2021 AMC 10A — Official Competition Problems (March 2021)
📅 2021 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
📋 答题说明
共 25 道题，每题从 A、B、C、D、E 五个选项中选一个答案，点击选项即可选择
答题过程中可随时更改选项，选完后点击底部「提交答案」统一批改
提交后显示对错、正确答案和简短解题思路
点击题目右侧 ⭐ 可收藏难题，方便后续复习
题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）
1
第 1 题
综合
What is the value of \[(2^2-2)-(3^2-3)+(4^2-4)\]
💡 解题思路
$(4-2)-(9-3)+(16-4)=2-6+12=8.$ This corresponds to $\boxed{\textbf{(D) } 8}.$
2
第 2 题
行程问题
Portia's high school has 3 times as many students as Lara's high school. The two high schools have a total of 2600 students. How many students does Portia's high school have?
💡 解题思路
The following system of equations can be formed with $P$ representing the number of students in Portia's high school and $L$ representing the number of students in Lara's high school: \begin{align*} P
3
第 3 题
数论
The sum of two natural numbers is 17{,}402 . One of the two numbers is divisible by 10 . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
💡 解题思路
The units digit of a multiple of $10$ will always be $0$ . We add a $0$ whenever we multiply by $10$ . So, removing the units digit is equal to dividing by $10$ .
4
第 4 题
行程问题
A cart rolls down a hill, travelling 5 inches the first second and accelerating so that during each successive 1 -second time interval, it travels 7 inches more than during the previous 1 -second interval. The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel?
💡 解题思路
Since \[\mathrm{Distance}=\mathrm{Speed}\cdot\mathrm{Time},\] we seek the sum \[5\cdot1+12\cdot1+19\cdot1+26\cdot1+\cdots=5+12+19+26+\cdots,\] in which there are $30$ terms.
5
第 5 题
统计
The quiz scores of a class with k > 12 students have a mean of 8 . The mean of a collection of 12 of these quiz scores is 14 . What is the mean of the remaining quiz scores in terms of k ?
💡 解题思路
The total score of the class is $8k,$ and the total score of the $12$ quizzes is $12\cdot14=168.$ Therefore, for the remaining quizzes ( $k-12$ of them), the total score is $8k-168.$ Their mean score
6
第 6 题
统计
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at 4 miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to 2 miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at 3 miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
💡 解题思路
Let $2d$ miles be the distance from the trailhead to the fire tower, where $d>0.$ When Chantal meets Jean, the two have traveled for \[\frac d4 + \frac d2 + \frac d3 = d\left(\frac 14 + \frac 12 + \fr
7
第 7 题
综合
Tom has a collection of 13 snakes, 4 of which are purple and 5 of which are happy. He observes that Which of these conclusions can be drawn about Tom's snakes? (A) Purple snakes can add. (B) Purple snakes are happy. (C) Snakes that can add are purple. (D) Happy snakes are not purple. (E) Happy snakes can't subtract.
💡 解题思路
We are given that \begin{align*} \text{happy}&\Longrightarrow\text{can add}, &(1) \\ \text{purple}&\Longrightarrow\text{cannot subtract}, \hspace{15mm} &(2) \\ \text{cannot subtract}&\Longrightarrow\t
8
第 8 题
分数与比例
When a student multiplied the number 66 by the repeating decimal, \[\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},\] where a and b are digits, he did not notice the notation and just multiplied 66 times \underline{1}.\underline{a} \ \underline{b}. Later he found that his answer is 0.5 less than the correct answer. What is the 2 -digit number \underline{a} \ \underline{b}?
💡 解题思路
We are given that $66\Bigl(\underline{1}.\overline{\underline{a} \ \underline{b}}\Bigr)-0.5=66\Bigl(\underline{1}.\underline{a} \ \underline{b}\Bigr),$ from which \begin{align*} 66\Bigl(\underline{1}.
9
第 9 题
综合
What is the least possible value of (xy-1)^2+(x+y)^2 for real numbers x and y ?
💡 解题思路
Expanding, we get that the expression is $x^2+2xy+y^2+x^2y^2-2xy+1$ or $x^2+y^2+x^2y^2+1$ . By the Trivial Inequality (all squares are nonnegative) the minimum value for this is $\boxed{\textbf{(D)} ~
10
第 10 题
综合
💡 解题思路
By multiplying the entire equation by $3-2=1$ , all the terms will simplify by difference of squares, and the final answer is $\boxed{\textbf{(C)} ~3^{128}-2^{128}}$ .
11
第 11 题
数论
For which of the following integers b is the base- b number 2021_b - 221_b not divisible by 3 ?
💡 解题思路
We have \begin{align*} 2021_b - 221_b &= (2021_b - 21_b) - (221_b - 21_b) \\ &= 2000_b - 200_b \\ &= 2b^3 - 2b^2 \\ &= 2b^2(b-1), \end{align*} which is divisible by $3$ unless $b\equiv2\pmod{3}.$ The
12
第 12 题
分数与比例
Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are 3 cm and 6 cm. Into each cone is dropped a spherical marble of radius 1 cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone? [图]
💡 解题思路
Initial Scenario
13
第 13 题
立体几何
What is the volume of tetrahedron ABCD with edge lengths AB = 2 , AC = 3 , AD = 4 , BC = √(13) , BD = 2√(5) , and CD = 5 ?
💡 解题思路
Drawing the tetrahedron out and testing side lengths, we realize that the $\triangle ACD, \triangle ABC,$ and $\triangle ABD$ are right triangles by the Converse of the Pythagorean Theorem. It is now
14
第 14 题
整数运算
All the roots of the polynomial z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16 are positive integers, possibly repeated. What is the value of B ?
💡 解题思路
By Vieta's formulas, the sum of the six roots is $10$ and the product of the six roots is $16$ . By inspection, we see the roots are $1, 1, 2, 2, 2,$ and $2$ , so the function is $(z-1)^2(z-2)^4=(z^2-
15
第 15 题
计数
Values for A,B,C, and D are to be selected from \{1, 2, 3, 4, 5, 6\} without replacement (i.e. no two letters have the same value). How many ways are there to make such choices so that the two curves y=Ax^2+B and y=Cx^2+D intersect? (The order in which the curves are listed does not matter; for example, the choices A=3, B=2, C=4, D=1 is considered the same as the choices A=4, B=1, C=3, D=2. )
💡 解题思路
Visualizing the two curves, we realize they are both parabolas with the same axis of symmetry. WLOG the first equation is above the second, since order doesn't matter. Then $C>A$ and $B>D$ . Therefore
16
第 16 题
统计
In the following list of numbers, the integer n appears n times in the list for 1 ≤ n ≤ 200 . \[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\] What is the median of the numbers in this list?
💡 解题思路
There are $1+2+..+199+200=\frac{(200)(201)}{2}=20100$ numbers in total. Let the median be $k$ . We want to find the median $k$ such that \[\frac{k(k+1)}{2}=20100/2,\] or \[k(k+1)=20100.\] Note that $\
17
第 17 题
几何·面积
Trapezoid ABCD has \overline{AB}\parallel\overline{CD},BC=CD=43 , and \overline{AD}\perp\overline{BD} . Let O be the intersection of the diagonals \overline{AC} and \overline{BD} , and let P be the midpoint of \overline{BD} . Given that OP=11 , the length of AD can be written in the form m√(n) , where m and n are positive integers and n is not divisible by the square of any prime. What is m+n ? [图] ~MRENTHUSIASM
💡 解题思路
Angle chasing* reveals that $\triangle BPC\sim\triangle BDA$ , therefore \[2=\frac{BD}{BP}=\frac{AB}{BC}=\frac{AB}{43},\] or $AB=86$ .
18
第 18 题
数论
Let f be a function defined on the set of positive rational numbers with the property that f(a· b)=f(a)+f(b) for all positive rational numbers a and b . Suppose that f also has the property that f(p)=p for every prime number p . For which of the following numbers x is f(x)<0 ?
💡 解题思路
From the answer choices, note that \begin{align*} f(25)&=f\left(\frac{25}{11}\cdot11\right) \\ &=f\left(\frac{25}{11}\right)+f(11) \\ &=f\left(\frac{25}{11}\right)+11. \end{align*} On the other hand,
19
第 19 题
几何·面积
The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\] is m+nπ , where m and n are integers. What is m + n ?
💡 解题思路
In order to attack this problem, we can use casework on the sign of $|x-y|$ and $|x+y|$ .
20
第 20 题
统计
In how many ways can the sequence 1,2,3,4,5 be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
💡 解题思路
By symmetry with respect to $3,$ note that $(x_1,x_2,x_3,x_4,x_5)$ is a valid sequence if and only if $(6-x_1,6-x_2,6-x_3,6-x_4,6-x_5)$ is a valid sequence. We enumerate the valid sequences that start
21
第 21 题
几何·面积
Let ABCDEF be an equiangular hexagon. The lines AB, CD, and EF determine a triangle with area 192√(3) , and the lines BC, DE, and FA determine a triangle with area 324√(3) . The perimeter of hexagon ABCDEF can be expressed as m +n√(p) , where m, n, and p are positive integers and p is not divisible by the square of any prime. What is m + n + p ? [图] ~MRENTHUSIASM
💡 解题思路
Let $P,Q,R,X,Y,$ and $Z$ be the intersections $\overleftrightarrow{AB}\cap\overleftrightarrow{CD},\overleftrightarrow{CD}\cap\overleftrightarrow{EF},\overleftrightarrow{EF}\cap\overleftrightarrow{AB},
22
第 22 题
统计
Hiram's algebra notes are 50 pages long and are printed on 25 sheets of paper; the first sheet contains pages 1 and 2 , the second sheet contains pages 3 and 4 , and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 19 . How many sheets were borrowed?
💡 解题思路
Suppose the roommate took sheets $a$ through $b$ , or equivalently, page numbers $2a-1$ through $2b$ . Because there are $(2b-2a+2)$ numbers taken, \[\frac{(2a-1+2b)(2b-2a+2)}{2}+19(50-(2b-2a+2))=\fra
23
第 23 题
几何·面积
Frieda the frog begins a sequence of hops on a 3 × 3 grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
💡 解题思路
We will use complementary counting. First, the frog can go left with probability $\frac14$ . We observe symmetry, so our final answer will be multiplied by 4 for the 4 directions, and since $4\cdot \f
24
第 24 题
几何·面积
The interior of a quadrilateral is bounded by the graphs of (x+ay)^2 = 4a^2 and (ax-y)^2 = a^2 , where a is a positive real number. What is the area of this region in terms of a , valid for all a > 0 ? Graph in Desmos: https://www.desmos.com/calculator/satawguqsc ~MRENTHUSIASM
💡 解题思路
The cases for $(x+ay)^2 = 4a^2$ are $x+ay = \pm2a,$ or two parallel lines. We rearrange each case and construct the table below: \[\begin{array}{c||c|c|c|c} & & & & \\ [-2.5ex] \textbf{Case} & \textbf
25
第 25 题
几何·面积
How many ways are there to place 3 indistinguishable red chips, 3 indistinguishable blue chips, and 3 indistinguishable green chips in the squares of a 3 × 3 grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
💡 解题思路
Call the different colors A,B,C. There are $3!=6$ ways to rearrange these colors to these three letters, so $6$ must be multiplied after the letters are permuted in the grid. WLOG assume that A is in