2010 AMC 10A — Official Competition Problems (February 2010)
📅 2010 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
统计
Mary's top book shelf holds five books with the following widths, in centimeters: 6 , \dfrac{1}{2} , 1 , 2.5 , and 10 . What is the average book width, in centimeters? (A)\ 1 (B)\ 2 (C)\ 3 (D)\ 4 (E)\ 5
💡 解题思路
To find the average, we add up the widths $6$ , $\dfrac{1}{2}$ , $1$ , $2.5$ , and $10$ , to get a total sum of $20$ . Since there are $5$ books, the average book width is $\frac{20}{5}=4$ The answer
2
第 2 题
几何·面积
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? (A)\ \dfrac{5}{4} (B)\ \dfrac{4}{3} (C)\ \dfrac{3}{2} (D)\ 2 (E)\ 3
💡 解题思路
Let the length of the small square be $x$ , intuitively, the length of the big square is $4x$ . It can be seen that the width of the rectangle is $3x$ . Thus, the length of the rectangle is $4x/3x = 4
3
第 3 题
综合
Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric? (A)\ 3 (B)\ 13 (C)\ 18 (D)\ 25 (E)\ 29
💡 解题思路
Let $x$ be the number of marbles Tyrone gave to Eric. Then, $97-x = 2\cdot(11+x)$ . Solving for $x$ yields $75=3x$ and $x = 25$ . The answer is $\boxed{D}$ .
4
第 4 题
规律与数列
A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain? (A)\ 50.2 (B)\ 51.5 (C)\ 52.4 (D)\ 53.8 (E)\ 55.2
💡 解题思路
Assuming that there are fractions of compact discs, it would take $412/56 ~= 7.357$ CDs to have equal reading time. However, since the number of discs must be a whole number, there are at least 8 CDs,
5
第 5 题
几何·面积
The area of a circle whose circumference is 24π is kπ . What is the value of k ? (A)\ 6 (B)\ 12 (C)\ 24 (D)\ 36 (E)\ 144
💡 解题思路
If the circumference of a circle is $24\pi$ , the radius would be $12$ . Since the area of a circle is $\pi r^2$ , the area is $144\pi$ . The answer is $\boxed{E}$ .
6
第 6 题
分数与比例
For positive numbers x and y the operation \spadesuit (x,y) is defined as \[\spadesuit (x,y) = x-\dfrac{1}{y}\] What is \spadesuit (2,\spadesuit (2,2)) ? (A)\ \dfrac{2}{3} (B)\ 1 (C)\ \dfrac{4}{3} (D)\ \dfrac{5}{3} (E)\ 2
💡 解题思路
$\spadesuit (2,2) =2-\frac{1}{2} =\frac{3}{2}$ . Then, $\spadesuit \left(2,\frac{3}{2}\right)$ is $2-\frac{1}{\frac{3}{2}} = 2- \frac{2}{3} = \frac{4}{3}$ The answer is $\boxed{C}$
7
第 7 题
综合
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? (A)\ 1 (B)\ √(2) (C)\ √(3) (D)\ 2 (E)\ 2√(2)
💡 解题思路
Crystal first runs north for one mile. Changing directions, she runs northeast for another mile. The angle difference between north and northeast is 45 degrees. She then switches directions to southea
8
第 8 题
时间问题
Tony works 2 hours a day and is paid 0.50 per hour for each full year of his age. During a six month period Tony worked 50 days and earned 630 . How old was Tony at the end of the six month period? (A)\ 9 (B)\ 11 (C)\ 12 (D)\ 13 (E)\ 14
💡 解题思路
Tony works $2$ hours a day and is paid $0.50$ dollars per hour for each full year of his age. This basically says that he gets a dollar for each year of his age. So if he is $12$ years old, he gets $1
9
第 9 题
规律与数列
A palindrome , such as 83438 , is a number that remains the same when its digits are reversed. The numbers x and x+32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x ?
💡 解题思路
$x$ is at most $999$ , so $x+32$ is at most $1031$ . The minimum value of $x+32$ is $1000$ . However, the only palindrome between $1000$ and $1032$ is $1001$ , which means that $x+32$ must be $1001$ .
10
第 10 题
综合
Marvin had a birthday on Tuesday, May 27 in the leap year 2008 . In what year will his birthday next fall on a Saturday? (A)\ 2011 (B)\ 2012 (C)\ 2013 (D)\ 2015 (E)\ 2017
💡 解题思路
There are $365$ days in a non-leap year. There are $7$ days in a week. Since $365 = 52 \cdot 7 + 1$ (or $365\equiv 1 \pmod{ 7}$ ), the same date (after February) moves "forward" one day in the subsequ
11
第 11 题
综合
The length of the interval of solutions of the inequality a \le 2x + 3 \le b is 10 . What is b - a ? (A)\ 6 (B)\ 10 (C)\ 15 (D)\ 20 (E)\ 30
💡 解题思路
Since we are given the range of the solutions, we must re-write the inequalities so that we have $x$ in terms of $a$ and $b$ .
12
第 12 题
统计
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
💡 解题思路
The water tower holds $\frac{100000}{0.1} = 1000000$ times more water than Logan's miniature. The volume of a sphere is: $V=\dfrac{4}{3}\pi r^3$ . Since we are comparing the heights (m), we should com
13
第 13 题
方程
Angelina drove at an average rate of 80 kmh and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kmh. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop?
💡 解题思路
The answer is $A$ because she drove at $80$ kmh for $t$ hours (the amount of time before the stop), and $100$ kmh for $\frac{8}{3}-t$ because she wasn't driving for $20$ minutes, or $\frac{1}{3}$ hour
14
第 14 题
几何·面积
Triangle ABC has AB=2 · AC . Let D and E be on \overline{AB} and \overline{BC} , respectively, such that \angle BAE = \angle ACD . Let F be the intersection of segments AE and CD , and suppose that \triangle CFE is equilateral. What is \angle ACB ?
💡 解题思路
Let $\angle BAE = \angle ACD = x$ .
15
第 15 题
计数
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements. Brian: "Mike and I are different species." Chris: "LeRoy is a frog." LeRoy: "Chris is a frog." Mike: "Of the four of us, at least two are toads." How many of these amphibians are frogs?
💡 解题思路
Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.
16
第 16 题
几何·面积
Nondegenerate \triangle ABC has integer side lengths, \overline{BD} is an angle bisector, AD = 3 , and DC=8 . What is the smallest possible value of the perimeter?
💡 解题思路
By the Angle Bisector Theorem , we know that $\frac{AB}{BC} = \frac{3}{8}$ . If we use the lowest possible integer values for $AB$ and $BC$ (the lengths of $AD$ and $DC$ , respectively), then $AB + BC
17
第 17 题
几何·面积
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
💡 解题思路
Imagine making the cuts one at a time. The first cut removes a box $2\times 2\times 3$ . The second cut removes two boxes, each of dimensions $2\times 2\times 0.5$ , and the third cut does the same as
18
第 18 题
概率
Bernardo randomly picks 3 distinct numbers from the set \{1,2,3,...,7,8,9\} and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set \{1,2,3,...,6,7,8\} and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
💡 解题思路
We can solve this by breaking the problem down into $2$ cases and adding up the probabilities.
19
第 19 题
几何·面积
Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r . The area of \triangle ACE is 70\% of the area of the hexagon. What is the sum of all possible values of r ?
💡 解题思路
It is clear that $\triangle ACE$ is an equilateral triangle. From the Law of Cosines on $\triangle ABC$ , we get that $AC^2 = r^2+1^2-2r\cos{\frac{2\pi}{3}} = r^2+r+1$ . Therefore, the area of $\trian
20
第 20 题
综合
A fly trapped inside a cubical box with side length 1 meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?
💡 解题思路
The distance of an interior diagonal in this cube is $\sqrt{3}$ and the distance of a diagonal on one of the square faces is $\sqrt{2}$ . It is not possible for the fly to travel any interior diagonal
21
第 21 题
整数运算
The polynomial x^3-ax^2+bx-2010 has three positive integer roots. What is the smallest possible value of a ?
💡 解题思路
By Vieta's Formulas , we know that $a$ is the sum of the three roots of the polynomial $x^3-ax^2+bx-2010$ . Again Vieta's Formulas tell us that $2010$ is the product of the three integer roots. Also,
22
第 22 题
几何·面积
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
💡 解题思路
To choose a chord, we know that two vertices must be chosen. This implies that for three chords to create a triangle and not intersect at a single point, six points need to be chosen. We also know tha
23
第 23 题
概率
Each of 2010 boxes in a line contains a single red marble, and for 1 \le k \le 2010 , the box in the kth position also contains k white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let P(n) be the probability that Isabella stops after drawing exactly n marbles. What is the smallest value of n for which P(n) < \frac{1}{2010} ?
💡 解题思路
The probability of drawing a white marble from box $k$ is $\frac{k}{k + 1}$ , and the probability of drawing a red marble from box $k$ is $\frac{1}{k+1}$ .
24
第 24 题
数论
The number obtained from the last two nonzero digits of 90! is equal to n . What is n ? Let P be the result of dividing 90! by tens such that P is not divisible by 10 . We want to consider P \mod 100 . But because 100 is not prime, and because P is obviously divisible by 4 (if in doubt, look at the answer choices), we only need to consider P \mod 25 . However, 25 is a very particular number. 1 · 2 · 3 · 4 \equiv -1 (mod 25) , and so is 6 · 7 · 8 · 9 . How can we group terms to take advantage of this fact? There might be a problem when you cancel out the 10 s from 90! . One method is to cancel out a factor of 2 from an existing number along with a factor of 5 . But this might prove cumbersome, as the grouping method will not be as effective. Instead, take advantage of inverses in modular arithmetic. Just leave the negative powers of 2 in a "storage base," and take care of the other terms first. Then, use Fermat's Little Theorem to solve for the power of 2 . Video Solution: https://youtu.be/30CamkkifHM?t=766
💡 解题思路
We will use the fact that for any integer $n$ , \begin{align*}(5n+1)(5n+2)(5n+3)(5n+4)&=[(5n+4)(5n+1)][(5n+2)(5n+3)]\\ &=(25n^2+25n+4)(25n^2+25n+6)\equiv 4\cdot 6\\ &=24\pmod{25}\equiv -1\pmod{25}.\en
25
第 25 题
几何·面积
Jim starts with a positive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with n = 55 , then his sequence contains 5 numbers: \[\begin{array}{ccccc} {}&{}&{}&{}&55 ; 55&-&7^2=&6 ; 6&-&2^2=&2 ; 2&-&1^2=&1 ; 1&-&1^2=&0 ; \end{array}\] Let N be the smallest number for which Jim’s sequence has 8 numbers. What is the units digit of N ? (A)\ 1 (B)\ 3 (C)\ 5 (D)\ 7 (E)\ 9
💡 解题思路
We can find the answer by working backwards. We begin with $1-1^2=0$ on the bottom row, then the $1$ goes to the right of the equal's sign in the row above. We find the smallest value $x$ for which $x