2021 AMC 10B — Official Competition Problems (March 2021)
📅 2021 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
整数运算
How many integer values of x satisfy |x|<3π ?
💡 解题思路
Since $3\pi\approx9.42$ , we multiply $9$ by $2$ for the integers from $1$ to $9$ and the integers from $-1$ to $-9$ and add $1$ to account for $0$ to get $\boxed{\textbf{(D)} ~19}$ .
2
第 2 题
综合
What is the value of √((3-2\sqrt{3))^2}+√((3+2\sqrt{3))^2} ?
💡 解题思路
Note that the square root of any number squared is always the absolute value of the squared number because the square root function will only return a nonnegative number. By squaring both $3$ and $2\s
3
第 3 题
综合
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the 28 students in the program, 25\% of the juniors as a class and 10\% of the seniors as a class are on the debate team. How many juniors are in the program?
💡 解题思路
Say there are $j$ juniors and $s$ seniors in the program. Converting percentages to fractions, $\frac{j}{4}$ and $\frac{s}{10}$ are on the debate team, and since an equal number of juniors and seniors
4
第 4 题
综合
At a math contest, 57 students are wearing blue shirts, and another 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
💡 解题思路
There are $46$ blue students paired with a blue partner. The other $11$ students wearing blue shirts must each be paired with a partner wearing a shirt of the opposite color. There are $64$ yellow stu
5
第 5 题
规律与数列
The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give 24 , while the other two multiply to 30 . What is the sum of the ages of Jonie's four cousins?
💡 解题思路
First look at the two cousins' ages that multiply to $24$ . Since the ages must be single-digit, the ages must either be $3 \text{ and } 8$ or $4 \text{ and } 6.$
6
第 6 题
分数与比例
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is 84 , and the afternoon class's mean score is 70 . The ratio of the number of students in the morning class to the number of students in the afternoon class is \frac{3}{4} . What is the mean of the scores of all the students?
💡 解题思路
Let there be $3x$ students in the morning class and $4x$ students in the afternoon class. The total number of students is $3x + 4x = 7x$ . The average is $\frac{3x\cdot84 + 4x\cdot70}{7x}=76$ . Theref
7
第 7 题
几何·面积
In a plane, four circles with radii 1,3,5, and 7 are tangent to line \ell at the same point A, but they may be on either side of \ell . Region S consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region S ?
💡 解题思路
Suppose that line $\ell$ is horizontal, and each circle lies either north or south to $\ell.$ We construct the circles one by one:
8
第 8 题
几何·面积
Mr. Zhou places all the integers from 1 to 225 into a 15 by 15 grid. He places 1 in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top? [图]
💡 解题思路
In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are $D$ and $E,$ respectively. Note that the numbers
9
第 9 题
计数
The point P(a,b) in the xy -plane is first rotated counterclockwise by 90^\circ around the point (1,5) and then reflected about the line y = -x . The image of P after these two transformations is at (-6,3) . What is b - a ?
💡 解题思路
The final image of $P$ is $(-6,3)$ . We know the reflection rule for reflecting over $y=-x$ is $(x,y) \rightarrow (-y, -x)$ . So before the reflection and after rotation the point is $(-3,6)$ .
10
第 10 题
行程问题
An inverted cone with base radius 12 cm and height 18 cm is full of water. The water is poured into a tall cylinder whose horizontal base has radius of 24 cm . What is the height in centimeters of the water in the cylinder?
💡 解题思路
The volume of a cone is $\frac{1}{3} \cdot\pi \cdot r^2 \cdot h$ where $r$ is the base radius and $h$ is the height. The water completely fills up the cone so the volume of the water is $\frac{1}{3}\c
11
第 11 题
几何·面积
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
💡 解题思路
Let the side lengths of the rectangular pan be $m$ and $n$ . It follows that $(m-2)(n-2) = \frac{mn}{2}$ , since half of the brownie pieces are in the interior. This gives $2(m-2)(n-2) = mn \iff mn -
12
第 12 题
分数与比例
Let N = 34 · 34 · 63 · 270 . What is the ratio of the sum of the odd divisors of N to the sum of the even divisors of N ?
💡 解题思路
Prime factorize $N$ to get $N=2^{3} \cdot 3^{5} \cdot 5\cdot 7\cdot 17^{2}$ . For each odd divisor $n$ of $N$ , there exist even divisors $2n, 4n, 8n$ of $N$ , therefore the ratio is $1:(2+4+8)=\boxed
13
第 13 题
数字运算
Let n be a positive integer and d be a digit such that the value of the numeral \underline{32d} in base n equals 263 , and the value of the numeral \underline{324} in base n equals the value of the numeral \underline{11d1} in base six. What is n + d ?
💡 解题思路
We can start by setting up an equation to convert $\underline{32d}$ base $n$ to base 10. To convert this to base 10, it would be $3{n}^2+2n+d.$ Because it is equal to 263, we can set this equation to
14
第 14 题
几何·面积
Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38,38, and 34 . What is the distance between two adjacent parallel lines?
💡 解题思路
[asy] size(8cm); pair O = (0, 0), A = (0, 3), B = (0, 9), R = (19, 3), L = (17, 9); draw(O--A--B); draw(O--R); draw(O--L); label("$A$", A, NE); label("$B$", B, N); label("$R$", R, NE); label("$L$", L,
15
第 15 题
方程
The real number x satisfies the equation x+\frac{1}{x} = √(5) . What is the value of x^{11}-7x^{7}+x^3?
💡 解题思路
We square $x+\frac{1}{x}=\sqrt5$ to get $x^2+2+\frac{1}{x^2}=5$ . We subtract 2 on both sides for $x^2+\frac{1}{x^2}=3$ and square again, and see that $x^4+2+\frac{1}{x^4}=9$ so $x^4+\frac{1}{x^4}=7$
16
第 16 题
数论
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, 1357, 89, and 5 are all uphill integers, but 32, 1240, and 466 are not. How many uphill integers are divisible by 15 ?
💡 解题思路
The divisibility rule of $15$ is that the number must be congruent to $0$ mod $3$ and congruent to $0$ mod $5$ . Being divisible by $5$ means that it must end with a $5$ or a $0$ . We can rule out the
17
第 17 题
规律与数列
Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given 2 cards out of a set of 10 cards numbered 1,2,3, \dots,10. The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon-- 11, Oscar-- 4, Aditi-- 7, Tyrone-- 16, Kim-- 17. Which of the following statements is true? (A) Ravon was given card 3.(B) Aditi was given card 3.(C) Ravon was given card 4.(D) Aditi was given card 4.(E) Tyrone was given card 7.
💡 解题思路
By logical deduction, we consider the scores from lowest to highest: \begin{align*} \text{Oscar's score is 4.} &\implies \text{Oscar is given cards 1 and 3.} \\ &\implies \text{Aditi is given cards 2
18
第 18 题
概率
A fair 6 -sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
💡 解题思路
Since 3 out of 6 of the numbers are even, there is a $\frac36$ chance that the first number we choose is even.
19
第 19 题
统计
Suppose that S is a finite set of positive integers. If the greatest integer in S is removed from S , then the average value (arithmetic mean) of the integers remaining is 32 . If the least integer in S is also removed, then the average value of the integers remaining is 35 . If the greatest integer is then returned to the set, the average value of the integers rises to 40 . The greatest integer in the original set S is 72 greater than the least integer in S . What is the average value of all the integers in the set S ?
💡 解题思路
We can then say that \( \frac{A+S(n)}{n+1} = 32 \), \( \frac{S(n)}{n} = 35 \), and \( \frac{B+S(n)}{n+1} = 40 \).
20
第 20 题
几何·面积
The figure is constructed from 11 line segments, each of which has length 2 . The area of pentagon ABCDE can be written as √(m) + √(n) , where m and n are positive integers. What is m + n ? [图]
💡 解题思路
[asy] /* Made by samrocksnature, adapted by Tucker, then adjusted by samrocksnature again, then adjusted by erics118 xD*/ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6
21
第 21 题
几何·面积
A square piece of paper has side length 1 and vertices A,B,C, and D in that order. As shown in the figure, the paper is folded so that vertex C meets edge \overline{AD} at point C' , and edge \overline{BC} intersects edge \overline{AB} at point E . Suppose that C'D = \frac{1}{3} . What is the perimeter of triangle \bigtriangleup AEC' ?
💡 解题思路
We can set the point on $CD$ where the fold occurs as point $F$ . Then, we can set $FD$ as $x$ , and $CF$ as $1-x$ because of symmetry due to the fold. It can be recognized that this is a right triang
22
第 22 题
数论
Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green; and there are 5 empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives 3 blocks all of the same color is \frac{m}{n} , where m and n are relatively prime positive integers. What is m + n ?
💡 解题思路
Let our denominator be $(5!)^3$ , so we consider all possible distributions.
23
第 23 题
概率
Note that the center of the coin can lie anywhere inside a green region, as shown below. [图] ~MRENTHUSIASM
💡 解题思路
To find the probability, we look at the $\frac{\text{success region}}{\text{total possible region}}$ . For the coin to be completely contained within the square, we must have the distance from the cen
24
第 24 题
分数与比例
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes 4 and 2 can be changed into any of the following by one move: (3,2),(2,1,2),(4),(4,1),(2,2), or (1,1,2). [图] Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?
💡 解题思路
We say that a game state is an N-position if it is winning for the next player (the player to move), and a P-position if it is winning for the other player. We are trying to find which of the given st
25
第 25 题
坐标几何
Let S be the set of lattice points in the coordinate plane, both of whose coordinates are integers between 1 and 30, inclusive. Exactly 300 points in S lie on or below a line with equation y=mx. The possible values of m lie in an interval of length \frac ab, where a and b are relatively prime positive integers. What is a+b?
💡 解题思路
First, we find a numerical representation for the number of lattice points in $S$ that are under the line $y=mx.$ For any value of $x,$ the highest lattice point under $y=mx$ is $\lfloor mx \rfloor.$