2021A AMC 12 — Official Competition Problems (February 2024)
📅 2024 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
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1
第 1 题
综合
💡 解题思路
We evaluate the given expression to get that \[2^{1+2+3}-(2^1+2^2+2^3)=2^6-(2^1+2^2+2^3)=64-2-4-8=50 \implies \boxed{\text{(B)}}\]
2
第 2 题
计数
Under what conditions is √(a^2+b^2)=a+b true, where a and b are real numbers? (A) It is never true. (B) It is true if and only if ab=0 . (C) It is true if and only if a+b\ge 0 . (D) It is true if and only if ab=0 and a+b\ge 0 . (E) It is always true.
💡 解题思路
One can square both sides to get $a^{2}+b^{2}=a^{2}+2ab+b^{2}$ . Then, $0=2ab\implies ab=0$ . Also, it is clear that both sides of the equation must be nonnegative. The answer is $\boxed{\textbf{(D)}}
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
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
5
第 5 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}.
6
第 6 题
概率
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is \frac13 . When 4 black cards are added to the deck, the probability of choosing red becomes \frac14 . How many cards were in the deck originally?
💡 解题思路
If the probability of choosing a red card is $\frac{1}{3}$ , the red and black cards are in ratio $1:2$ . This means at the beginning there are $x$ red cards and $2x$ black cards.
7
第 7 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)} ~
8
第 8 题
规律与数列
A sequence of numbers is defined by D_0=0,D_1=0,D_2=1 and D_n=D_{n-1}+D_{n-3} for n\ge 3 . What are the parities (evenness or oddness) of the triple of numbers (D_{2021},D_{2022},D_{2023}) , where E denotes even and O denotes odd?
💡 解题思路
We construct the following table: \[\begin{array}{c||c|c|c|c|c|c|c|c|c|c|c} &&&&&&&&&&& \\ [-2.5ex] \textbf{Term} &\boldsymbol{D_0}&\boldsymbol{D_1}&\boldsymbol{D_2}&\boldsymbol{D_3}&\boldsymbol{D_4}&
9
第 9 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 $\boxed{\textbf{(C)} ~3^{128}-2^{128}}$ .
10
第 10 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
11
第 11 题
行程问题
A laser is placed at the point (3,5) . The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the y -axis, then hit and bounce off the x -axis, then hit the point (7,5) . What is the total distance the beam will travel along this path? [图] ~MRENTHUSIASM
💡 解题思路
Let $A=(3,5)$ and $D=(7,5).$ Suppose that the beam hits and bounces off the $y$ -axis at $B,$ then hits and bounces off the $x$ -axis at $C.$
12
第 12 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-
13
第 13 题
综合
Of the following complex numbers z , which one has the property that z^5 has the greatest real part?
💡 解题思路
First, $\textbf{(B)}$ is $2\text{cis}(150)$ , $\textbf{(C)}$ is $2\text{cis}(135)$ , $\textbf{(D)}$ is $2\text{cis}(120)$ .
14
第 14 题
规律与数列
What is the value of \[(\sum_{k=1}^{20} \log_{5^k} 3^{k^2})·(\sum_{k=1}^{100} \log_{9^k} 25^k)?\]
💡 解题思路
We will apply the following logarithmic identity: \[\log_{p^n}{q^n}=\log_{p}{q},\] which can be proven by the Change of Base Formula: \[\log_{p^n}{q^n}=\frac{\log_{p}{q^n}}{\log_{p}{p^n}}=\frac{n\log_
15
第 15 题
数论
A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of 4 , and the group must have at least one singer. Let N be the number of different groups that could be selected. What is the remainder when N is divided by 100 ?
💡 解题思路
Suppose that $t$ tenors and $b$ basses are selected. The requirements are $t\equiv b\pmod{4}$ and $(t,b)\neq(0,0).$
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
方程
How many solutions does the equation \sin ( \frac{π}2 \cos x)=\cos ( \frac{π}2 \sin x) have in the closed interval [0,π] ?
💡 解题思路
The ranges of $\frac{\pi}2 \sin x$ and $\frac{\pi}2 \cos x$ are both $\left[-\frac{\pi}2, \frac{\pi}2 \right],$ which is included in the range of $\arcsin,$ so we can use it with no issues. \begin{ali
20
第 20 题
规律与数列
Suppose that on a parabola with vertex V and a focus F there exists a point A such that AF=20 and AV=21 . What is the sum of all possible values of the length FV?
💡 解题思路
Let $\ell$ be the directrix of $\mathcal P$ ; recall that $\mathcal P$ is the set of points $T$ such that the distance from $T$ to $\ell$ is equal to $TF$ . Let $P$ and $Q$ be the orthogonal projectio
21
第 21 题
数论
The five solutions to the equation \[(z-1)(z^2+2z+4)(z^2+4z+6)=0\] may be written in the form x_k+y_ki for 1\le k\le 5, where x_k and y_k are real. Let \mathcal E be the unique ellipse that passes through the points (x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4), and (x_5,y_5) . The eccentricity of \mathcal E can be written in the form √(\frac mn) , where m and n are relatively prime positive integers. What is m+n ? (Recall that the eccentricity of an ellipse \mathcal E is the ratio \frac ca , where 2a is the length of the major axis of \mathcal E and 2c is the is the distance between its two foci.)
💡 解题思路
The solutions to this equation are $z = 1$ , $z = -1 \pm i\sqrt 3$ , and $z = -2\pm i\sqrt 2$ . Consider the five points $(1,0)$ , $\left(-1,\pm\sqrt 3\right)$ , and $\left(-2,\pm\sqrt 2\right)$ ; the
22
第 22 题
几何·角度
Suppose that the roots of the polynomial P(x)=x^3+ax^2+bx+c are \cos \frac{2π}7,\cos \frac{4π}7, and \cos \frac{6π}7 , where angles are in radians. What is abc ?
💡 解题思路
Let $z=e^{\frac{2\pi i}{7}}.$ Since $z$ is a $7$ th root of unity, we have $z^7=1.$ For all integers $k,$ note that \[\cos\frac{2k\pi}{7}=\operatorname{Re}\left(z^k\right)=\operatorname{Re}\left(z^{-k
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
Semicircle \Gamma has diameter \overline{AB} of length 14 . Circle \Omega lies tangent to \overline{AB} at a point P and intersects \Gamma at points Q and R . If QR=3\sqrt3 and \angle QPR=60^\circ , then the area of \triangle PQR equals \tfrac{a√(b)}{c} , where a and c are relatively prime positive integers, and b is a positive integer not divisible by the square of any prime. What is a+b+c ? [图] ~MRENTHUSIASM
💡 解题思路
Let $O=\Gamma$ be the center of the semicircle and $X=\Omega$ be the center of the circle.
25
第 25 题
规律与数列
Let d(n) denote the number of positive integers that divide n , including 1 and n . For example, d(1)=1,d(2)=2, and d(12)=6 . (This function is known as the divisor function.) Let \[f(n)=\frac{d(n)}{\sqrt [3]n}.\] There is a unique positive integer N such that f(N)>f(n) for all positive integers n\ne N . What is the sum of the digits of N?
💡 解题思路
We consider the prime factorization of $n:$ \[n=\prod_{i=1}^{k}p_i^{e_i}.\] By the Multiplication Principle, we have \[d(n)=\prod_{i=1}^{k}(e_i+1).\] Now, we rewrite $f(n)$ as \[f(n)=\frac{d(n)}{\sqrt