2017 AMC 10A — Official Competition Problems (February 2017)
📅 2017 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
What is the value of (2(2(2(2(2(2+1)+1)+1)+1)+1)+1) ?
💡 解题思路
Notice this is the term $a_6$ in a recursive sequence, defined recursively as $a_1 = 3, a_n = 2a_{n-1} + 1.$ Thus: \[\begin{split} a_2 = 3 \cdot 2 + 1 = 7.\\ a_3 = 7 \cdot 2 + 1 = 15.\\ a_4 = 15 \cdot
2
第 2 题
综合
Pablo buys popsicles for his friends. The store sells single popsicles for \1 each, 3 -popsicle boxes for \2 each, and 5 -popsicle boxes for \3 . What is the greatest number of popsicles that Pablo can buy with \8 ?
💡 解题思路
$\$3$ boxes give us the most popsicles/dollar, so we want to buy as many of those as possible. After buying $2$ , we have $\$2$ left. We cannot buy a third $\$3$ box, so we opt for the $\$2$ box inste
3
第 3 题
几何·面积
Tamara has three rows of two 6 -feet by 2 -feet flower beds in her garden. The beds are separated and also surrounded by 1 -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet? [图]
💡 解题思路
Finding the area of the shaded walkway can be achieved by computing the total area of Tamara's garden and then subtracting the combined area of her six flower beds.
4
第 4 题
行程问题
Mia is "helping" her mom pick up 30 toys that are strewn on the floor. Mia’s mom manages to put 3 toys into the toy box every 30 seconds, but each time immediately after those 30 seconds have elapsed, Mia takes 2 toys out of the box. How much time, in minutes, will it take Mia and her mom to put all 30 toys into the box for the first time?
💡 解题思路
Every $30$ seconds, $3$ toys are put in the box and $2$ toys are taken out, so the number of toys in the box increases by $3-2=1$ every $30$ seconds. Then after $27 \times 30 = 810$ seconds (or $13 \f
5
第 5 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 the two real numbers be $x,y$ . We are given that $x+y=4xy,$ and dividing both sides by $xy$ , $\frac{x}{xy}+\frac{y}{xy}=4.$
6
第 6 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
Rewriting the given statement: "if someone got all the multiple choice questions right on the upcoming exam then he or she would receive an A on the exam." If that someone is Lewis the statement becom
7
第 7 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 Silvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define t
8
第 8 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 within the group?
💡 解题思路
Each one of the ten people has to shake hands with all the $20$ other people they don’t know. So $10\cdot20 = 200$ . From there, we calculate how many handshakes occurred between the people who don’t
9
第 9 题
行程问题
Minnie rides on a flat road at 20 kilometers per hour (kph), downhill at 30 kph, and uphill at 5 kph. Penny rides on a flat road at 30 kph, downhill at 40 kph, and uphill at 10 kph. Minnie goes from town A to town B , a distance of 10 km all uphill, then from town B to town C , a distance of 15 km all downhill, and then back to town A , a distance of 20 km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the 45 -km ride than it takes Penny?
💡 解题思路
The distance from town $A$ to town $B$ is $10$ km uphill, and since Minnie rides uphill at a speed of $5$ kph, it will take her $2$ hours. Next, she will ride from town $B$ to town $C$ , a distance of
10
第 10 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 triangle inequality generalizes to all polygons, so $x < 3+7+15$ and $15<x+3+7$ yields $5<x<25$ . Now, we know that there are $19$ numbers between $5$ and $25$ exclusive, but we must subtract $2$
11
第 11 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 216π . What is the length \textit{AB} ?
💡 解题思路
In order to solve this problem, we must first visualize what the region looks like. We know that, in a three dimensional space, the region consisting of all points within $3$ units of a point would be
12
第 12 题
坐标几何
Let S be a 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 this common value. Which of the following is a correct description for S?
💡 解题思路
If the two equal values are $3$ and $x+2$ , then $x=1$ . Also, $y-4\le 3$ because $3$ is the common value. Solving for $y$ , we get $y \le 7$ . Therefore the portion of the line $x=1$ where $y \le 7$
13
第 13 题
数论
Define a sequence recursively by F_{0}=0,~F_{1}=1, and F_{n}= the remainder when F_{n-1}+F_{n-2} is divided by 3, for all n≥ 2. Thus the sequence starts 0,1,1,2,0,2,\ldots What is F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}?
💡 解题思路
A pattern starts to emerge as the function is continued. The repeating pattern is $0,1,1,2,0,2,2,1\ldots$ The problem asks for the sum of eight consecutive terms in the sequence. Because there are eig
14
第 14 题
分数与比例
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was A dollars. The cost of his movie ticket was 20\% of the difference between A and the cost of his soda, while the cost of his soda was 5\% of the difference between A and the cost of his movie ticket. To the nearest whole percent, what fraction of A did Roger pay for his movie ticket and soda?
💡 解题思路
Let $m$ = cost of movie ticket Let $s$ = cost of soda
15
第 15 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
We can use geometric probability to solve this. Suppose a point $(x,y)$ lies in the $xy$ -plane. Let $x$ be Chloe's number and $y$ be Laurent's number. Then obviously we want $y>x$ , which basically g
16
第 16 题
规律与数列
There are 10 horses, named Horse 1 , Horse 2 , . . . , 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
If we have horses, $a_1, a_2, \ldots, a_n$ , then any number that is a multiple of all those numbers is a time when all horses will meet at the starting point. The least of these numbers is the LCM. T
17
第 17 题
几何·面积
Distinct points P , Q , R , S lie on the circle x^{2}+y^{2}=25 and have integer coordinates. The distances PQ and RS are irrational numbers. What is the greatest possible value of the ratio \frac{PQ}{RS} ?
💡 解题思路
Because $P$ , $Q$ , $R$ , and $S$ are lattice points, there are only a few coordinates that actually satisfy the equation. The coordinates are $(\pm 3,\pm 4), (\pm 4, \pm 3), (0,\pm 5),$ and $(\pm 5,0
18
第 18 题
数论
Amelia has a coin that lands heads with probability \frac{1}{3} , and Blaine has a coin that lands on heads with probability \frac{2}{5} . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is \frac{p}{q} , where p and q are relatively prime positive integers. What is q-p ?
💡 解题思路
Let $P$ be the probability Amelia wins. Note that $P = \text{chance she wins on her first turn} + \text{chance she gets to her turn again}\cdot P$ , since if she gets to her turn again, she is back wh
19
第 19 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
Let Alice be A, Bob be B, Carla be C, Derek be D, and Eric be E. We can split this problem up into two cases:
20
第 20 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 $n \equiv S(n) \pmod{9}$ . This can be seen from the fact that $\sum_{k=0}^{n}10^{k}a_k \equiv \sum_{k=0}^{n}a_k \pmod{9}$ . Thus, if $S(n) = 1274$ , then $n \equiv 5 \pmod{9}$ , and thus $n
21
第 21 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 \dfrac{x}{y} ?
💡 解题思路
Analyze the first right triangle.
22
第 22 题
几何·面积
Sides \overline{AB} and \overline{AC} of equilateral triangle ABC are tangent to a circle at points B and C respectively. What fraction of the area of \triangle ABC lies outside the circle?
How many triangles with positive area have all their vertices at points (i,j) in the coordinate plane, where i and j are integers between 1 and 5 , inclusive?
💡 解题思路
We can solve this by finding all the combinations, then subtracting the ones that are on the same line. There are $25$ points in all, from $(1,1)$ to $(5,5)$ , so $\dbinom{25}3$ is $\frac{25\cdot 24\c
24
第 24 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
$f(x)$ must have four roots, three of which are roots of $g(x)$ . Using the fact that every polynomial has a unique factorization into its roots, and since the leading coefficient of $f(x)$ and $g(x)$
25
第 25 题
数论
How many integers between 100 and 999, inclusive, have the property that some permutation of its digits is a multiple of 11 between 100 and 999? For example, both 121 and 211 have this property.
💡 解题思路
There are 81 multiples of 11 between $100$ and $999$ inclusive. Some have digits repeated twice, making 3 permutations.