2017B 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
行程问题
Kymbrea's comic book collection currently has 30 comic books in it, and she is adding to her collection at the rate of 2 comic books per month. LaShawn's collection currently has 10 comic books in it, and he is adding to his collection at the rate of 6 comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
💡 解题思路
Kymbrea has $30$ comic books initially and every month, she adds two. This can be represented as $30 + 2x$ where x is the number of months elapsed. LaShawn's collection, similarly, is $10 + 6x$ . To f
2
第 2 题
整数运算
Real numbers x , y , and z satisfy the inequalities 0<x<1 , -1<y<0 , and 1<z<2 . Which of the following numbers is necessarily positive?
💡 解题思路
Notice that $y+z$ must be positive because $|z|>|y|$ . Therefore the answer is $\boxed{\textbf{(E) } y+z}$ .
3
第 3 题
综合
Supposed that x and y are nonzero real numbers such that \frac{3x+y}{x-3y}=-2 . What is the value of \frac{x+3y}{3x-y} ?
💡 解题思路
Rearranging, we find $3x+y=-2x+6y$ , or $5x=5y\implies x=y$ . Substituting, we can convert the second equation into $\frac{x+3x}{3x-x}=\frac{4x}{2x}=\boxed{\textbf{(D)}\ 2}$ .
4
第 4 题
统计
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
💡 解题思路
Let's call the distance that Samia had to travel in total as $2x$ , so that we can avoid fractions. We know that the length of the bike ride and how far she walked are equal, so they are both $\frac{2
5
第 5 题
统计
The data set [6, 19, 33, 33, 39, 41, 41, 43, 51, 57] has median Q_2 = 40 , first quartile Q_1 = 33 , and third quartile Q_3 = 43 . An outlier in a data set is a value that is more than 1.5 times the interquartile range below the first quartle ( Q_1 ) or more than 1.5 times the interquartile range above the third quartile ( Q_3 ), where the interquartile range is defined as Q_3 - Q_1 . How many outliers does this data set have?
💡 解题思路
The interquartile range is defined as $Q3 - Q1$ , which is $43 - 33 = 10$ . $1.5$ times this value is $15$ , so all values more than $15$ below $Q1$ = $33 - 15 = 18$ is an outlier. The only one that f
6
第 6 题
几何·面积
The circle having (0,0) and (8,6) as the endpoints of a diameter intersects the x -axis at a second point. What is the x -coordinate of this point?
💡 解题思路
Because the two points are on a diameter, the center must be halfway between them at the point $(4,3)$ . The distance from $(0,0)$ to $(4,3)$ is 5 so the circle has radius 5. Thus, the equation of the
7
第 7 题
函数
The functions \sin(x) and \cos(x) are periodic with least period 2π . What is the least period of the function \cos(\sin(x)) ?
💡 解题思路
Start by noting that $\cos(-x)=\cos(x)$ . Then realize that under this function, the negative sine values yield the same as their positive value, so take the absolute value of the sine function to get
8
第 8 题
几何·面积
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
💡 解题思路
Let $a$ be the short side of the rectangle, and $b$ be the long side of the rectangle. The diagonal, therefore, is $\sqrt{a^2 + b^2}$ . We can get the equation $\frac{a}{b} = \frac{b}{\sqrt{a^2 + b^2}
9
第 9 题
几何·面积
A circle has center (-10, -4) and has radius 13 . Another circle has center (3, 9) and radius √(65) . The line passing through the two points of intersection of the two circles has equation x+y=c . What is c ?
💡 解题思路
The equations of the two circles are $(x+10)^2+(y+4)^2=169$ and $(x-3)^2+(y-9)^2=65$ . Rearrange them to $(x+10)^2+(y+4)^2-169=0$ and $(x-3)^2+(y-9)^2-65=0$ , respectively. Their intersection points a
10
第 10 题
分数与比例
At Typico High School, 60\% of the students like dancing, and the rest dislike it. Of those who like dancing, 80\% say that they like it, and the rest say that they dislike it. Of those who dislike dancing, 90\% say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
💡 解题思路
WLOG, let there be $100$ students. $60$ of them like dancing, and $40$ do not. Of those who like dancing, $20\%$ , or $12$ of them say they dislike dancing. Of those who dislike dancing, $90\%$ , or $
11
第 11 题
规律与数列
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3 , 23578 , and 987620 are monotonous, but 88 , 7434 , and 23557 are not. How many monotonous positive integers are there?
💡 解题思路
Case 1: monotonous numbers with digits in ascending order
12
第 12 题
规律与数列
What is the sum of the roots of z^{12}=64 that have a positive real part?
💡 解题思路
The root of any polynomial of the form $z^n = a$ will have all $n$ of it roots will have magnitude $\sqrt[n]{a}$ and be the vertices of a regular $n$ -gon in the complex plane (This concept is known a
13
第 13 题
综合
In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible? [图]
💡 解题思路
Looking at the answer choices, we see that the possibilities are indeed countable. Thus, we will utilize that approach in the form of two separate cases, as rotation and reflection take care of numero
14
第 14 题
立体几何
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
💡 解题思路
The top cone has radius 2 and height 4 so it has volume $\dfrac{1}{3} \pi (2)^2 \times 4$ .
15
第 15 题
几何·面积
Let ABC be an equilateral triangle. Extend side \overline{AB} beyond B to a point B' so that BB'=3AB . Similarly, extend side \overline{BC} beyond C to a point C' so that CC'=3BC , and extend side \overline{CA} beyond A to a point A' so that AA'=3CA . What is the ratio of the area of \triangle A'B'C' to the area of \triangle ABC ? [图]
💡 解题思路
Note that by symmetry, $\triangle A'B'C'$ is also equilateral. Therefore, we only need to find one of the sides of $A'B'C'$ to determine the area ratio. WLOG, let $AB = BC = CA = 1$ . Therefore, $BB'
16
第 16 题
概率
The number 21!=51,090,942,171,709,440,000 has over 60,000 positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
💡 解题思路
We can consider a factor of $21!$ to be odd if it does not contain a $2$ ; hence, finding the exponent of $2$ in the prime factorization of $21!$ will help us find our answer. We can start off with al
17
第 17 题
概率
A coin is biased in such a way that on each toss the probability of heads is \frac{2}{3} and the probability of tails is \frac{1}{3} . The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B? (A) The probability of winning Game A is \frac{4}{81} less than the probability of winning Game B. (B) The probability of winning Game A is \frac{2}{81} less than the probability of winning Game B. (C) The probabilities are the same. (D) The probability of winning Game A is \frac{2}{81} greater than the probability of winning Game B. (E) The probability of winning Game A is \frac{4}{81} greater than the probability of winning Game B.
💡 解题思路
The probability of winning Game A is the sum of the probabilities of getting three tails and getting three heads which is $\left(\frac{2}{3}\right)^3 + \left(\frac{1}{3}\right)^3 = \frac{8}{27} + \fra
18
第 18 题
几何·面积
The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD=3 . Point E is chosen so that ED=5 and line ED is perpendicular to line AD . Segment AE intersects the circle at a point C between A and E . What is the area of \triangle ABC ?
💡 解题思路
[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(8.865514650638614cm); real labelscalefactor = 0.5; /* changes
19
第 19 题
数论
Let N=123456789101112\dots4344 be the 79 -digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when N is divided by 45 ?
💡 解题思路
We will consider this number $\bmod\ 5$ and $\bmod\ 9$ . By looking at the last digit, it is obvious that the number is $\equiv 4\bmod\ 5$ . To calculate the number $\bmod\ 9$ , note that
20
第 20 题
概率
Real numbers x and y are chosen independently and uniformly at random from the interval (0,1) . What is the probability that \lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor ?
💡 解题思路
First let us take the case that $\lfloor \log_2{x} \rfloor = \lfloor \log_2{y} \rfloor = -1$ . In this case, both $x$ and $y$ lie in the interval $[{1\over2}, 1)$ . The probability of this is $\frac{1
21
第 21 题
统计
Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
💡 解题思路
First, remove all the 90s, since they make no impact. So, we have numbers from $1$ to $10$ . Then, $5$ is the 7th number. Let the sum of the first $6$ numbers be $k$ . Then, $k\equiv 0 \mod 6$ and $k\
22
第 22 题
概率
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
💡 解题思路
It amounts to filling in a $4 \times 4$ matrix. Columns $C_1 - C_4$ are the random draws each round; rowof each player. Also, let $\%R_A$ be the number of nonzero elements in $R_A$ . Sidenote: (Not th
23
第 23 题
坐标几何
The graph of y=f(x) , where f(x) is a polynomial of degree 3 , contains points A(2,4) , B(3,9) , and C(4,16) . Lines AB , AC , and BC intersect the graph again at points D , E , and F , respectively, and the sum of the x -coordinates of D , E , and F is 24. What is f(0) ?
💡 解题思路
Note that $f(x) - x^2$ has roots $2, 3$ , and $4$ . Therefore, we may write $f(x) = a(x-2)(x-3)(x-4) +x^2$ . Now we find that lines $AB$ , $AC$ , and $BC$ are defined by the equations $y = 5x - 6$ , $
24
第 24 题
几何·面积
Quadrilateral ABCD has right angles at B and C , \triangle ABC \sim \triangle BCD , and AB > BC . There is a point E in the interior of ABCD such that \triangle ABC \sim \triangle CEB and the area of \triangle AED is 17 times the area of \triangle CEB . What is \tfrac{AB}{BC} ?
💡 解题思路
Let $CD=1$ , $BC=x$ , and $AB=x^2$ . Note that $AB/BC=x$ . By the Pythagorean Theorem, $BD=\sqrt{x^2+1}$ . Since $\triangle BCD \sim \triangle ABC \sim \triangle CEB$ , the ratios of side lengths must
25
第 25 题
统计
A set of n people participate in an online video basketball tournament. Each person may be a member of any number of 5 -player teams, but no two teams may have exactly the same 5 members. The site statistics show a curious fact: The average, over all subsets of size 9 of the set of n participants, of the number of complete teams whose members are among those 9 people is equal to the reciprocal of the average, over all subsets of size 8 of the set of n participants, of the number of complete teams whose members are among those 8 people. How many values n , 9≤ n≤ 2017 , can be the number of participants?
💡 解题思路
Let there be $T$ teams. For each team, there are ${n-5\choose 4}$ different subsets of $9$ players that includes a given full team, so the total number of team-(group of 9) pairs is