2017A 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
综合
Pablo buys popsicles for his friends. The store sells single popsicles for \1 each, 3-popsicle boxes for \2 , and 5-popsicle boxes for \3 . What is the greatest number of popsicles that Pablo can buy with \8 ?
💡 解题思路
We can take two 5-popsicle boxes and one 3-popsicle box with $\$8$ . Note that it is optimal since one popsicle is at the rate of $\$1$ per popsicle, three popsicles at $\$\frac{2}{3}$ per popsicle an
2
第 2 题
规律与数列
The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
💡 解题思路
Let $x, y$ be our two numbers. Then $x+y = 4xy$ . Thus,
3
第 3 题
数论
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
💡 解题思路
Taking the contrapositive of the statement "if he got all of them right, he got an A" yields "if he didn't get an A, he didn't get all of them right", yielding the answer $\boxed{\textbf{(B)}}$ .
4
第 4 题
几何·面积
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
💡 解题思路
Let $j$ represent how far Jerry walked, and $s$ represent how far Sylvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define t
5
第 5 题
综合
At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
💡 解题思路
All of the handshakes will involve at least one person from the $10$ who knows no one. Label these ten people $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , $G$ , $H$ , $I$ , $J$ .
6
第 6 题
几何·面积
Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm . She places the rods with lengths 3 cm , 7 cm , and 15 cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
💡 解题思路
The quadrilateral cannot be a straight line. Thus, the fourth side must be longer than $15 - (3 + 7) = 5$ and shorter than $15 + 3 + 7 = 25$ . This means Joy can use the $19$ possible integer rod leng
7
第 7 题
函数
Define a function on the positive integers recursively by f(1) = 2 , f(n) = f(n-1) + 1 if n is even, and f(n) = f(n-2) + 2 if n is odd and greater than 1 . What is f(2017) ?
💡 解题思路
This is a recursive function, which means the function refers back to itself to calculate subsequent terms. To solve this, we must identify the base case, $f(1)=2$ . We also know that when $n$ is odd,
8
第 8 题
立体几何
The region consisting of all points in three-dimensional space within 3 units of line segment \overline{AB} has volume 216 π . What is the length AB ?
💡 解题思路
Let the length $AB$ be $L$ . Then, we see that the region is just the union of the cylinder with central axis $\overline{AB}$ and radius $3$ and the two hemispheres connected to each face of the cylin
9
第 9 题
坐标几何
Let S be the set of points (x,y) in the coordinate plane such that two of the three quantities 3 , x+2 , and y-4 are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of S ?
💡 解题思路
If the two equal values are $3$ and $x+2$ , then $x=1$ . Also, $y-4\leqslant 3$ because $3$ is the common value. Solving for $y$ , we get $y\leqslant 7$ . Therefore the portion of the line $x=1$ where
10
第 10 题
概率
Chloe chooses a real number uniformly at random from the interval [ 0,2017 ] . Independently, Laurent chooses a real number uniformly at random from the interval [ 0 , 4034 ] . What is the probability that Laurent's number is greater than Chloe's number?
💡 解题思路
Suppose Laurent's number is in the interval $[ 0, 2017 ]$ . Then, by symmetry, the probability of Laurent's number being greater is $\dfrac{1}{2}$ . Next, suppose Laurent's number is in the interval $
11
第 11 题
几何·面积
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of 2017 . She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
💡 解题思路
We know that the sum of the interior angles of the polygon is a multiple of $180$ . Note that $\left\lceil\frac{2017}{180}\right\rceil = 12$ and $180\cdot 12 = 2160$ , so the angle Claire forgot is $\
12
第 12 题
规律与数列
There are 10 horses, named Horse 1, Horse 2, \ldots , Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse k runs one lap in exactly k minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time S > 0 , in minutes, at which all 10 horses will again simultaneously be at the starting point is S = 2520 . Let T>0 be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of T ?
💡 解题思路
We know that Horse $k$ will be at the starting point after $n$ minutes if $k|n$ . Thus, we are looking for the smallest $n$ such that at least $5$ of the numbers $\{1,2,\cdots,10\}$ divide $n$ . Thus,
13
第 13 题
行程问题
Driving at a constant speed, Sharon usually takes 180 minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving \frac{1}{3} of the way, she hits a bad snowstorm and reduces her speed by 20 miles per hour. This time the trip takes her a total of 276 minutes. How many miles is the drive from Sharon's house to her mother's house?
💡 解题思路
Let total distance be $x$ . Her speed in miles per minute is $\tfrac{x}{180}$ . Then, the distance that she drove before hitting the snowstorm is $\tfrac{x}{3}$ . Her speed in snowstorm is reduced $20
14
第 14 题
计数
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?
💡 解题思路
Alice may sit in the center chair, in an end chair, or in a next-to-end chair. Suppose she sits in the center chair. The 2nd and 4th chairs (next to her) must be occupied by Derek and Eric, in either
15
第 15 题
方程
Let f(x) = \sin{x} + 2\cos{x} + 3\tan{x} , using radian measure for the variable x . In what interval does the smallest positive value of x for which f(x) = 0 lie?
💡 解题思路
We must first get an idea of what $f(x)$ looks like:
16
第 16 题
几何·面积
In the figure below, semicircles with centers at A and B and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter JK . The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at P is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at P ? [图]
💡 解题思路
Connect the centers of the tangent circles! (call the center of the large circle $C$ )
17
第 17 题
综合
There are 24 different complex numbers z such that z^{24}=1 . For how many of these is z^6 a real number?
💡 解题思路
Note that these $z$ such that $z^{24}=1$ are $e^{\frac{ni\pi}{12}}$ for integer $0\leq n<24$ . So
18
第 18 题
规律与数列
Let S(n) equal the sum of the digits of positive integer n . For example, S(1507) = 13 . For a particular positive integer n , S(n) = 1274 . Which of the following could be the value of S(n+1) ?
💡 解题思路
Note that $n\equiv S(n)\bmod 9$ , so $S(n+1)-S(n)\equiv n+1-n = 1\bmod 9$ . So, since $S(n)=1274\equiv 5\bmod 9$ , we have that $S(n+1)\equiv 6\bmod 9$ . The only one of the answer choices $\equiv 6\b
19
第 19 题
几何·面积
A square with side length x is inscribed in a right triangle with sides of length 3 , 4 , and 5 so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length y is inscribed in another right triangle with sides of length 3 , 4 , and 5 so that one side of the square lies on the hypotenuse of the triangle. What is \frac{x}{y} ?
💡 解题思路
Analyze the first right triangle.
20
第 20 题
方程
How many ordered pairs (a,b) such that a is a positive real number and b is an integer between 2 and 200 , inclusive, satisfy the equation (\log_b a)^{2017}=\log_b(a^{2017})?
💡 解题思路
By the properties of logarithms, we can rearrange the equation to read $x^{2017}=2017x$ with $x=\log_b a$ . If $x\neq 0$ , we may divide by it and get $x^{2016}=2017$ , which implies $x=\pm \root{2016
21
第 21 题
逻辑推理
A set S is constructed as follows. To begin, S = \{0,10\} . Repeatedly, as long as possible, if x is an integer root of some polynomial a_{n}x^n + a_{n-1}x^{n-1} + \dots + a_{1}x + a_0 for some n≥{1} , all of whose coefficients a_i are elements of S , then x is put into S . When no more elements can be added to S , how many elements does S have?
💡 解题思路
At first, $S=\{0,10\}$ .
22
第 22 题
几何·面积
A square is drawn in the Cartesian coordinate plane with vertices at (2, 2) , (-2, 2) , (-2, -2) , (2, -2) . A particle starts at (0,0) . Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is 1/8 that the particle will move from (x, y) to each of (x, y + 1) , (x + 1, y + 1) , (x + 1, y) , (x + 1, y - 1) , (x, y - 1) , (x - 1, y - 1) , (x - 1, y) , or (x - 1, y + 1) . The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is m/n , where m and n are relatively prime positive integers. What is m + n ?
💡 解题思路
We let $c, e,$ and $m$ be the probability of reaching a corner before an edge when starting at an "inside corner" (e.g. $(1, 1)$ ), an "inside edge" (e.g. $(1, 0)$ ), and the middle respectively.
23
第 23 题
函数
For certain real numbers a , b , and c , the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of g(x) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is f(1) ?
💡 解题思路
Let $r_1,r_2,$ and $r_3$ be the roots of $g(x)$ . Let $r_4$ be the additional root of $f(x)$ . Then from Vieta's formulas on the quadratic term of $g(x)$ and the cubic term of $f(x)$ , we obtain the f
24
第 24 题
几何·面积
Quadrilateral ABCD is inscribed in circle O and has side lengths AB=3, BC=2, CD=6 , and DA=8 . Let X and Y be points on \overline{BD} such that \frac{DX}{BD} = \frac{1}{4} and \frac{BY}{BD} = \frac{11}{36} . Let E be the intersection of line AX and the line through Y parallel to \overline{AD} . Let F be the intersection of line CX and the line through E parallel to \overline{AC} . Let G be the point on circle O other than C that lies on line CX . What is XF· XG ? [图] ~raxu, put in by fuzimiao2013
💡 解题思路
Using the given ratios, note that $\frac{XY}{BD} = 1 - \frac{1}{4} - \frac{11}{36} = \frac{4}{9}.$
25
第 25 题
概率
The vertices V of a centrally symmetric hexagon in the complex plane are given by \[V=\{ √(2)i,-√(2)i, \frac{1}{√(8)}(1+i),\frac{1}{√(8)}(-1+i),\frac{1}{√(8)}(1-i),\frac{1}{√(8)}(-1-i) \}.\] For each j , 1≤ j≤ 12 , an element z_j is chosen from V at random, independently of the other choices. Let P={\prod}_{j=1}^{12}z_j be the product of the 12 numbers selected. What is the probability that P=-1 ?
💡 解题思路
It is possible to solve this problem using elementary counting methods. This solution proceeds by a cleaner generating function.