (B)\ (\frac{1}{5}+\frac{1}{7})t+1=1 (C)\ (\frac{1}{5}+\frac{1}{7})t=1 ; (D)\ (\frac{1}{5}+\frac{1}{7})(t-1)=1 (E)\ (5+7)t=1
💡 解题思路
Doug can paint $\frac{1}{5}$ of a room per hour, Dave can paint $\frac{1}{7}$ of a room per hour, and the time they spend working together is $t-1$ .
11
第 11 题
规律与数列
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the 13 visible numbers have the greatest possible sum. What is that sum? [图] (A)\ 154 (B)\ 159 (C)\ 164 (D)\ 167 (E)\ 189
💡 解题思路
To maximize the sum of the $13$ faces that are showing, we can minimize the sum of the numbers of the $5$ faces that are not showing.
12
第 12 题
统计
A function f has domain [0,2] and range [0,1] . (The notation [a,b] denotes \{x:a \le x \le b \} .) What are the domain and range, respectively, of the function g defined by g(x)=1-f(x+1) ? (A)\ [-1,1],[-1,0] (B)\ [-1,1],[0,1] (C)\ [0,2],[-1,0] (D)\ [1,3],[-1,0] (E)\ [1,3],[0,1]
💡 解题思路
$g(x)$ is defined if $f(x + 1)$ is defined. Thus the domain is all $x| x + 1 \in [0,2] \rightarrow x \in [ - 1,1]$ .
13
第 13 题
几何·面积
Points A and B lie on a circle centered at O , and \angle AOB = 60^\circ . A second circle is internally tangent to the first and tangent to both \overline{OA} and \overline{OB} . What is the ratio of the area of the smaller circle to that of the larger circle? (A)\ \frac{1}{16} (B)\ \frac{1}{9} (C)\ \frac{1}{8} (D)\ \frac{1}{6} (E)\ \frac{1}{4}
💡 解题思路
[asy]size(200); defaultpen(fontsize(10)); pair O=(0,0), A=(3,0), B=(3/2,3/2*3^.5), C=(3^.5,1), D=(3^.5,0), F=(1.5*3^.5,1.5), G=(2*3^.5,2); picture p = new picture; draw(p,Circle(O,0.2)); clip(p,O--C--
14
第 14 题
几何·面积
What is the area of the region defined by the inequality |3x-18|+|2y+7|\le3 ? (A)\ 3 (B)\ \frac{7}{2} (C)\ 4 (D)\ \frac{9}{2} (E)\ 5
💡 解题思路
Area is invariant under translation, so after translating left $6$ and up $7/2$ units, we have the inequality
15
第 15 题
数字运算
Let k={2008}^{2}+{2}^{2008} . What is the units digit of k^2+2^k ? (A)\ 0 (B)\ 2 (C)\ 4 (D)\ 6 (E)\ 8
The numbers \log(a^3b^7) , \log(a^5b^{12}) , and \log(a^8b^{15}) are the first three terms of an arithmetic sequence , and the 12^th term of the sequence is \log{b^n} . What is n ? (A)\ 40 (B)\ 56 (C)\ 76 (D)\ 112 (E)\ 143
💡 解题思路
Let $A = \log(a)$ and $B = \log(b)$ .
17
第 17 题
规律与数列
Let a_1,a_2,\ldots be a sequence determined by the rule a_n=a_{n-1}/2 if a_{n-1} is even and a_n=3a_{n-1}+1 if a_{n-1} is odd. For how many positive integers a_1 \le 2008 is it true that a_1 is less than each of a_2 , a_3 , and a_4 ? (A)\ 250 (B)\ 251 (C)\ 501 (D)\ 502 (E) 1004
💡 解题思路
All positive integers can be expressed as $4n$ , $4n+1$ , $4n+2$ , or $4n+3$ , where $n$ is a nonnegative integer.
18
第 18 题
几何·面积
Triangle ABC , with sides of length 5 , 6 , and 7 , has one vertex on the positive x -axis, one on the positive y -axis, and one on the positive z -axis. Let O be the origin . What is the volume of tetrahedron OABC ? (A)\ √(85) (B)\ √(90) (C)\ √(95) (D)\ 10 (E)\ √(105)
💡 解题思路
Without loss of generality, let $A$ be on the $x$ axis, $B$ be on the $y$ axis, and $C$ be on the $z$ axis, and let $AB, BC, CA$ have respective lengths of 5, 6, and 7. Let $a,b,c$ denote the lengths
19
第 19 题
综合
In the expansion of \[(1 + x + x^2 + ·s + x^{27})(1 + x + x^2 + ·s + x^{14})^2,\] what is the coefficient of x^{28} ? (A)\ 195 (B)\ 196 (C)\ 224 (D)\ 378 (E)\ 405
💡 解题思路
Let $A = \left(1 + x + x^2 + \cdots + x^{14}\right)$ and $B = \left(1 + x + x^2 + \cdots + x^{27}\right)$ . We are expanding $A \cdot A \cdot B$ .
20
第 20 题
几何·面积
Triangle ABC has AC=3 , BC=4 , and AB=5 . Point D is on \overline{AB} , and \overline{CD} bisects the right angle. The inscribed circles of \triangle ADC and \triangle BCD have radii r_a and r_b , respectively. What is r_a/r_b ? (A)\ \frac{1}{28}(10-√(2)) (B)\ \frac{3}{56}(10-√(2)) (C)\ \frac{1}{14}(10-√(2)) (D)\ \frac{5}{56}(10-√(2)) ; (E)\ \frac{3}{28}(10-√(2))
💡 解题思路
By the Angle Bisector Theorem , \[\frac{BD}{4} = \frac{5-BD}{3} \Longrightarrow BD = \frac{20}7\] By Law of Sines on $\triangle BCD$ , \[\frac{BD}{\sin 45^{\circ}} = \frac{CD}{\sin \angle B} \Longrigh
21
第 21 题
计数
A permutation (a_1,a_2,a_3,a_4,a_5) of (1,2,3,4,5) is heavy-tailed if a_1 + a_2 < a_4 + a_5 . What is the number of heavy-tailed permutations? (A)\ 36 (B)\ 40 (C)\ 44 (D)\ 48 (E)\ 52
💡 解题思路
There are $5!=120$ total permutations.
22
第 22 题
应用题
A round table has radius 4 . Six rectangular place mats are placed on the table. Each place mat has width 1 and length x as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length x . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is x ? [图] (A)\ 2√(5)-√(3) (B)\ 3 (C)\ \frac{3√(7)-√(3)}{2} (D)\ 2√(3) (E)\ \frac{5+2√(3)}{2}
💡 解题思路
Let one of the mats be $ABCD$ , and the center be $O$ as shown:
23
第 23 题
几何·面积
The solutions of the equation z^4+4z^3i-6z^2-4zi-i=0 are the vertices of a convex polygon in the complex plane. What is the area of the polygon? (A)\ 2^{\frac{5}{8}} (B)\ 2^{\frac{3}{4}} (C)\ 2 (D)\ 2^{\frac{5}{4}} (E)\ 2^{\frac{3}{2}}
💡 解题思路
Looking at the coefficients, we are immediately reminded of the binomial expansion of ${\left(x+1\right)}^{4}$ .
24
第 24 题
几何·面积
Triangle ABC has \angle C = 60^{\circ} and BC = 4 . Point D is the midpoint of BC . What is the largest possible value of \tan{\angle BAD} ? (A)\ \frac{√(3)}{6} (B)\ \frac{√(3)}{3} (C)\ \frac{√(3)}{2√(2)} (D)\ \frac{√(3)}{4√(2)-3} (E)\ 1
💡 解题思路
[asy]unitsize(12mm); pair C=(0,0), B=(4 * dir(60)), A = (8,0), D=(2 * dir(60)); pair E=(1,0), F=(2,0); draw(C--B--A--C); draw(A--D);draw(D--E);draw(B--F); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); la
25
第 25 题
坐标几何
A sequence (a_1,b_1) , (a_2,b_2) , (a_3,b_3) , \ldots of points in the coordinate plane satisfies (a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n) for n = 1,2,3,\ldots . Suppose that (a_{100},b_{100}) = (2,4) . What is a_1 + b_1 ? (A)\ -\frac{1}{2^{97}} (B)\ -\frac{1}{2^{99}} (C)\ 0 (D)\ \frac{1}{2^{98}} (E)\ \frac{1}{2^{96}}
💡 解题思路
This sequence can also be expressed using matrix multiplication as follows:
