2018B 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
Kate bakes a 20 -inch by 18 -inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
💡 解题思路
The area of the pan is $20\cdot18=360$ . Since the area of each piece is $2\cdot2=4$ , there are $\frac{360}{4} = \boxed{\textbf{(A) } 90}$ pieces.
2
第 2 题
统计
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?
💡 解题思路
Suppose that Sam's average speed during the last $30$ minutes was $x$ mph.
3
第 3 题
坐标几何
A line with slope 2 intersects a line with slope 6 at the point (40,30) . What is the distance between the x -intercepts of these two lines?
💡 解题思路
Using point slope form, we get the equations $y-30 = 6(x-40)$ and $y-30 = 2(x-40)$ . Simplifying, we get $6x-y=210$ and $2x-y=50$ . Letting $y=0$ in both equations and solving for $x$ gives the $x$ -i
4
第 4 题
几何·面积
A circle has a chord of length 10 , and the distance from the center of the circle to the chord is 5 . What is the area of the circle?
💡 解题思路
Let $O$ be the center of the circle, $\overline{AB}$ be the chord, and $M$ be the midpoint of $\overline{AB},$ as shown below. [asy] /* Made by MRENTHUSIASM */ size(200); pair O, A, B, M; O = (0,0); A
5
第 5 题
数论
How many subsets of \{2,3,4,5,6,7,8,9\} contain at least one prime number?
💡 解题思路
Since an element of a subset is either in or out, the total number of subsets of the $8$ -element set is $2^8 = 256$ . However, since we are only concerned about the subsets with at least $1$ prime in
6
第 6 题
应用题
Suppose S cans of soda can be purchased from a vending machine for Q quarters. Which of the following expressions describes the number of cans of soda that can be purchased for D dollars, where 1 dollar is worth 4 quarters?
💡 解题思路
Each can of soda costs $\frac QS$ quarters, or $\frac{Q}{4S}$ dollars. Therefore, $D$ dollars can purchase $\frac{D}{\left(\tfrac{Q}{4S}\right)}=\boxed{\textbf{(B) } \frac{4DS}{Q}}$ cans of soda.
7
第 7 题
综合
💡 解题思路
From the Change of Base Formula, we have \[\frac{\prod_{i=3}^{13} \log (2i+1)}{\prod_{i=1}^{11}\log (2i+1)} = \frac{\log 25 \cdot \log 27}{\log 3 \cdot \log 5} = \frac{(2\log 5)\cdot(3\log 3)}{\log 3
8
第 8 题
几何·面积
Line segment \overline{AB} is a diameter of a circle with AB = 24 . Point C , not equal to A or B , lies on the circle. As point C moves around the circle, the centroid (center of mass) of \triangle ABC traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
💡 解题思路
By the Inscribed Angle Theorem, $\triangle ABC$ is a right triangle with $\angle C=90^{\circ}.$ So, its circumcenter is the midpoint of $\overline{AB},$ and its median from $C$ is half as long as $\ov
9
第 9 题
规律与数列
What is \[\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ?\]
💡 解题思路
Recall that the sum of the first $100$ positive integers is $\sum^{100}_{k=1} k = \frac{101\cdot100}{2}=5050.$ It follows that \begin{align*} \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) &= \sum^{100}_{i=1
10
第 10 题
统计
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?
💡 解题思路
To minimize the number of distinct values, we want to maximize the number of times a number appears. So, we could have $223$ numbers appear $9$ times, $1$ number appear once, and the mode appear $10$
11
第 11 题
几何·面积
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point A in the figure on the right. The box has base length w and height h . What is the area of the sheet of wrapping paper? [图]
💡 解题思路
Consider one-quarter of the image (the wrapping paper is divided up into $4$ congruent squares). The length of each dotted line is $h$ . The area of the rectangle that is $w$ by $h$ is $wh$ . The comb
12
第 12 题
几何·面积
Side \overline{AB} of \triangle ABC has length 10 . The bisector of angle A meets \overline{BC} at D , and CD = 3 . The set of all possible values of AC is an open interval (m,n) . What is m+n ?
💡 解题思路
Let $AC=x.$ By Angle Bisector Theorem, we have $\frac{AB}{AC}=\frac{BD}{CD},$ from which $BD=CD\cdot\frac{AB}{AC}=\frac{30}{x}.$
13
第 13 题
几何·面积
Square ABCD has side length 30 . Point P lies inside the square so that AP = 12 and BP = 26 . The centroids of \triangle{ABP} , \triangle{BCP} , \triangle{CDP} , and \triangle{DAP} are the vertices of a convex quadrilateral. What is the area of that quadrilateral? [图]
💡 解题思路
As shown below, let $M_1,M_2,M_3,M_4$ be the midpoints of $\overline{AB},\overline{BC},\overline{CD},\overline{DA},$ respectively, and $G_1,G_2,G_3,G_4$ be the centroids of $\triangle{ABP},\triangle{B
14
第 14 题
数论
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
💡 解题思路
Suppose that Chloe is $c$ years old today, so Joey is $c+1$ years old today. After $n$ years, Chloe and Zoe will be $n+c$ and $n+1$ years old, respectively. We are given that \[\frac{n+c}{n+1}=1+\frac
15
第 15 题
数论
How many odd positive 3 -digit integers are divisible by 3 but do not contain the digit 3 ?
💡 解题思路
Let $\underline{ABC}$ be one such odd positive $3$ -digit integer with hundreds digit $A,$ tens digit $B,$ and ones digit $C.$ Since $\underline{ABC}\equiv0\pmod3,$ we need $A+B+C\equiv0\pmod3$ by the
16
第 16 题
几何·面积
The solutions to the equation (z+6)^8=81 are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled A,B, and C . What is the least possible area of \triangle ABC?
💡 解题思路
Recall that translations preserve the shapes and the sizes for all objects. We translate the solutions to the given equation $6$ units right, so they become the solutions to the equation $z^8=81.$
17
第 17 题
整数运算
Let p and q be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and q is as small as possible. What is q-p ?
💡 解题思路
More generally, let $a,b,c,d,p,$ and $q$ be positive integers such that $bc-ad=1$ and \[\frac ab 0,$ or \[bp-aq\geq1. \hspace{15mm} (1)\] From $\frac pq 0,$ or \[cq-dp\geq1. \hspace{15mm} (2)\] Since
18
第 18 题
函数
A function f is defined recursively by f(1)=f(2)=1 and \[f(n)=f(n-1)-f(n-2)+n\] for all integers n ≥ 3 . What is f(2018) ?
💡 解题思路
For all integers $n \geq 7,$ note that \begin{align*} f(n)&=f(n-1)-f(n-2)+n \\ &=[f(n-2)-f(n-3)+n-1]-f(n-2)+n \\ &=-f(n-3)+2n-1 \\ &=-[f(n-4)-f(n-5)+n-3]+2n-1 \\ &=-f(n-4)+f(n-5)+n+2 \\ &=-[f(n-5)-f(n
19
第 19 题
数字运算
Mary chose an even 4 -digit number n . She wrote down all the divisors of n in increasing order from left to right: 1,2,\ldots,\dfrac{n}{2},n . At some moment Mary wrote 323 as a divisor of n . What is the smallest possible value of the next divisor written to the right of 323 ?
💡 解题思路
Let $d$ be the next divisor written to the right of $323.$
20
第 20 题
几何·面积
Let ABCDEF be a regular hexagon with side length 1 . Denote by X , Y , and Z the midpoints of sides \overline {AB} , \overline{CD} , and \overline{EF} , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of \triangle ACE and \triangle XYZ ? [图] ~MRENTHUSIASM
💡 解题思路
[asy] /* Made by MRENTHUSIASM */ size(200); draw(polygon(6)); pair A, B, C, D, E, F, X, Y, Z, M, N, O, P, Q, R; A = dir(120); B = dir(60); C = dir(0); D = dir(300); E = dir(240); F = dir(180); X = mid
21
第 21 题
几何·面积
In \triangle{ABC} with side lengths AB = 13 , AC = 12 , and BC = 5 , let O and I denote the circumcenter and incenter, respectively. A circle with center M is tangent to the legs AC and BC and to the circumcircle of \triangle{ABC} . What is the area of \triangle{MOI} ? [图] ~MRENTHUSIASM
💡 解题思路
In this solution, let the brackets denote areas.
22
第 22 题
几何·角度
Consider polynomials P(x) of degree at most 3 , each of whose coefficients is an element of \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} . How many such polynomials satisfy P(-1) = -9 ?
💡 解题思路
Suppose that $P(x)=ax^3+bx^2+cx+d.$ This problem is equivalent to counting the ordered quadruples $(a,b,c,d),$ where all of $a,b,c,$ and $d$ are integers from $0$ through $9$ such that \[P(-1)=-a+b-c+
23
第 23 题
规律与数列
Ajay is standing at point A near Pontianak, Indonesia, 0^\circ latitude and 110^\circ E longitude. Billy is standing at point B near Big Baldy Mountain, Idaho, USA, 45^\circ N latitude and 115^\circ W longitude. Assume that Earth is a perfect sphere with center C. What is the degree measure of \angle ACB? [图] ~MRENTHUSIASM
💡 解题思路
This solution refers to the Diagram section.
24
第 24 题
方程
Let \lfloor x \rfloor denote the greatest integer less than or equal to x . How many real numbers x satisfy the equation x^2 + 10,000\lfloor x \rfloor = 10,000x ?
💡 解题思路
This rewrites itself to $x^2=10,000\{x\}$ where $\lfloor x \rfloor + \{x\} = x$ .
25
第 25 题
几何·面积
Circles \omega_1 , \omega_2 , and \omega_3 each have radius 4 and are placed in the plane so that each circle is externally tangent to the other two. Points P_1 , P_2 , and P_3 lie on \omega_1 , \omega_2 , and \omega_3 respectively such that P_1P_2=P_2P_3=P_3P_1 and line P_iP_{i+1} is tangent to \omega_i for each i=1,2,3 , where P_4 = P_1 . See the figure below. The area of \triangle P_1P_2P_3 can be written in the form √(a)+√(b) for positive integers a and b . What is a+b ? [图]
💡 解题思路
Let $O_1$ and $O_2$ be the centers of $\omega_1$ and $\omega_2$ respectively and draw $O_1O_2$ , $O_1P_1$ , and $O_2P_2$ . Note that $\angle{O_1P_1P_2}$ and $\angle{O_2P_2P_3}$ are both right. Further