2003B 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
综合
Which of the following is the same as \[\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?\] \text {(A) } -1 \text {(B) } -\frac{2}{3} \text {(C) } \frac{2}{3} \text {(D) } 1 \text {(E) } \frac{14}{3}
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs \1 more than a pink pill, and Al's pills cost a total of \textdollar 546 for the two weeks. How much does one green pill cost?
💡 解题思路
Because there are $14$ days in two weeks, Al spends $546/14 = 39$ dollars per day for the cost of a green pill and a pink pill. If the green pill costs $x$ dollars and the pink pill $x-1$ dollars, the
3
第 3 题
几何·面积
Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost \1 each, begonias \1.50 each, cannas \2 each, dahlias \2.50 each, and Easter lilies \3 each. What is the least possible cost, in dollars, for her garden? [图]
💡 解题思路
The areas of the five regions from greatest to least are $21,20,15,6$ and $4$ .
4
第 4 题
统计
Moe uses a mower to cut his rectangular 90 -foot by 150 -foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. He walks at the rate of 5000 feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow the lawn? Easy problem if you understand what swath means, not hard at all
5
第 5 题
几何·面积
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is 4:3 . The horizontal length of a " 27 -inch" television screen is closest, in inches, to which of the following?
💡 解题思路
If you divide the television screen into two right triangles, the legs are in the ratio of $4 : 3$ , and we can let one leg be $4x$ and the other be $3x$ . Then we can use the Pythagorean Theorem.
6
第 6 题
规律与数列
The second and fourth terms of a geometric sequence are 2 and 6 . Which of the following is a possible first term?
💡 解题思路
Let the first term be $a$ and the common ratio be $r$ . Therefore,
7
第 7 题
概率
Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels,dimes, and quarters, whose total value is 8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank? \text {(A) } 0 \text {(B) } 13 \text {(C) } 37 \text {(D) } 64 \text {(E) } 83$
💡 解题思路
Where $a,b,c$ is the number of nickels, dimes, and quarters, respectively, we can set up two equations:
8
第 8 题
规律与数列
Let \clubsuit(x) denote the sum of the digits of the positive integer x . For example, \clubsuit(8)=8 and \clubsuit(123)=1+2+3=6 . For how many two-digit values of x is \clubsuit(\clubsuit(x))=3 ?
💡 解题思路
Let $y=\clubsuit (x)$ . Since $x \leq 99$ , we have $y \leq 18$ . Thus if $\clubsuit (y)=3$ , then $y=3$ or $y=12$ . The 3 values of $x$ for which $\clubsuit (x)=3$ are 12, 21, and 30, and the 7 value
9
第 9 题
函数
Let f be a linear function for which f(6) - f(2) = 12. What is f(12) - f(2)?\text {(A) } 12 \text {(B) } 18 \text {(C) } 24 \text {(D) } 30 \text {(E) } 36
💡 解题思路
Since $f$ is a linear function with slope $m$ ,
10
第 10 题
几何·面积
Several figures can be made by attaching two equilateral triangles to the regular pentagon ABCDE in two of the five positions shown. How many non-congruent figures can be constructed in this way? \text {(A) } 1 \text {(B) } 2 \text {(C) } 3 \text {(D) } 4 \text {(E) } 5
💡 解题思路
Place the first triangle. Now, we can place the second triangle either adjacent to the first, or with one side between them, for a total of $\boxed{\text{(B) }2}$
11
第 11 题
规律与数列
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM? \text {(A) 10:22 PM and 24 seconds} \text {(B) 10:24 PM} \text {(C) 10:25 PM} \text {(D) 10:27 PM} \text {(E) 10:30 PM}
💡 解题思路
For every $60$ minutes that pass by in actual time, $57+\frac{36}{60}=57.6$ minutes pass by on Cassandra's watch. When her watch first reads, 10:00 pm, $10(60)=600$ minutes have passed by on her watch
12
第 12 题
整数运算
What is the largest integer that is a divisor of \[(n+1)(n+3)(n+5)(n+7)(n+9)\] for all positive even integers n ? \text {(A) } 3 \text {(B) } 5 \text {(C) } 11 \text {(D) } 15 \text {(E) } 165
💡 解题思路
For all consecutive odd integers, one of every five is a multiple of 5 and one of every three is a multiple of 3. The answer is $3 \cdot 5 = 15$ , so ${\boxed{\textbf{(D)15}}}$ is the correct answer.
13
第 13 题
分数与比例
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies 75\% of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius ? (A)\ 2:1 (B)\ 3:1 (C)\ 4:1 (D)\ 16:3 (E)\ 6:1
💡 解题思路
Let $r$ be the common radius of the sphere and the cone, and $h$ be the cone’s height. Then \[75\% \cdot \left(\frac 43 \pi r^3\right) = \frac 13 \pi r^2 h \Longrightarrow h = 3r\] Thus $h:r = 3:1$ an
14
第 14 题
几何·面积
In rectangle ABCD, AB=5 and BC=3 . Points F and G are on \overline{CD} so that DF=1 and GC=2 . Lines AF and BG intersect at E . Find the area of \triangle AEB .
💡 解题思路
$\triangle EFG \sim \triangle EAB$ because $FG \parallel AB.$ The ratio of $\triangle EFG$ to $\triangle EAB$ is $2:5$ since $AB=5$ and $FG=2$ from subtraction. If we let $h$ be the height of $\triang
15
第 15 题
几何·面积
A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF ? [图] https://www.youtube.com/watch?v=LREcUjK-56U&feature=youtu.be
💡 解题思路
Here is an easy way to look at this, where $p$ is the perimeter, and $a$ is the apothem :
16
第 16 题
几何·面积
Three semicircles of radius 1 are constructed on diameter \overline{AB} of a semicircle of radius 2 . The centers of the small semicircles divide \overline{AB} into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [图]
If \log (xy^3) = 1 and \log (x^2y) = 1 , what is \log (xy) ? (A)\ -\frac 12 (B)\ 0 (C)\ \frac 12 (D)\ \frac 35 (E)\ 1
💡 解题思路
Since \begin{align*} &\log(xy) +2\log y = 1 \\ \log(xy) + \log x = 1 \quad \Longrightarrow \quad &2\log(xy) + 2\log x = 2 \end{align*} Summing gives \[3\log(xy) + 2\log y + 2\log x = 3 \Longrightarrow
18
第 18 题
数论
Let x and y be positive integers such that 7x^5 = 11y^{13}. The minimum possible value of x has a prime factorization a^cb^d. What is a + b + c + d?
💡 解题思路
Substitute $a^cb^d$ into $x$ . We then have $7(a^{5c}b^{5d}) = 11y^{13}$ . Divide both sides by $7$ , and it follows that:
19
第 19 题
概率
Let S be the set of permutations of the sequence 1,2,3,4,5 for which the first term is not 1 . A permutation is chosen randomly from S . The probability that the second term is 2 , in lowest terms, is a/b . What is a+b ? (A)\ 5 (B)\ 6 (C)\ 11 (D)\ 16 (E)\ 19
💡 解题思路
There are $4$ choices for the first element of $S$ , and for each of these choices there are $4!$ ways to arrange the remaining elements. If the second element must be $2$ , then there are only $3$ ch
20
第 20 题
坐标几何
Part of the graph of f(x) = ax^3 + bx^2 + cx + d is shown. What is b ? (A)\ -4 (B)\ -2 (C)\ 0 (D)\ 2 (E)\ 4
💡 解题思路
Since \begin{align*} -f(-1) = a - b + c - d = 0 = f(1) = a + b + c + d \end{align*}
21
第 21 题
概率
An object moves 8 cm in a straight line from A to B , turns at an angle \alpha , measured in radians and chosen at random from the interval (0,π) , and moves 5 cm in a straight line to C . What is the probability that AC < 7 ? (A)\ \frac{1}{6} (B)\ \frac{1}{5} (C)\ \frac{1}{4} (D)\ \frac{1}{3} (E)\ \frac{1}{2}
💡 解题思路
By the Law of Cosines , \begin{align*} AB^2 + BC^2 - 2 AB \cdot BC \cos \alpha = 89 - 80 \cos \alpha = AC^2 & \frac 12\\ \end{align*}
22
第 22 题
几何·角度
Let ABCD be a rhombus with AC = 16 and BD = 30 . Let N be a point on \overline{AB} , and let P and Q be the feet of the perpendiculars from N to \overline{AC} and \overline{BD} , respectively. Which of the following is closest to the minimum possible value of PQ ? (A)\ 6.5 (B)\ 6.75 (C)\ 7 (D)\ 7.25 (E)\ 7.5
💡 解题思路
Let $\overline{AC}$ and $\overline{BD}$ intersect at $O$ . Since $ABCD$ is a rhombus, then $\overline{AC}$ and $\overline{BD}$ are perpendicular bisectors . Thus $\angle POQ = 90^{\circ}$ , so $OPNQ$
23
第 23 题
坐标几何
The number of x -intercepts on the graph of y=\sin(1/x) in the interval (0.0001,0.001) is closest to (A)\ 2900 (B)\ 3000 (C)\ 3100 (D)\ 3200 (E)\ 3300
💡 解题思路
The function $f(x) = \sin x$ has roots in the form of $\pi n$ for all integers $n$ . Therefore, we want $\frac{1}{x} = \pi n$ on $\frac{1}{10000} \le x \le \frac{1}{1000}$ , so $1000 \le \frac 1x = \p
24
第 24 题
方程
Positive integers a,b, and c are chosen so that a<b<c , and the system of equations has exactly one solution. What is the minimum value of c ? (A)\ 668 (B)\ 669 (C)\ 1002 (D)\ 2003 (E)\ 2004
💡 解题思路
Consider the graph of $f(x)=|x-a|+|x-b|+|x-c|$ .
25
第 25 题
几何·面积
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle? (A)\ \dfrac{1}{36} (B)\ \dfrac{1}{24} (C)\ \dfrac{1}{18} (D)\ \dfrac{1}{12} (E)\ \dfrac{1}{9}
💡 解题思路
The first point is placed anywhere on the circle, because it doesn't matter where it is chosen.