2017 AMC 10B — Official Competition Problems (February 2017)
📅 2017 B 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
数字运算
Mary thought of a positive two-digit number. She multiplied it by 3 and added 11 . Then she switched the digits of the result, obtaining a number between 71 and 75 , inclusive. What was Mary's number?
💡 解题思路
Let her $2$ -digit number be $x$ . Multiplying by $3$ makes it a multiple of $3$ , meaning that the sum of its digits is divisible by $3$ . Adding on $11$ increases the sum of the digits by $1+1 = 2,$
2
第 2 题
统计
Sofia ran 5 laps around the 400 -meter track at her school. For each lap, she ran the first 100 meters at an average speed of 4 meters per second and the remaining 300 meters at an average speed of 5 meters per second. How much time did Sofia take running the 5 laps?
💡 解题思路
If Sofia ran the first $100$ meters of each lap at $4$ meters per second and the remaining $300$ meters of each lap at $5$ meters per second, then she took $\frac{100}{4}+\frac{300}{5}=25+60=85$ secon
3
第 3 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 $\boxed{\textbf{(E) } y+z}$ .
4
第 4 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 $\frac{x+3x}{3x-x}=\frac{4x}{2x}=\boxed{\textbf{(D)}\ 2}$ .
5
第 5 题
行程问题
Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?
💡 解题思路
Denote the number of blueberry and cherry jelly beans as $b$ and $c$ respectively. Then $b = 2c$ and $b-10 = 3(c-10)$ . Substituting, we have $2c-10 = 3c-30$ , so $c=20$ , $b=\boxed{\textbf{(D) } 40}$
6
第 6 题
行程问题
What is the largest number of solid 2-in × 2-in × 1-in blocks that can fit in a 3-in × 2-in×3-in box?
💡 解题思路
We find that the volume of the larger block is $18$ , and the volume of the smaller block is $4$ . Dividing the two, we see that only a maximum of four $2$ by $2$ by $1$ blocks can fit inside the $3$
7
第 7 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
8
第 8 题
几何·面积
Points A(11, 9) and B(2, -3) are vertices of \triangle ABC with AB=AC . The altitude from A meets the opposite side at D(-1, 3) . What are the coordinates of point C ?
💡 解题思路
Since $AB = AC$ , then $\triangle ABC$ is isosceles, so $BD = CD$ . Therefore, the coordinates of $C$ are $(-1 - 3, 3 + 6) = \boxed{\textbf{(C) } (-4,9)}$ .
9
第 9 题
数论
A radio program has a quiz consisting of 3 multiple-choice questions, each with 3 choices. A contestant wins if he or she gets 2 or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
💡 解题思路
There are two ways the contestant can win.
10
第 10 题
方程
The lines with equations ax-2y=c and 2x+by=-c are perpendicular and intersect at (1, -5) . What is c ?
💡 解题思路
Writing each equation in slope-intercept form, we get $y=\frac{a}{2}x-\frac{1}{2}c$ and $y=-\frac{2}{b}x-\frac{c}{b}$ . We observe the slope of each equation is $\frac{a}{2}$ and $-\frac{2}{b}$ , resp
11
第 11 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
$60\% \cdot 20\% = 12\%$ of the people that claim that they dislike dancing actually like it, and $40\% \cdot 90\% = 36\%$ of the people that claim that they dislike dancing actually dislike it. There
12
第 12 题
分数与比例
Elmer's new car gives 50\% percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is 20\% more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?
💡 解题思路
Suppose that his old car runs at $x$ km per liter. Then his new car runs at $\frac{3}{2}x$ km per liter, or $x$ km per $\frac{2}{3}$ of a liter. Let the cost of the old car's fuel be $c$ , so the trip
13
第 13 题
综合
There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?
💡 解题思路
By PIE (Property of Inclusion/Exclusion), we have
14
第 14 题
数论
An integer N is selected at random in the range 1≤ N ≤ 2020 . What is the probability that the remainder when N^{16} is divided by 5 is 1 ?
💡 解题思路
Notice that we can rewrite $N^{16}$ as $(N^{4})^4$ . By Fermat's Little Theorem , we know that $N^{(5-1)} \equiv 1 \pmod {5}$ if $N \not \equiv 0 \pmod {5}$ . Therefore for all $N \not \equiv 0 \pmod
15
第 15 题
几何·面积
Rectangle ABCD has AB=3 and BC=4 . Point E is the foot of the perpendicular from B to diagonal \overline{AC} . What is the area of \triangle AED ?
How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0 ?
💡 解题思路
We can use complementary counting. There are $2017$ positive integers in total to consider, and there are $9$ one-digit integers, $9 \cdot 9 = 81$ two digit integers without a zero, $9 \cdot 9 \cdot 9
17
第 17 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
18
第 18 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
First we figure out the number of ways to put the $3$ blue disks. Denote the spots to put the disks as $1-6$ from left to right, top to bottom. The cases to put the blue disks are $(1,2,3),(1,2,4),(1,
19
第 19 题
几何·面积
Let ABC be an equilateral triangle. Extend side \overline{AB} beyond B to a point B' so that BB'=3 · AB . Similarly, extend side \overline{BC} beyond C to a point C' so that CC'=3 · BC , and extend side \overline{CA} beyond A to a point A' so that AA'=3 · CA . 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'
20
第 20 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 note that the only thing that affects the parity of the factor are the powers of 2. There are $10+5+2+1 = 18$ factors of 2 in the number. Thus, there are $18$ cases in which a factor of $21!$ would
21
第 21 题
几何·面积
In \triangle ABC , AB=6 , AC=8 , BC=10 , and D is the midpoint of \overline{BC} . What is the sum of the radii of the circles inscribed in \triangle ADB and \triangle ADC ?
💡 解题思路
We note that by the converse of the Pythagorean Theorem, $\triangle ABC$ is a right triangle with a right angle at $A$ . Also, the median to the hypotenuse will be half of the hypotenuse. Therefore, $
22
第 22 题
几何·面积
The diameter \overline{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 \overline{AE} intersects the circle at a point C between A and E . What is the area of \triangle ABC ?
💡 解题思路
Notice that $ADE$ and $ABC$ are right triangles. Then $AE = \sqrt{7^2+5^2} = \sqrt{74}$ . $\sin{DAE} = \frac{5}{\sqrt{74}} = \sin{BAE} = \sin{BAC} = \frac{BC}{4}$ , so $BC = \frac{20}{\sqrt{74}}$ . We
23
第 23 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 only need to find the remainders of N when divided by 5 and 9 to determine the answer. By inspection, $N \equiv 4 \text{ (mod 5)}$ . The remainder when $N$ is divided by $9$ is $1+2+3+4+ \cdots +1+
24
第 24 题
几何·面积
The vertices of an equilateral triangle lie on the hyperbola xy=1 , and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? [图]
💡 解题思路
WLOG, let the centroid of $\triangle ABC$ be $I = (-1,-1)$ . The centroid of an equilateral triangle is the same as the circumcenter. It follows that the circumcircle must intersect the graph exactly
25
第 25 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
Let the sum of the scores of Isabella's first $6$ tests be $S$ . Because the mean of her first $7$ scores is an integer, then $S + 95 \equiv 0 \text{ (mod 7)} \Rightarrow S \equiv3 \text{ (mod 7)}$ .