2005 AMC 10A — Official Competition Problems (January 2005)
📅 2005 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
📋 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
应用题
While eating out, Mike and Joe each tipped their server \2 . Mike tipped 10\% of his bill and Joe tipped 20\%$ of his bill. What was the difference, in dollars, between their bills?
💡 解题思路
Let $m$ be Mike's bill and $j$ be Joe's bill (both in dollars). We then have $\frac{10}{100}m = 2 \iff m = 20$ and $\frac{20}{100}j = 2 \iff j = 10$ , so the desired difference is $\left\lvert m-j\rig
2
第 2 题
分数与比例
For each pair of real numbers a ≠ b , define the operation \star as \[(a \star b) = \frac{a+b}{a-b}.\] What is the value of ((1 \star 2) \star 3) ?
The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution x . What is the value of b ?
💡 解题思路
$2x + 7 = 3 \iff x = -2$ , so we require $-2b-10 = -2 \iff -2b = 8 \iff b = \boxed{\textbf{(B) } -4}$ .
4
第 4 题
几何·面积
A rectangle with a diagonal of length x is twice as long as it is wide. What is the area of the rectangle?
💡 解题思路
Let's set the length to $2$ and the width to $1$ , so the rectangle has area $2 \cdot 1 = 2$ and diagonal $x = \sqrt{1^2+2^2} = \sqrt{5}$ (by Pythagoras' theorem).
5
第 5 题
行程问题
A store normally sells windows at \100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
💡 解题思路
The store's offer means that every $5$ th window is free, so Dave would get $\left\lfloor\frac{7}{5}\right\rfloor=1$ free window and Doug would also get $\left\lfloor\frac{8}{5}\right\rfloor=1$ free w
6
第 6 题
统计
The average (mean) of 20 numbers is 30 , and the average of 30 other numbers is 20 . What is the average of all 50 numbers?
💡 解题思路
Since the average of the first $20$ numbers is $30$ , their sum is $20\cdot30=600$ .
7
第 7 题
行程问题
Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
💡 解题思路
Let $m$ be the distance in miles that Mike rode.
8
第 8 题
几何·面积
In the figure, the length of side AB of square ABCD is √(50) , E is between B and H , and BE = 1 . What is the area of the inner square EFGH ? [图]
💡 解题思路
We see that side $BE$ , which we know is $1$ , is also the shorter leg of one of the four right triangles (which are obviously congruent, using the symmetry of the diagram). So $AH = 1$ , and hence $H
9
第 9 题
概率
Three tiles are marked X and two other tiles are marked O . The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX ?
💡 解题思路
There are $\frac{5!}{2!3!} = 10$ distinct arrangements of $3$ $X$ s and $2$ $O$ s, and only $1$ distinct arrangement that reads $XOXOX$ .
10
第 10 题
方程
There are two values of a for which the equation 4x^2 + ax + 8x + 9 = 0 has only one solution for x . What is the sum of those values of a ?
💡 解题思路
A quadratic equation has exactly $1$ distinct root if and only if the left-hand side is a perfect square. So we require
11
第 11 题
立体几何
A wooden cube n units on a side is painted red on all six faces and then cut into n^3 unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n ?
💡 解题思路
Since there are $n^2$ little faces on each face of the big wooden cube, there are a total of $6n^2$ little faces painted red. Moreover, as each unit cube has $6$ faces, there are $6n^3$ little faces i
12
第 12 题
几何·面积
The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length 2 ? [图]
💡 解题思路
The area of the trefoil is equal to the area of an equilateral triangle with side length $2$ , plus the area of $4$ segments. Each segment has area equal to that of a $60^{\circ}$ sector with radius $
13
第 13 题
整数运算
How many positive integers n satisfy the following condition: \[(130n)^{50} > n^{100} > 2^{200} \ ?\]
💡 解题思路
Since $n > 0$ , all $3$ terms of the inequality are positive, so we may take the $50$ th root, yielding
14
第 14 题
统计
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
💡 解题思路
If the middle digit is the average of the first and last digits, twice the middle digit must be equal to the sum of the first and last digits.
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6 . How many two-digit numbers have this property?
💡 解题思路
Let the number be $10a+b$ , where $a$ is its tens digit and $b$ is its units digit. Then $(10a+b)-(a+b) = 9a$ must have a units digit of $6$ , and as $a$ is the tens digit, we can only have $1 \leq a
17
第 17 题
规律与数列
In the five-sided star shown, the letters A , B , C , D , and E are replaced by the numbers 3 , 5 , 6 , 7 , and 9 , although not necessarily in this order. The sums of the numbers at the ends of the line segments \overline{AB} , \overline{BC} , \overline{CD} , \overline{DE} , and \overline{EA} form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence? [图]
💡 解题思路
Each of $A$ , $B$ , $C$ , $D$ , and $E$ forms part of exactly $2$ sums along line segments (e.g. $A$ forms part of the sums of $\overline{AB}$ and $\overline{EA}$ ). Thus, the sum of all these line se
18
第 18 题
概率
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?
💡 解题思路
There are at most $5$ games played.
19
第 19 题
几何·面积
Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45^{\circ} , as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point B from the line on which the bases of the original squares were placed? [图]
💡 解题思路
The rotated middle square is lowered until it touches both the adjoining squares, so since the horizontal distance between those squares is $1$ inch, the middle square will stop being lowered once the
20
第 20 题
几何·面积
An equiangular octagon has four sides of length 1 and four sides of length √(2)/2 , arranged so that no two consecutive sides have the same length. What is the area of the octagon?
💡 解题思路
The sum of the octagon's angles is $180\cdot(8-2)^{\circ} = 1080^{\circ}$ , so since it is equiangular, each angle is $\frac{1080^{\circ}}{8} = 135^{\circ}$ , i.e. the same as in a regular octagon. Th
21
第 21 题
整数运算
For how many positive integers n does 1+2+\dotsb+n evenly divide 6n ?
💡 解题思路
By a standard result, $1+2+\dotsb+n = \frac{n(n+1)}{2}$ , so this will evenly divide $6n$ precisely if $\frac{6n}{\left(\frac{n(n+1)}{2}\right)} = \frac{12}{n+1}$ is an integer, or equivalently, $(n+1
22
第 22 题
数论
Let S be the set of the 2005 smallest positive multiples of 4 , and let T be the set of the 2005 smallest positive multiples of 6 . How many elements are common to S and T ?
💡 解题思路
Since the least common multiple of $4$ and $6$ is $12$ , the elements that are common to $S$ and $T$ are all multiples of $12$ . Moreover, as the largest element of $S$ is $4 \cdot 2005$ , while that
23
第 23 题
几何·面积
Let \overline{AB} be a diameter of a circle and C be a point on \overline{AB} with 2 · AC = BC . Let D and E be points on the circle such that \overline{DC} \perp \overline{AB} and \overline{DE} is a second diameter. What is the ratio of the area of \triangle DCE to the area of \triangle ABD ? [图]
💡 解题思路
WLOG, let us assume that the diameter is of length $1$ .
24
第 24 题
数论
For each positive integer m > 1 , let P(m) denote the greatest prime factor of m . For how many positive integers n is it true that both P(n) = √(n) and P(n+48) = √(n+48) ?
💡 解题思路
If $P(n) = \sqrt{n}$ , then $n = p_{1}^{2}$ , where $p_{1}$ is a prime number .
25
第 25 题
几何·面积
In \triangle ABC we have AB = 25 , BC = 39 , and AC = 42 . Points D and E are on \overline{AB} and \overline{AC} respectively, with AD = 19 and AE = 14 . What is the ratio of the area of triangle ADE to the area of the quadrilateral BCED ?
💡 解题思路
We have \[\frac{[ADE]}{[ABC]} = \frac{AD}{AB} \cdot \frac{AE}{AC} = \frac{19}{25} \cdot \frac{14}{42} = \frac{19}{75}.\]