2023B 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
分数与比例
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only \frac{1}{3} full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
💡 解题思路
The first three glasses each have a full glass. Let's assume that each glass has "1 unit" of juice. It won't matter exactly how much juice everyone has because we're dealing with ratios, and that woul
2
第 2 题
计数
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20\% on every pair of shoes. Carlos also knew that he had to pay a 7.5\% sales tax on the discounted price. He had \43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
💡 解题思路
Let the price originally be \( x \). Then, after a \(20\) percent discount, the price is now \( x - \frac{1}{5}x = \frac{4}{5}x \).
3
第 3 题
几何·面积
A 3-4-5 right triangle is inscribed in circle A , and a 5-12-13 right triangle is inscribed in circle B . What is the ratio of the area of circle A to the area of circle B ?
💡 解题思路
Because the triangles are right triangles, we know the hypotenuses are diameters of circles $A$ and $B$ . Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply $\pi$ to
4
第 4 题
几何·面积
Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint?
💡 解题思路
$6.5$ millimeters is equal to $0.65$ centimeters. $25$ meters is $2500$ centimeters. The answer is $0.65 \times 2500$ , so the answer is $\boxed{\textbf{(C) 1,625}}$ .
5
第 5 题
几何·面积
You are playing a game. A 2 × 1 rectangle covers two adjacent squares (oriented either horizontally or vertically) of a 3 × 3 grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
💡 解题思路
Notice that the $3\times3$ square grid has a total of $12$ possible $2\times1$ rectangles.
6
第 6 题
整数运算
When the roots of the polynomial \[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 · · · · (x-10)^{10}\] are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is P(x) positive?
💡 解题思路
The expressions to the power of even powers are always positive, so we don't need to care about those. We only need to care about $(x-1)^1(x-3)^3(x-5)^5(x-7)^7(x-9)^9$ . We need 0, 2, or 4 of the expr
7
第 7 题
整数运算
For how many integers n does the expression \[√(\frac{\log (n^2) - (\log n)^2){\log n - 3}}\] represent a real number, where log denotes the base 10 logarithm?
💡 解题思路
We have \begin{align*} \sqrt{\frac{\log \left( n^2 \right) - \left( \log n \right)^2}{\log n - 3}} & = \sqrt{\frac{2 \log n - \left( \log n \right)^2}{\log n - 3}} \\ & = \sqrt{\frac{\left( \log n \ri
8
第 8 题
综合
How many nonempty subsets B of \{0, 1, 2, 3, ·s, 12\} have the property that the number of elements in B is equal to the least element of B ? For example, B = \{4, 6, 8, 11\} satisfies the condition.
💡 解题思路
There is no way to have a set with 0. If a set is to have its lowest element as 1, it must have only 1 element: 1. If a set is to have its lowest element as 2, it must have 2, and the other element wi
9
第 9 题
几何·面积
What is the area of the region in the coordinate plane defined by | | x | - 1 | + | | y | - 1 | \le 1 ? (A)\ 2 (B)\ 8 (C)\ 4 (D)\ 15 (E)\ 12 ~diagram by grogg007
💡 解题思路
First consider, $|x-1|+|y-1| \le 1.$ We can see that it is a square with a side length of $\sqrt{2}$ (diagonal $2$ ). The area of the square is $\sqrt{2}^2 = 2.$
10
第 10 题
几何·面积
In the xy -plane, a circle of radius 4 with center on the positive x -axis is tangent to the y -axis at the origin, and a circle with radius 10 with center on the positive y -axis is tangent to the x -axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?
💡 解题思路
The center of the first circle is $(4,0)$ . The center of the second circle is $(0,10)$ . Thus, the slope of the line that passes through these two centers is $- \frac{10}{4} = - \frac{5}{2}$ .
11
第 11 题
几何·面积
What is the maximum area of an isosceles trapezoid that has legs of length 1 and one base twice as long as the other?
💡 解题思路
Let the trapezoid be $ABCD$ with $AD = BC = 1, \; AB = x, CD = 2x$ . Extend $AD$ and $BC$ to meet at point $E$ . Then, notice $\triangle ABE \sim \triangle DCE$ with side length ratio $1:2$ and $AE =
12
第 12 题
分数与比例
For complex number u = a+bi and v = c+di (where i=√(-1) ), define the binary operation u \otimes v = ac + bdi Suppose z is a complex number such that z\otimes z = z^{2}+40 . What is |z| ?
💡 解题思路
let $z$ = $a+bi$ .
13
第 13 题
几何·面积
A rectangular box \mathcal{P} has distinct edge lengths a , b , and c . The sum of the lengths of all 12 edges of \mathcal{P} is 13 , the areas of all 6 faces of \mathcal{P} is \frac{11}{2} , and the volume of \mathcal{P} is \frac{1}{2} . What is the length of the longest interior diagonal connecting two vertices of \mathcal{P} ?
💡 解题思路
[asy] import geometry; pair A = (-3, 4); pair B = (-3, 5); pair C = (-1, 4); pair D = (-1, 5); pair AA = (0, 0); pair BB = (0, 1); pair CC = (2, 0); pair DD = (2, 1); draw(D--AA,dashed); draw(A--B); d
14
第 14 题
整数运算
For how many ordered pairs (a,b) of integers does the polynomial x^3+ax^2+bx+6 have 3 distinct integer roots?
💡 解题思路
Denote three roots as $r_1 < r_2 < r_3$ . Following from Vieta's formula, $r_1r_2r_3 = -6$ .
15
第 15 题
数论
Suppose a , b , and c are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true? I. If \gcd(a,14)=1 or \gcd(b,15)=1 or both, then \gcd(c,210)=1 . II. If \gcd(c,210)=1 , then \gcd(a,14)=1 or \gcd(b,15)=1 or both. III. \gcd(c,210)=1 if and only if \gcd(a,14)=\gcd(b,15)=1 .
💡 解题思路
We examine each of the conditions.
16
第 16 题
概率
In the state of Coinland, coins have values 6,10, and 15 cents. Suppose x is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of x?
💡 解题思路
This problem asks to find largest $x$ that cannot be written as \[6 a + 10 b + 15 c = x, \hspace{1cm} (1)\]
17
第 17 题
几何·面积
Triangle ABC has side lengths in arithmetic progression, and the smallest side has length 6. If the triangle has an angle of 120^\circ, what is the area of ABC ?
💡 解题思路
The length of the side opposite to the angle with $120^\circ$ is longest. We denote its value as $x$ .
18
第 18 题
统计
Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was 3 points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was 18 points higher than her average for the first semester and was again 3 points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true? (A) Yolanda's quiz average for the academic year was 22 points higher than Zelda's. (B) Zelda's quiz average for the academic year was higher than Yolanda's. (C) Yolanda's quiz average for the academic year was 3 points higher than Zelda's. (D) Zelda's quiz average for the academic year equaled Yolanda's. (E) If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.
💡 解题思路
Denote by $A_i$ the average of person with initial $A$ in semester $i \in \left\{1, 2 \right\}$ Thus, $Y_1 = Z_1 + 3$ , $Y_2 = Y_1 + 18$ , $Y_2 = Z_2 + 3$ .
19
第 19 题
概率
Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? Some of the solutions below are not quite correct, though they ultimately lead to the same answer choice. See the Talk page for more details .
💡 解题思路
Because each bin will have an odd number, they will have at least one ball. So we can put one ball in each bin prematurely. We then can add groups of 2 balls into each bin, meaning we now just have to
20
第 20 题
概率
Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position?
💡 解题思路
Let Cyrus's starting position be $S$ . WLOG, let the place Cyrus lands at for his first jump be $O$ . From $O$ , Cyrus can reach all the points on $\odot O$ . The probability that Cyrus will land less
21
第 21 题
综合
A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is 3\sqrt3 inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey? [图]
💡 解题思路
We augment the frustum to a circular cone. Denote by $O$ the apex of the cone. Denote by $A$ the bug and $B$ the honey.
22
第 22 题
函数
A real-valued function f has the property that for all real numbers a and b, \[f(a + b) + f(a - b) = 2f(a) f(b).\] Which one of the following cannot be the value of f(1)?
💡 解题思路
Substituting $a = b$ we get \[f(2a) + f(0) = 2f(a)^2\] Substituting $a= 0$ we find \[2f(0) = 2f(0)^2 \implies f(0) \in \{0, 1\}.\] This gives \[f(2a) = 2f(a)^2 - f(0) \geq 0-1\] Plugging in $a = \frac
23
第 23 题
概率
When n standard six-sided dice are rolled, the product of the numbers rolled can be any of 936 possible values. What is n ?
💡 解题思路
We start by trying to prove a function of $n$ , and then we can apply the function and equate it to $936$ to find the value of $n$ .
24
第 24 题
数论
Suppose that a , b , c and d are positive integers satisfying all of the following relations. \[abcd=2^6· 3^9· 5^7\] \[lcm(a,b)=2^3· 3^2· 5^3\] \[lcm(a,c)=2^3· 3^3· 5^3\] \[lcm(a,d)=2^3· 3^3· 5^3\] \[lcm(b,c)=2^1· 3^3· 5^2\] \[lcm(b,d)=2^2· 3^3· 5^2\] \[lcm(c,d)=2^2· 3^3· 5^2\] What is gcd(a,b,c,d) ?
💡 解题思路
Denote by $\nu_p (x)$ the number of prime factor $p$ in number $x$ .
25
第 25 题
几何·面积
A regular pentagon with area √(5)+1 is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
💡 解题思路
Since $A$ is folded onto $O$ , $AM = MO$ where $M$ is the intersection of $AO$ and the creaseline between $A$ and $O$ . Note that the inner pentagon is regular, and therefore similar to the original p