2021B 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
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
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
3
第 3 题
综合
Suppose \[2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.\] What is the value of x?
💡 解题思路
Subtracting $2$ from both sides and taking reciprocals gives $1+\frac{1}{2+\frac{2}{3+x}}=\frac{53}{38}$ . Subtracting $1$ from both sides and taking reciprocals again gives $2+\frac{2}{3+x}=\frac{38}
4
第 4 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
5
第 5 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 $(-3,6)$ .
6
第 6 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
7
第 7 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 $1:(2+4+8)=\boxed
8
第 8 题
几何·面积
Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38,38, and 34 . What is the distance between two adjacent parallel lines?
💡 解题思路
[asy] size(8cm); pair O = (0, 0), A = (0, 3), B = (0, 9), R = (19, 3), L = (17, 9); draw(O--A--B); draw(O--R); draw(O--L); label("$A$", A, NE); label("$B$", B, N); label("$R$", R, NE); label("$L$", L,
Two distinct numbers are selected from the set \{1,2,3,4,\dots,36,37\} so that the sum of the remaining 35 numbers is the product of these two numbers. What is the difference of these two numbers?
💡 解题思路
The sum of the first $n$ integers is given by $\frac{n(n+1)}{2}$ , so $\frac{37(37+1)}{2}=703$ .
11
第 11 题
几何·面积
Triangle ABC has AB=13,BC=14 and AC=15 . Let P be the point on \overline{AC} such that PC=10 . There are exactly two points D and E on line BP such that quadrilaterals ABCD and ABCE are trapezoids. What is the distance DE? [图] ~Brendanb4321
💡 解题思路
Toss on the Cartesian plane with $A=(5, 12), B=(0, 0),$ and $C=(14, 0)$ . Then $\overline{AD}\parallel\overline{BC}, \overline{AB}\parallel\overline{CE}$ by the trapezoid condition, where $D, E\in\ove
12
第 12 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 \( \frac{B+S(n)}{n+1} = 40 \).
13
第 13 题
综合
💡 解题思路
We rearrange to get \[5\cos3\theta = 3\sin\theta-1.\] We can graph two functions in this case: $y=5\cos{3x}$ and $y=3\sin{x} -1$ . Using transformation of functions, we know that $5\cos{3x}$ is just a
14
第 14 题
几何·面积
Let ABCD be a rectangle and let \overline{DM} be a segment perpendicular to the plane of ABCD . Suppose that \overline{DM} has integer length, and the lengths of \overline{MA},\overline{MC}, and \overline{MB} are consecutive odd positive integers (in this order). What is the volume of pyramid MABCD?
💡 解题思路
Let $MA=a$ and $MD=d.$ It follows that $MC=a+2$ and $MB=a+4.$
15
第 15 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
16
第 16 题
规律与数列
Let g(x) be a polynomial with leading coefficient 1, whose three roots are the reciprocals of the three roots of f(x)=x^3+ax^2+bx+c, where 1<a<b<c. What is g(1) in terms of a,b, and c?
💡 解题思路
Note that $f(1/x)$ has the same roots as $g(x)$ , if it is multiplied by some monomial so that the leading term is $x^3$ they will be equal. We have \[f(1/x) = \frac{1}{x^3} + \frac{a}{x^2}+\frac{b}{x
17
第 17 题
几何·面积
Let ABCD be an isosceles trapezoid having parallel bases \overline{AB} and \overline{CD} with AB>CD. Line segments from a point inside ABCD to the vertices divide the trapezoid into four triangles whose areas are 2, 3, 4, and 5 starting with the triangle with base \overline{CD} and moving clockwise as shown in the diagram below. What is the ratio \frac{AB}{CD}? [图]
💡 解题思路
Without the loss of generality, let $\mathcal T$ have vertices $A$ , $B$ , $C$ , and $D$ , with $AB = r$ and $CD = s$ . Also denote by $P$ the point in the interior of $\mathcal T$ .
18
第 18 题
综合
Let z be a complex number satisfying 12|z|^2=2|z+2|^2+|z^2+1|^2+31. What is the value of z+\frac 6z?
💡 解题思路
Using the fact $z\bar{z}=|z|^2$ , the equation rewrites itself as \begin{align*} 12z\bar{z}&=2(z+2)(\bar{z}+2)+(z^2+1)(\bar{z}^2+1)+31 \\ -12z\bar{z}+2z\bar{z}+4(z+\bar{z})+8+z^2\bar{z}^2+(z^2+\bar{z}
19
第 19 题
概率
Two fair dice, each with at least 6 faces are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is \frac34 of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is \frac{1}{12} . What is the least possible number of faces on the two dice combined?
💡 解题思路
Suppose the dice have $a$ and $b$ faces, and WLOG $a\geq{b}$ . Since each die has at least $6$ faces, there will always be $6$ ways to sum to $7$ . As a result, there must be $\tfrac{4}{3}\cdot6=8$ wa
20
第 20 题
几何·角度
Let Q(z) and R(z) be the unique polynomials such that \[z^{2021}+1=(z^2+z+1)Q(z)+R(z)\] and the degree of R is less than 2. What is R(z)?
💡 解题思路
Let $z=s$ be a root of $z^2+z+1$ so that $s^2+s+1=0.$ It follows that \[(s-1)\left(s^2+s+1\right)=s^3-1=0,\] from which $s^3=1,$ but $s\neq1.$
21
第 21 题
规律与数列
Let S be the sum of all positive real numbers x for which \[x^{2^{\sqrt2}}=\sqrt2^{2^x}.\] Which of the following statements is true?
💡 解题思路
Note that \begin{align*} x^{2^{\sqrt{2}}} &= {\sqrt{2}}^{2^x} \\ 2^{\sqrt{2}} \log_2 x &= 2^{x} \log_2 \sqrt{2}. \end{align*} (At this point we see by inspection that $x=\sqrt{2}$ is a solution.)
22
第 22 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
23
第 23 题
数论
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin i is 2^{-i} for i=1,2,3,.... More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is \frac pq, where p and q are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins 3,17, and 10. ) What is p+q?
💡 解题思路
"Evenly spaced" just means the bins form an arithmetic sequence.
24
第 24 题
几何·面积
Let ABCD be a parallelogram with area 15 . Points P and Q are the projections of A and C, respectively, onto the line BD; and points R and S are the projections of B and D, respectively, onto the line AC. See the figure, which also shows the relative locations of these points. [图] Suppose PQ=6 and RS=8, and let d denote the length of \overline{BD}, the longer diagonal of ABCD. Then d^2 can be written in the form m+n\sqrt 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?
💡 解题思路
Let $X$ denote the intersection point of the diagonals $AC$ and $BD$ . Remark that by symmetry $X$ is the midpoint of both $\overline{PQ}$ and $\overline{RS}$ , so $XP = XQ = 3$ and $XR = XS = 4$ . No
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 $\lfloor mx \rfloor.$