2000 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
规律与数列
In the year 2001 , the United States will host the International Mathematical Olympiad . Let I,M, and O be distinct positive integers such that the product I · M · O = 2001 . What is the largest possible value of the sum I + M + O ?
💡 解题思路
First, we need to recognize that a number is going to be largest only if, of the $3$ factors , two of them are small. If we want to make sure that this is correct, we could test with a smaller number,
2
第 2 题
综合
2000(2000^{2000}) = .
💡 解题思路
We can use an elementary exponents rule to solve our problem. We know that $a^b\cdot a^c = a^{b+c}$ . Hence, $2000(2000^{2000}) = (2000^{1})(2000^{2000}) = 2000^{2000+1} = 2000^{2001} \Rightarrow \box
3
第 3 题
综合
Each day, Jenny ate 20\% of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, 32 remained. How many jellybeans were in the jar originally?
💡 解题思路
We can begin by labeling the number of initial jellybeans $x$ . If she ate $20\%$ of the jellybeans, then $80\%$ is remaining. Hence, after day 1, there are: $0.8 * x$
4
第 4 题
规律与数列
The Fibonacci sequence 1,1,2,3,5,8,13,21,\ldots starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
💡 解题思路
Note that any digits other than the units digit will not affect the answer. So to make computation quicker, we can just look at the Fibonacci sequence in $\bmod{10}$ :
5
第 5 题
逻辑推理
If |x - 2| = p , where x < 2 , then x - p =
💡 解题思路
When $x < 2,$ $x-2$ is negative so $|x - 2| = 2-x = p$ and $x = 2-p$ .
6
第 6 题
数论
Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
💡 解题思路
Any two prime numbers between 4 and 18 have an odd product and an even sum. Any odd number minus an even number is an odd number, so we can eliminate A, B, and D. Since the highest two prime numbers w
7
第 7 题
整数运算
How many positive integers b have the property that \log_{b} 729 is a positive integer? (A) \ 0 (B) \ 1 (C) \ 2 (D) \ 3 (E) \ 4
💡 解题思路
If $\log_{b} 729 = n$ , then $b^n = 729$ . Since $729 = 3^6$ , $b$ must be $3$ to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of $b \Longrightarrow \boxed{\mathrm{E}}$ .
8
第 8 题
几何·面积
Figures 0 , 1 , 2 , and 3 consist of 1 , 5 , 13 , and 25 nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? [图] https://www.youtube.com/watch?v=HVP6qjKAkjA&t=2s
💡 解题思路
We can attempt $0^2+1^2=1$ and $1^2+2^2=5$ , so the pattern here looks like the number of squares in the $n$ -th figure is $n^2+(n+1)^2$ . When we plug in 100 for $n$ , we get $100^2+101^2=10000+10201
9
第 9 题
概率
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71 , 76 , 80 , 82 , and 91 . What was the last score Mrs. Walter entered?
💡 解题思路
The first number is divisible by $1$ .
10
第 10 题
坐标几何
The point P = (1,2,3) is reflected in the xy -plane, then its image Q is rotated by 180^\circ about the x -axis to produce R , and finally, R is translated by 5 units in the positive- y direction to produce S . What are the coordinates of S ? \text {(A) } (1,7, - 3) \text {(B) } ( - 1,7, - 3) \text {(C) } ( - 1, - 2,8) \text {(D) } ( - 1,3,3) \text {(E) } (1,3,3)
💡 解题思路
Step 1: Reflect in the $xy$ -plane. Replace $z$ with its additive inverse: $(1,2,-3)$
11
第 11 题
综合
Two non-zero real numbers , a and b, satisfy ab = a - b . Which of the following is a possible value of \frac {a}{b} + \frac {b}{a} - ab ?
💡 解题思路
$\frac {a}{b} + \frac {b}{a} - ab = \frac{a^2 + b^2}{ab} - (a - b) = \frac{a^2 + b^2}{a-b} - \frac{(a-b)^2}{(a-b)} = \frac{2ab}{a-b} = \frac{2(a-b)}{a-b} =2 \Rightarrow \boxed{\text{E}}$ .
12
第 12 题
整数运算
Let A, M, and C be nonnegative integers such that A + M + C = 12 . What is the maximum value of A · M · C + A · M + M · C + A · C ?
💡 解题思路
It is not hard to see that \[(A+1)(M+1)(C+1)=\] \[AMC+AM+AC+MC+A+M+C+1\] Since $A+M+C=12$ , we can rewrite this as \[(A+1)(M+1)(C+1)=\] \[AMC+AM+AC+MC+13\] So we wish to maximize \[(A+1)(M+1)(C+1)-13\
13
第 13 题
综合
One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? \text {(A)}\ 3 \text {(B)}\ 4 \text {(C)}\ 5 \text {(D)}\ 6 \text {(E)}\ 7
💡 解题思路
Let $c$ be the total amount of coffee, $m$ of milk, and $p$ the number of people in the family. Then each person drinks the same total amount of coffee and milk (8 ounces), so \[\left(\frac{c}{6} + \f
14
第 14 题
统计
When the mean , median , and mode of the list \[10,2,5,2,4,2,x\] are arranged in increasing order, they form a non-constant arithmetic progression . What is the sum of all possible real values of x ? \text {(A)}\ 3 \text {(B)}\ 6 \text {(C)}\ 9 \text {(D)}\ 17 \text {(E)}\ 20
💡 解题思路
We apply casework upon the median:
15
第 15 题
规律与数列
Let f be a function for which f(\dfrac{x}{3}) = x^2 + x + 1 . Find the sum of all values of z for which f(3z) = 7 . \[\text {(A)}\ -1/3 \text {(B)}\ -1/9 \text {(C)}\ 0 \text {(D)}\ 5/9 \text {(E)}\ 5/3\]
💡 解题思路
Let $y = \frac{x}{3}$ ; then $f(y) = (3y)^2 + 3y + 1 = 9y^2 + 3y+1$ . Thus $f(3z)-7=81z^2+9z-6=3(9z-2)(3z+1)=0$ , and $z = -\frac{1}{3}, \frac{2}{9}$ . These sum up to $\boxed{\textbf{(B) }-\frac19}$
16
第 16 题
几何·面积
A checkerboard of 13 rows and 17 columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered 1,2,\ldots,17 , the second row 18,19,\ldots,34 , and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1,2,\ldots,13, , the second column 14,15,\ldots,26 and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system). \text {(A)}\ 222 \text {(B)}\ 333 \text {(C)}\ 444 \text {(D)}\ 555 \text {(E)}\ 666
💡 解题思路
Index the rows with $i = 1, 2, 3, ..., 13$ Index the columns with $j = 1, 2, 3, ..., 17$
17
第 17 题
几何·面积
A circle centered at O has radius 1 and contains the point A . The segment AB is tangent to the circle at A and \angle AOB = \theta . If point C lies on \overline{OA} and \overline{BC} bisects \angle ABO , then OC = [图] \text {(A)}\ \sec^2 \theta - \tan \theta \text {(B)}\ \frac 12 \text {(C)}\ \frac{\cos^2 \theta}{1 + \sin \theta} \text {(D)}\ \frac{1}{1+\sin\theta} \text {(E)}\ \frac{\sin \theta}{\cos^2 \theta}
💡 解题思路
Since $\overline{AB}$ is tangent to the circle, $\triangle OAB$ is a right triangle. This means that $OA = 1$ , $AB = \tan \theta$ and $OB = \sec \theta$ . By the Angle Bisector Theorem , \[\frac{OB}{
18
第 18 题
综合
In year N , the 300^{th} day of the year is a Tuesday. In year N+1 , the 200^{th} day is also a Tuesday. On what day of the week did the 100 th day of year N-1 occur? \text {(A)}\ Thursday \text {(B)}\ Friday \text {(C)}\ Saturday \text {(D)}\ Sunday \text {(E)}\ Monday
💡 解题思路
There are either \[65 + 200 = 265\] or \[66 + 200 = 266\] days between the first two dates depending upon whether or not year $N+1$ is a leap year (since the February 29th of the leap year would come
19
第 19 题
几何·面积
In triangle ABC , AB = 13 , BC = 14 , AC = 15 . Let D denote the midpoint of \overline{BC} and let E denote the intersection of \overline{BC} with the bisector of angle BAC . Which of the following is closest to the area of the triangle ADE ? \text {(A)}\ 2 \text {(B)}\ 2.5 \text {(C)}\ 3 \text {(D)}\ 3.5 \text {(E)}\ 4
💡 解题思路
The answer is exactly $3$ , choice $\mathrm{(C)}$ . We can find the area of triangle $ADE$ by using the simple formula $\frac{bh}{2}$ . Dropping an altitude from $A$ , we see that it has length $12$ (
20
第 20 题
逻辑推理
If x,y, and z are positive numbers satisfying \[x + \frac{1}{y} = 4, y + \frac{1}{z} = 1, and z + \frac{1}{x} = \frac{7}{3}\] Then what is the value of xyz ? \text {(A)}\ \frac{2}{3} \text {(B)}\ 1 \text {(C)}\ \frac{4}{3} \text {(D)}\ 2 \text {(E)}\ \frac{7}{3}
💡 解题思路
We multiply all given expressions to get: \[(1)xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz} = \frac{28}{3}\] Adding all the given expressions gives that \[(2) x + y + z +
21
第 21 题
几何·面积
Through a point on the hypotenuse of a right triangle , lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is m times the area of the square. The ratio of the area of the other small right triangle to the area of the square is \textbf {(A)}\ \frac{1}{2m+1} \textbf {(B)}\ m \textbf {(C)}\ 1-m \textbf {(D)}\ \frac{1}{4m} \textbf {(E)}\ \frac{1}{8m^2}
💡 解题思路
WLOG, let a side of the square be $1$ . Simple angle chasing shows that the two right triangles are similar . Thus the ratio of the sides of the triangles are the same. Since $A = \frac{1}{2}bh = \fra
22
第 22 题
坐标几何
The graph below shows a portion of the curve defined by the quartic polynomial P(x) = x^4 + ax^3 + bx^2 + cx + d . Which of the following is the smallest?
💡 解题思路
Note that there are 3 maxima/minima. Hence we know that the rest of the graph is greater than 10. We approximate each of the above expressions:
23
第 23 题
概率
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from 1 through 46 , inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer . It so happens that the integers on the winning ticket have the same property— the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? \textbf {(A)}\ 1/5 \textbf {(B)}\ 1/4 \textbf {(C)}\ 1/3 \textbf {(D)}\ 1/2 \textbf {(E)}\ 1
💡 解题思路
The product of the numbers has to be a power of $10$ in order to have an integer base ten logarithm. Thus all of the numbers must be in the form $2^m5^n$ . Listing out such numbers from $1$ to $46$ ,
24
第 24 题
几何·面积
If circular arcs AC and BC have centers at B and A , respectively, then there exists a circle tangent to both \overarc {AC} and \overarc{BC} , and to \overline{AB} . If the length of \overarc{BC} is 12 , then the circumference of the circle is [图] \textbf {(A)}\ 24 \textbf {(B)}\ 25 \textbf {(C)}\ 26 \textbf {(D)}\ 27 \textbf {(E)}\ 28
💡 解题思路
First, note the triangle $ABC$ is equilateral. Next, notice that since the arc $BC$ has length 12, it follows that we can find the radius of the sector centered at $A$ . $\frac {1}{6}({2}{\pi})AB=12 \
25
第 25 题
几何·面积
Eight congruent equilateral triangles , each of a different color, are used to construct a regular octahedron . How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) \textbf {(A)}\ 210 \textbf {(B)}\ 560 \textbf {(C)}\ 840 \textbf {(D)}\ 1260 \textbf {(E)}\ 1680
💡 解题思路
Since the octahedron is indistinguishable by rotations, without loss of generality fix a face to be red.