2000 AMC 10 — Official Competition Problems (January 2000)
📅 2000 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
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
行程问题
Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was \12.48 , but in January her bill was \17.54 because she used twice as much connect time as in December. What is the fixed monthly fee?
💡 解题思路
Let $x$ be the fixed fee, and $y$ be the amount she pays for the minutes she used in the first month.
5
第 5 题
几何·面积
Points M and N are the midpoints of sides PA and PB of \triangle PAB . As P moves along a line that is parallel to side AB , how many of the four quantities listed below change? (a) the length of the segment MN (b) the perimeter of \triangle PAB (c) the area of \triangle PAB (d) the area of trapezoid ABNM [图]
💡 解题思路
(a) Triangles $ABP$ and $MNP$ are similar, and since $PM=\frac{1}{2}AP$ , $MN=\frac{1}{2}AB$ .
6
第 6 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 $\bmod{10}$ :
7
第 7 题
几何·面积
In rectangle ABCD , AD=1 , P is on \overline{AB} , and \overline{DB} and \overline{DP} trisect \angle ADC . What is the perimeter of \triangle BDP ? [图]
At Olympic High School, \frac{2}{5} of the freshmen and \frac{4}{5} of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true? (A) There are five times as many sophomores as freshmen. (B) There are twice as many sophomores as freshmen. (C) There are as many freshmen as sophomores. (D) There are twice as many freshmen as sophomores. (E) There are five times as many freshmen as sophomores.
💡 解题思路
Let $f$ be the number of freshman and $s$ be the number of sophomores.
9
第 9 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 $x = 2-p$ .
10
第 10 题
几何·面积
The sides of a triangle with positive area have lengths 4 , 6 , and x . The sides of a second triangle with positive area have lengths 4 , 6 , and y . What is the smallest positive number that is not a possible value of |x-y| ?
💡 解题思路
Since $6$ and $4$ are fixed sides, the smallest possible side has to be larger than $6-4=2$ and the largest possible side has to be smaller than $6+4=10$ . This gives us the triangle inequality $2<x<1
11
第 11 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
12
第 12 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
13
第 13 题
统计
There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color? [图] https://www.youtube.com/watch?v=IP3aip6lT40 https://youtu.be/Ca6WSj9NN7k?t=510
💡 解题思路
In each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.
14
第 14 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 $1$ .
15
第 15 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
$\frac {a}{b} + \frac {b}{a} - ab = \frac{a^2 + b^2}{ab} - (a - b) = \frac{a^2 + b^2}{a-b} - \frac{(a-b)^2}{(a-b)} = \frac{2ab}{a-b} = \frac{2(a-b)}{a-b} =2 \Rightarrow \boxed{\text{E}}$ .
16
第 16 题
综合
The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment AB meets segment CD at E . Find the length of segment AE . [图] https://youtu.be/oWxqYyW926I
💡 解题思路
Let $l_1$ be the line containing $A$ and $B$ and let $l_2$ be the line containing $C$ and $D$ . If we set the bottom left point at $(0,0)$ , then $A=(0,3)$ , $B=(6,0)$ , $C=(4,2)$ , and $D=(2,0)$ .
17
第 17 题
概率
Boris has an incredible coin-changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?
💡 解题思路
If Boris puts a quarter or nickel in the machine, he neither gains nor loses any money. If he puts a penny in the machine, he gains $\$1.24$ .
18
第 18 题
几何·面积
Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number? https://youtu.be/j7Hi5I8INII - Happytwin
💡 解题思路
The area she sees looks at follows:
19
第 19 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
20
第 20 题
整数运算
Let A , M , and C be nonnegative integers such that A+M+C=10 . What is the maximum value of A· M· C+A· M+M· C+C· A ?
💡 解题思路
The trick is to realize that the sum $AMC+AM+MC+CA$ is similar to the product $(A+1)(M+1)(C+1)$ . If we multiply $(A+1)(M+1)(C+1)$ , we get \[(A+1)(M+1)(C+1) = AMC + AM + AC + MC + A + M + C + 1.\] We
21
第 21 题
逻辑推理
If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? \[\textrm{I. All alligators are creepy crawlers.}\] \[\textrm{II. Some ferocious creatures are creepy crawlers.}\] \[\textrm{III. Some alligators are not creepy crawlers.}\] (A)\ I only (B)\ II only (C)\ III only (D)\ II and III only (E)\ None must be true
💡 解题思路
We interpret the problem statement as a query about three abstract concepts denoted as "alligators", "creepy crawlers" and "ferocious creatures". In answering the question, we may NOT refer to reality
22
第 22 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
23
第 23 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:
24
第 24 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 $\boxed{\textbf{(B) }-\frac19}$
25
第 25 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