2012B 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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第 1 题
综合
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?
💡 解题思路
Multiplying $18$ and $2$ by $4$ we get $72$ students and $8$ rabbits. We then subtract: $72 - 8 = \boxed{\textbf{(C)}\ 64}.$
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第 2 题
几何·面积
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? [图]
💡 解题思路
If the radius is $5$ , then the width is $10$ , hence the length is $20$ . $10\times20= \boxed{\textbf{(E)}\ 200}.$
3
第 3 题
综合
For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?
💡 解题思路
If $x$ is the number of holes that the chipmunk dug, then $3x=4(x-4)$ , so $3x=4x-16$ , and $x=16$ . The number of acorns hidden by the chipmunk is equal to $3x = \boxed{\textbf{(D)}\ 48}$
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第 4 题
分数与比例
Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?
💡 解题思路
The ratio $\frac{400 \text{ euros}}{500 \text{ dollars}}$ can be simplified using conversion factors: \[\frac{400 \text{ euros}}{500 \text{ dollars}} \cdot \frac{1.3 \text{ dollars}}{1 \text{ euro}} =
5
第 5 题
规律与数列
Two integers have a sum of 26 . when two more integers are added to the first two, the sum is 41 . Finally, when two more integers are added to the sum of the previous 4 integers, the sum is 57 . What is the minimum number of even integers among the 6 integers?
💡 解题思路
Since, $x + y = 26$ , $x$ can equal $15$ , and $y$ can equal $11$ , so no even integers are required to make 26. To get to $41$ , we have to add $41 - 26 = 15$ . If $a+b=15$ , at least one of $a$ and
6
第 6 题
逻辑推理
In order to estimate the value of x-y where x and y are real numbers with x>y>0 , Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
💡 解题思路
The original expression $x-y$ now becomes $(x+k) - (y-k)=(x-y)+2k>x-y$ , where $k$ is a positive constant, hence the answer is $\boxed{\textbf{(A)}}$ .
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第 7 题
规律与数列
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the 3rd red light and the 21st red light? Note: 1 foot is equal to 12 inches.
💡 解题思路
We know the repeating section is made of $2$ red lights and $3$ green lights. The 3rd red light would appear in the 2nd section of this pattern, and the 21st red light would appear in the 11th section
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第 8 题
综合
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
💡 解题思路
We can count the number of possible foods for each day and then multiply to enumerate the number of combinations.
9
第 9 题
行程问题
It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many seconds would it take Clea to ride the escalator down when she is not walking?
💡 解题思路
She walks at a rate of $x$ units per second to travel a distance $y$ . As $vt=d$ , we find $60x=y$ and $24*(x+k)=y$ , where $k$ is the speed of the escalator. Setting the two equations equal to each o
10
第 10 题
几何·面积
What is the area of the polygon whose vertices are the points of intersection of the curves x^2 + y^2 =25 and (x-4)^2 + 9y^2 = 81 ?
💡 解题思路
The first curve is a circle with radius $5$ centered at the origin, and the second curve is an ellipse with center $(4,0)$ and end points of $(-5,0)$ and $(13,0)$ . Finding points of intersection, we
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第 11 题
方程
In the equation below, A and B are consecutive positive integers, and A , B , and A+B represent number bases: \[132_A+43_B=69_{A+B}.\] What is A+B ?
💡 解题思路
Change the equation to base 10: \[A^2 + 3A +2 + 4B +3= 6A + 6B + 9\] \[A^2 - 3A - 2B - 4=0\]
12
第 12 题
规律与数列
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
💡 解题思路
There are $\binom{20}{2}$ selections; however, we count these twice, therefore
13
第 13 题
方程
Two parabolas have equations y= x^2 + ax +b and y= x^2 + cx +d , where a, b, c, and d are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common?
💡 解题思路
Set the two equations equal to each other: $x^2 + ax + b = x^2 + cx + d$ . Now remove the x squared and get $x$ 's on one side: $ax-cx=d-b$ . Now factor $x$ : $x(a-c)=d-b$ . If $a$ cannot equal $c$ ,
14
第 14 题
规律与数列
Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernardo. The winner is the last person who produces a number less than 1000 . Let N be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of N ?
💡 解题思路
The last number that Bernardo says has to be between 950 and 999. Note that $1\rightarrow 2\rightarrow 52\rightarrow 104\rightarrow 154\rightarrow 308\rightarrow 358\rightarrow 716\rightarrow 766$ con
15
第 15 题
分数与比例
Jesse cuts a circular disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of 120. He makes two circular cones using each sector to form the lateral surface of each cone. What is the ratio of the volume of the smaller cone to the larger cone?
💡 解题思路
If the original radius is $12$ , then the circumference is $24\pi$ ; since arcs are defined by the central angles, the smaller arc, a $120$ degree angle, is half the size of the larger sector. so the
16
第 16 题
计数
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
💡 解题思路
Let the ordered triple $(a,b,c)$ denote that $a$ songs are liked by Amy and Beth, $b$ songs by Beth and Jo, and $c$ songs by Jo and Amy. The only possible triples are $(1,1,1), (2,1,1), (1,2,1)(1,1,2)
17
第 17 题
几何·面积
Square PQRS lies in the first quadrant. Points (3,0), (5,0), (7,0), and (13,0) lie on lines SP, RQ, PQ , and SR , respectively. What is the sum of the coordinates of the center of the square PQRS ? [图] (diagram by MSTang)
Let (a_1,a_2, \dots ,a_{10}) be a list of the first 10 positive integers such that for each 2 \le i \le 10 either a_i+1 or a_i-1 or both appear somewhere before a_i in the list. How many such lists are there?
💡 解题思路
Let $1\leq k\leq 10$ . Assume that $a_1=k$ . If $k<10$ , the first number appear after $k$ that is greater than $k$ must be $k+1$ , otherwise if it is any number $x$ larger than $k+1$ , there will be
19
第 19 题
立体几何
A unit cube has vertices P_1,P_2,P_3,P_4,P_1',P_2',P_3', and P_4' . Vertices P_2 , P_3 , and P_4 are adjacent to P_1 , and for 1\le i\le 4, vertices P_i and P_i' are opposite to each other. A regular octahedron has one vertex in each of the segments P_1P_2 , P_1P_3 , P_1P_4 , P_1'P_2' , P_1'P_3' , and P_1'P_4' . What is the octahedron's side length? [图]
💡 解题思路
Observe the diagram above. Each dot represents one of the six vertices of the regular octahedron. Three dots have been placed exactly x units from the $(0,0,0)$ corner of the unit cube. The other thre
20
第 20 题
几何·面积
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of r_1√(n_1)+r_2√(n_2)+r_3 , where r_1 , r_2 , and r_3 are rational numbers and n_1 and n_2 are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to r_1+r_2+r_3+n_1+n_2 ?
💡 解题思路
Name the trapezoid $ABCD$ , where $AB$ is parallel to $CD$ , $ABBC$ , so there are only three possible trapezoids:
21
第 21 题
几何·面积
Square AXYZ is inscribed in equiangular hexagon ABCDEF with X on \overline{BC} , Y on \overline{DE} , and Z on \overline{EF} . Suppose that AB=40 , and EF=41(√(3)-1) . What is the side-length of the square? [图] (diagram by djmathman)
💡 解题思路
We can, $\textsc{wlog}$ , assume $Y$ coincides with $D$ and $CD\parallel AF$ as before. In which case, we will have $BC=EF=41(\sqrt{3}-1)$ . So we have square $AXDZ$ inscribed in equiangular hexagon $
22
第 22 题
行程问题
A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there? [图]
Consider all polynomials of a complex variable, P(z)=4z^4+az^3+bz^2+cz+d , where a,b,c, and d are integers, 0\le d\le c\le b\le a\le 4 , and the polynomial has a zero z_0 with |z_0|=1. What is the sum of all values P(1) over all the polynomials with these properties?
💡 解题思路
Since $z_0$ is a root of $P$ , and $P$ has integer coefficients, $z_0$ must be algebraic. Since $z_0$ is algebraic and lies on the unit circle, $z_0$ must be a root of unity (Comment: this is not true
24
第 24 题
数论
Define the function f_1 on the positive integers by setting f_1(1)=1 and if n=p_1^{e_1}p_2^{e_2}·s p_k^{e_k} is the prime factorization of n>1 , then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}·s (p_k+1)^{e_k-1}.\] For every m\ge 2 , let f_m(n)=f_1(f_{m-1}(n)) . For how many N s in the range 1\le N\le 400 is the sequence (f_1(N),f_2(N),f_3(N),\dots ) unbounded? Note: A sequence of positive numbers is unbounded if for every integer B , there is a member of the sequence greater than B .
💡 解题思路
First of all, notice that for any odd prime $p$ , the largest prime that divides $p+1$ is no larger than $\frac{p+1}{2}$ , therefore eventually the factorization of $f_k(N)$ does not contain any prime
25
第 25 题
几何·面积
Let S=\{(x,y) : x\in \{0,1,2,3,4\}, y\in \{0,1,2,3,4,5\}, and (x,y)\ne (0,0)\} . Let T be the set of all right triangles whose vertices are in S . For every right triangle t=\triangle{ABC} with vertices A , B , and C in counter-clockwise order and right angle at A , let f(t)=\tan(\angle{CBA}) . What is \[\prod_{t\in T} f(t)?\]
💡 解题思路
Consider reflections. For any right triangle $ABC$ with the right labeling described in the problem, any reflection $A'B'C'$ labeled that way will give us $\tan CBA \cdot \tan C'B'A' = 1$ . First we c