2020 AMC 10B — Official Competition Problems (January 2020)
📅 2020 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
综合
What is the value of \[1-(-2)-3-(-4)-5-(-6)?\]
💡 解题思路
We know that when we subtract negative numbers, $a-(-b)=a+b$ .
2
第 2 题
立体几何
Carl has 5 cubes each having side length 1 , and Kate has 5 cubes each having side length 2 . What is the total volume of these 10 cubes?
💡 解题思路
A cube with side length $1$ has volume $1^3=1$ , so $5$ of these will have a total volume of $5\cdot1=5$ .
3
第 3 题
分数与比例
The ratio of w to x is 4:3 , the ratio of y to z is 3:2 , and the ratio of z to x is 1:6 . What is the ratio of w to y?
💡 解题思路
We have \[\frac wy = \frac wx \cdot \frac xz \cdot \frac zy = \frac43\cdot\frac61\cdot\frac23=\frac{16}{3},\] from which $w:y=\boxed{\textbf{(E)}\ 16:3}.$
4
第 4 题
几何·面积
The acute angles of a right triangle are a^{\circ} and b^{\circ} , where a>b and both a and b are prime numbers. What is the least possible value of b ?
💡 解题思路
Since the three angles of a triangle add up to $180^{\circ}$ and one of the angles is $90^{\circ}$ because it's a right triangle, $a^{\circ} + b^{\circ} = 90^{\circ}$ .
5
第 5 题
统计
How many distinguishable arrangements are there of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)
💡 解题思路
Let's first find how many possibilities there would be if they were all distinguishable, then divide out the ones we overcounted.
6
第 6 题
统计
Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome-it reads the same forward and backward. Then 2 hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this 2 -hour period?
💡 解题思路
In order to get the smallest palindrome greater than $15951$ , we need to raise the middle digit. If we were to raise any of the digits after the middle, we would be forced to also raise a digit befor
7
第 7 题
几何·面积
How many positive even multiples of 3 less than 2020 are perfect squares?
💡 解题思路
Any even multiple of $3$ is a multiple of $6$ , so we need to find multiples of $6$ that are perfect squares and less than $2020$ . Any solution that we want will be in the form $(6n)^2$ , where $n$ i
8
第 8 题
几何·面积
Points P and Q lie in a plane with PQ=8 . How many locations for point R in this plane are there such that the triangle with vertices P , Q , and R is a right triangle with area 12 square units?
💡 解题思路
Let the brackets denote areas. We are given that \[[PQR]=\frac12\cdot PQ\cdot h_R=12.\] Since $PQ=8,$ it follows that $h_R=3.$
9
第 9 题
方程
How many ordered pairs of integers (x, y) satisfy the equation \[x^{2020}+y^2=2y?\]
💡 解题思路
Rearranging the terms and and completing the square for $y$ yields the result $x^{2020}+(y-1)^2=1$ . Then, notice that $x$ can only be $0$ , $1$ and $-1$ because any value of $x^{2020}$ that is greate
10
第 10 题
几何·面积
A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? [图]
💡 解题思路
Notice that when the cone is created, the 2 shown radii when merged will become the slant height of the cone and the intact circumference of the circle will become the circumference of the base of the
11
第 11 题
概率
Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select?
💡 解题思路
We don't care about which books Harold selects. We just care that Betty picks $2$ books from Harold's list and $3$ that aren't on Harold's list.
12
第 12 题
分数与比例
The decimal representation of \[\dfrac{1}{20^{20}}\] consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
Andy the Ant lives on a coordinate plane and is currently at (-20, 20) facing east (that is, in the positive x -direction). Andy moves 1 unit and then turns 90^{\circ} left. From there, Andy moves 2 units (north) and then turns 90^{\circ} left. He then moves 3 units (west) and again turns 90^{\circ} left. Andy continues his progress, increasing his distance each time by 1 unit and always turning left. What is the location of the point at which Andy makes the 2020 th left turn?
💡 解题思路
Andy makes a total of $2020$ moves: $1010$ horizontal ( $505$ left and $505$ right) and $1010$ vertical ( $505$ up and $505$ down).
14
第 14 题
几何·面积
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles?
💡 解题思路
[asy] real x=sqrt(3); real y=2sqrt(3); real z=3.5; real a=x/2; real b=0.5; real c=3a; pair A, B, C, D, E, F; A = (1,0); B = (3,0); C = (4,x); D = (3,y); E = (1,y); F = (0,x); fill(A--B--C--D--E--F--A-
15
第 15 题
规律与数列
Steve wrote the digits 1 , 2 , 3 , 4 , and 5 in order repeatedly from left to right, forming a list of 10,000 digits, beginning 123451234512\ldots. He then erased every third digit from his list (that is, the 3 rd, 6 th, 9 th, \ldots digits from the left), then erased every fourth digit from the resulting list (that is, the 4 th, 8 th, 12 th, \ldots digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions 2019, 2020, 2021 ?
💡 解题思路
Note that cycles exist initially and after each round of erasing.
16
第 16 题
计数
Bela and Jenn play the following game on the closed interval [0, n] of the real number line, where n is a fixed integer greater than 4 . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval [0, n] . Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
💡 解题思路
We use game theory.
17
第 17 题
几何·面积
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
💡 解题思路
Consider the $10$ people to be standing in a circle, where two people opposite of each other form a diameter of the circle.
18
第 18 题
分数与比例
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
💡 解题思路
Suppose that we have a deck, currently containing just one black card. We then insert $n$ red cards one-by-one into the deck at random positions. It is easy to see using induction, that the black card
19
第 19 题
数字运算
In a certain card game, a player is dealt a hand of 10 cards from a deck of 52 distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as 158A00A4AA0 . What is the digit A ?
Let B be a right rectangular prism (box) with edges lengths 1,3, and 4 , together with its interior. For real r≥0 , let S(r) be the set of points in 3 -dimensional space that lie within a distance r of some point in B . The volume of S(r) can be expressed as ar^{3} + br^{2} + cr +d , where a,b,c, and d are positive real numbers. What is \frac{bc}{ad}?
💡 解题思路
Split $S(r)$ into 4 regions:
21
第 21 题
几何·面积
In square ABCD , points E and H lie on \overline{AB} and \overline{DA} , respectively, so that AE=AH. Points F and G lie on \overline{BC} and \overline{CD} , respectively, and points I and J lie on \overline{EH} so that \overline{FI} \perp \overline{EH} and \overline{GJ} \perp \overline{EH} . See the figure below. Triangle AEH , quadrilateral BFIE , quadrilateral DHJG , and pentagon FCGJI each has area 1. What is FI^2 ? [图]
💡 解题思路
Since the total area is $4$ , the side length of square $ABCD$ is $2$ . We see that since triangle $HAE$ is a right isosceles triangle with area 1, we can determine sides $HA$ and $AE$ both to be $\sq
22
第 22 题
数论
What is the remainder when 2^{202} +202 is divided by 2^{101}+2^{51}+1 ?
💡 解题思路
Completing the square, then difference of squares:
23
第 23 题
几何·面积
Square ABCD in the coordinate plane has vertices at the points A(1,1), B(-1,1), C(-1,-1), and D(1,-1). Consider the following four transformations: \bullet L, a rotation of 90^{\circ} counterclockwise around the origin; \bullet R, a rotation of 90^{\circ} clockwise around the origin; \bullet H, a reflection across the x -axis; and \bullet V, a reflection across the y -axis. Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying R and then V would send the vertex A at (1,1) to (-1,-1) and would send the vertex B at (-1,1) to itself. How many sequences of 20 transformations chosen from \{L, R, H, V\} will send all of the labeled vertices back to their original positions? (For example, R, R, V, H is one sequence of 4 transformations that will send the vertices back to their original positions.)
💡 解题思路
For each transformation:
24
第 24 题
整数运算
How many positive integers n satisfy \[\dfrac{n+1000}{70} = \lfloor √(n) \rfloor?\] (Recall that \lfloor x\rfloor is the greatest integer not exceeding x .)
💡 解题思路
We can first consider the equation without a floor function:
25
第 25 题
数论
Let D(n) denote the number of ways of writing the positive integer n as a product \[n = f_1· f_2·s f_k,\] where k\ge1 , the f_i are integers strictly greater than 1 , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as 6 , 2· 3 , and 3·2 , so D(6) = 3 . What is D(96) ?
💡 解题思路
Note that $96 = 2^5 \cdot 3$ . Since there are at most six not necessarily distinct factors $>1$ multiplying to $96$ , we have six cases: $k=1, 2, ..., 6.$ Now we look at each of the six cases.