2020 AMC amc10b 真题 答案 详解

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中英双语真题

2020 AMC10B 2020 AMC10B Problem 1 What is the value of 1-(-2)-3-(-4)-5-(-6)? 下列算式的值为多少 1-(-2)-3-(-4)-5-(-6)? (A) -20 (B) -3 (C) 3 (D) 5 (E) 21 Problem 2 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? Carl有5个边长为1的立方体，Kate有5个边长为2的立方体。这10个立方体的总体积是多少？ (A) 24 (B) 25 (C) 28 (D) 40 (E) 45 Problem 3 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? w比x的比值是4:3，y比z的比值是3:2，z比x的比值是1:6。则w比y是多少？ (A) 4:3 (B) 3:2 (C) 8:3 (D) 4:1 (E) 16:3 Problem 4 The acute angles of a right triangle are a° and b°, where a > b and both a and b are prime numbers. What is the least possible value of b? 一个直角三角形的两个锐角的度数分别为a°和b°，其中a>b，且a和b均为质数。则b的最小值是多少？ (A) 2 (B) 3 (C) 5 (D) 7 (E) 11

英文真题

2020 AMC 10B Problems Problem 1 What is the value of $1-(-2)-3-(-4)-5-(-6)?$ (A) -20 (B) -3 (C) 3 (D) 5 (E) 21 Problem 2 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? (A) 24 (B) 25 (C) 28 (D) 40 (E) 45 Problem 3 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? (A) 4:3 (B) 3:2 (C) 8:3 (D) 4:1 (E) 16:3 Problem 4 The acute angles of a right triangle are a° and b°, where a > b and both a and b are prime numbers. What is the least possible value of b? (A) 2 (B) 3 (C) 5 (D) 7 (E) 11 Problem 5 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.) (A) 210 (B) 420 (C) 630 (D) 840 (E) 1050 Problem 6 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? (A) 50 (B) 55 (C) 60 (D) 65 (E) 70 Problem 7 How many positive even multiples of 3 less than 2020 are perfect squares? (A) 7 (B) 8 (C) 9 (D) 10 (E) 12 Problem 8 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? (A) 2 (B) 4 (C) 6 (D) 8 (E) 12 Problem 9 How many ordered pairs of integers (x, y) satisfy the equation $$x^{2020}+y^2=2y$$ ? (A) 1 (B) 2 (C) 3 (D) 4 (E) infinitely many

真题文字详解(英文)

Problem 1 What is the value of $$1-(-2)-3-(-4)-5-(-6)?$$
(A) -20 (B) -3 (C) 3 (D) 5 (E) 21

Solution 1 We know that when we subtract negative numbers, $a-(-b)=a+b$. The equation becomes $1+2-3+4-5+6=\boxed{\text{(D)}\ 5}.$

Solution 2 Like Solution 1, we know that when we subtract $a-(-b)$, that will equal $a+b$ as the opposite/negative of a negative is a positive. Thus, $1-(-2)-3-(-4)-5-(-6)=1+2-3+4-5+6$. We can group together a few terms to make our computation a bit simpler. $1+(2-3)+4+(-5+6)=1+(-1)+4+1=\boxed{\text{(D)}\ 5}$.

Solution 3 Notice $1-(-2)-3-(-4)-5-(-6)$ has three groups: $1-(-2)$, $-3-(-4)$, $-5-(-6)$. The first group, $1-(-2)$, can be expressed as $a-\left[-(a+1)\right]$. Simplify, $a-\left[-(a+1)\right]$ is $2a+1$. The second and third groups can be expressed as $-c-\left[-(c+1)\right]$. Simplify, $-c-\left[-(c+1)\right]$ is $1$. Thus, the sum of the three groups is $2\cdot1+1+1+1=5$ or $\boxed{\text{(D)}\ 5}$.

Problem 2 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?
(A) 24 (B) 25 (C) 28 (D) 40 (E) 45

Solution 1 A cube with side length $1$ has volume $1^3=1$, so $5$ of these will have a total volume of $5\cdot1=5$. A cube with side length $2$ has volume $2^3=8$, so $5$ of these will have a total volume of $5\cdot8=40$. $5+40=\boxed{\text{(E)}\ 45}$.

Solution 2 (more in depth about Solution 1) The total volume of Carl's cubes is $5$. This is because to find the volume of a cube or rectangular prism, you have to multiply the height by the length by the width. So in this question, it would be $1\times1\times1$. This is equal to $1$. Since Carl has $5$ cubes, you will have to multiply $1$ by $5$, to account for all the $5$ cubes. Next, to find the total volume of Kate's cubes you have to do the same thing. Except this time, the height, the width, and the length are all $2$, so it will be $2\times2\times2=8$. Now you have to multiply by $5$ to account for all the $5$ blocks. This is $40$. So the total volume of Kate's cubes is $40$. Lastly, to find the total of Carl's and Kate's cubes, you must add the total volume of their cubes together. This is going to be $5+40=\boxed{\text{(E)}\ 45}$.

Problem 3 The following problem is from both the 2020 AMC 10B #3 and 2020 AMC 12B #3, so both problems redirect to this page. 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 $y$?
(A) $4:3$ (B) $3:2$ (C) $8:3$ (D) $4:1$ (E) $16:3$

Solution 1 (One Sentence) We have $$\frac{w}{y} = \frac{w}{x}\cdot\frac{x}{z}\cdot\frac{z}{y} = \frac{4}{3}\cdot\frac{6}{1}\cdot\frac{2}{3} = \frac{16}{3},$$ from which $w:y=\boxed{\text{(E)}\ 16:3}$.

Solution

真题文字详解(中英双语)

Problem 1 What is the value of 1 - (-2) - 3 - (-4) - 5 - (-6)? (A) -20 (B) -3 (C) 3 (D) 5 (E) 21 Solution 1 We know that when we subtract negative numbers, a - (-b) = a + b The equation becomes 1 + 2 - 3 + 4 - 5 + 6 = (D) 5 Solution 2 Like Solution 1, we know that when we subtract a - (-b), that will equal a + b as the opposite/negative of a negative is a positive. Thus, 1 - (-2) - 3 - (-4) - 5 - (-6) = 1 + 2 - 3 + 4 - 5 + 6 We can group together a few terms to make our computation a bit simpler. 1 + (2 - 3) + 4 + (-5 + 6) = 1 + (-1) + 4 + 1 = (D) 5 Solution 3 Notice 1 - (-2) - 3 - (-4) - 5 - (-6) has three groups: 1 - (-2), -3 - (-4), -5 - (-6) The first group, 1 - (-2), can be expressed as a - [-(a + 1)]. Simplify, a - [-(a + 1)] is 2a + 1 The second and third groups can be expressed as -c - [-(c + 1)]. Simplify, -c - [-(c + 1)] is 1 Thus, the sum of the three groups is 2 · 1 + 1 + 1 + 1 = 5 or (D) 5 Problem 2 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? (A) 24 (B) 25 (C) 28 (D) 40 (E) 45 Solution 1 A cube with side length 1 has volume 1^3 = 1, so 5 of these will have a total volume of 5 · 1 = 5. A cube with side length 2 has volume 2^3 = 8, so 5 of these will have a total volume of 5 · 8 = 40. 5 + 40 = (E) 45 Solution 2 (more in depth about Solution 1) The total volume of Carl's cubes is 5. This is because to find the volume of a cube or a rectangular prism, you have to multiply the height by the length by the width. So in this question, it would be 1 × 1 × 1. This is equal to 1. Since Carl has 5 cubes, you will have to multiply 1 by 5, to account for all the 5 cubes. Next, to find the total volume of Kate's cubes you have to do the same thing. Except, this time, the height, the width, and the length, are all 2, so it will be 2 × 2 × 2 = 8. Now you have to multiply by 5 to account for all the 5 blocks. This is 40. So the total volume of Kate's cubes is 40. Lastly, to find the total of Carl's and Kate's cubes, you must add the total volume of their cubes together. This is going to be 5 + 40 = (E) 45 Problem 3 The following problem is from both the 2020 AMC 10B #3 and 2020 AMC 12B #3, so both problems redirect to this page. 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? (A) 4 : 3 (B) 3 : 2 (C) 8 : 3 (D) 4 : 1 (E) 16 : 3 Solution 1 (One Sentence) We have w/x = w/x . z/y = 4/3 . 2/3 = 16/3' from which w : y = (E) 16 : 3 Solution 2 We need to somehow link all three of the ratios together. We can start by connecting the last two ratios together by multiplying the last ratio by two.