2013 AMC 10A — Official Competition Problems (February 2013)
📅 2013 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
行程问题
A taxi ride costs \1.50 plus \0.25 per mile traveled. How much does a 5 -mile taxi ride cost?
💡 解题思路
There are five miles which need to be traveled. The cost of these five miles is $(0.25\cdot5) = 1.25$ . Adding this to $1.50$ , we get $\boxed{\textbf{(C) }2.75}$
2
第 2 题
行程问题
Alice is making a batch of cookies and needs 2\frac{1}{2} cups of sugar. Unfortunately, her measuring cup holds only \frac{1}{4} cup of sugar. How many times must she fill that cup to get the correct amount of sugar?
💡 解题思路
To get how many cups we need, we realize that we simply need to divide the number of cups needed by the number of cups collected in her measuring cup each time. Thus, we need to evaluate the fraction
3
第 3 题
几何·面积
Square ABCD has side length 10 . Point E is on \overline{BC} , and the area of \bigtriangleup ABE is 40 . What is BE ? [图]
💡 解题思路
We are given that the area of $\triangle ABE$ is $40$ , and that $AB = 10$ .
4
第 4 题
综合
A softball team played ten games, scoring 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
💡 解题思路
We know that, for the games where they scored an odd number of runs, they cannot have scored twice as many runs as their opponents, as odd numbers are not divisible by $2$ . Thus, from this, we know t
5
第 5 题
应用题
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid \105, Dorothy paid \125, and Sammy paid \175. In order to share costs equally, Tom gave Sammy t dollars, and Dorothy gave Sammy d dollars. What is t-d$ ?
💡 解题思路
The total amount paid is $105 + 125 + 175 = 405$ . To get how much each should have paid, we do $405/3 = 135$ .
6
第 6 题
规律与数列
Joey and his five brothers are ages 3 , 5 , 7 , 9 , 11 , and 13 . One afternoon two of his brothers whose ages sum to 16 went to the movies, two brothers younger than 10 went to play baseball, and Joey and the 5 -year-old stayed home. How old is Joey?
💡 解题思路
Because the $5$ -year-old stayed home, we know that the $11$ -year-old did not go to the movies, as the $5$ -year-old did not and $11+5=16$ . Also, the $11$ -year-old could not have gone to play baseb
7
第 7 题
计数
A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
💡 解题思路
Let us split this up into two cases.
8
第 8 题
综合
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?\]
💡 解题思路
Factoring out, we get: $\frac{2^{2012}(2^2 + 1)}{2^{2012}(2^2-1)}$ .
9
第 9 题
应用题
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20\% of her three-point shots and 30\% of her two-point shots. Shenille attempted 30 shots. How many points did she score?
💡 解题思路
Let the number of attempted three-point shots be $x$ and the number of attempted two-point shots be $y$ . We know that $x+y=30$ , and we need to evaluate $3(0.2x) + 2(0.3y)$ , as we know that the thre
10
第 10 题
分数与比例
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
💡 解题思路
Let the total amount of flowers be $x$ . Thus, the number of pink flowers is $0.6x$ , and the number of red flowers is $0.4x$ . The number of pink carnations is $\frac{2}{3}(0.6x) = 0.4x$ and the numb
11
第 11 题
计数
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?
💡 解题思路
Let the number of students on the council be $x$ . To select a two-person committee, we can select a "first person" and a "second person." There are $x$ choices to select a first person; subsequently,
12
第 12 题
几何·面积
In \triangle ABC , AB=AC=28 and BC=20 . Points D,E, and F are on sides \overline{AB} , \overline{BC} , and \overline{AC} , respectively, such that \overline{DE} and \overline{EF} are parallel to \overline{AC} and \overline{AB} , respectively. What is the perimeter of parallelogram ADEF ? [图]
💡 解题思路
Note that because $\overline{DE}$ and $\overline{EF}$ are parallel to the sides of $\triangle ABC$ , the internal triangles $\triangle BDE$ and $\triangle EFC$ are similar to $\triangle ABC$ , and are
13
第 13 题
数论
How many three-digit numbers are not divisible by 5 , have digits that sum to less than 20 , and have the first digit equal to the third digit?
💡 解题思路
We use a casework approach to solve the problem. These three digit numbers are of the form $\overline{xyx}$ .( $\overline{abc}$ denotes the number $100a+10b+c$ ). We see that $x\neq 0$ and $x\neq 5$ ,
14
第 14 题
立体几何
A solid cube of side length 1 is removed from each corner of a solid cube of side length 3 . How many edges does the remaining solid have?
💡 解题思路
We can use Euler's polyhedron formula that says that $F+V=E+2$ . We know that there are originally $6$ faces on the cube, and each corner cube creates $3$ more. $6+8(3) = 30$ . In addition, each cube
15
第 15 题
几何·面积
Two sides of a triangle have lengths 10 and 15 . The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?
💡 解题思路
The shortest side length has the longest altitude perpendicular to it. The average of the two altitudes given will be between the lengths of the two altitudes, therefore the length of the side perpend
16
第 16 题
几何·面积
A triangle with vertices (6, 5) , (8, -3) , and (9, 1) is reflected about the line x=8 to create a second triangle. What is the area of the union of the two triangles?
💡 解题思路
Let $A$ be at $(6, 5)$ , B be at $(8, -3)$ , and $C$ be at $(9, 1)$ . Reflecting over the line $x=8$ , we see that $A' = D = (10,5)$ , $B' = B$ (as the x-coordinate of B is 8), and $C' = E = (7, 1)$ .
17
第 17 题
综合
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365 -day period will exactly two friends visit her?
💡 解题思路
The $365$ -day time period can be split up into $6$ , $60$ -day time periods, because after $60$ days, all three of them visit again (Least common multiple of $3$ , $4$ , and $5$ ). You can find how m
18
第 18 题
几何·面积
Let points A = (0, 0) , B = (1, 2) , C=(3, 3) , and D = (4, 0) . Quadrilateral ABCD is cut into equal area pieces by a line passing through A . This line intersects \overline{CD} at point (\frac{p}{q}, \frac{r}{s}) , where these fractions are in lowest terms. What is p+q+r+s ?
💡 解题思路
First, we shall find the area of quadrilateral $ABCD$ . This can be done in any of three ways:
19
第 19 题
数字运算
In base 10 , the number 2013 ends in the digit 3 . In base 9 , on the other hand, the same number is written as (2676)_{9} and ends in the digit 6 . For how many positive integers b does the base- b -representation of 2013 end in the digit 3 ?
💡 解题思路
We want the integers $b$ such that $2013\equiv 3\pmod{b} \Rightarrow b$ is a factor of $2010$ . Since $2010=2 \cdot 3 \cdot 5 \cdot 67$ , it has $(1+1)(1+1)(1+1)(1+1)=2^4=16$ factors. Since $b$ cannot
20
第 20 题
几何·面积
A unit square is rotated 45^\circ about its center. What is the area of the region swept out by the interior of the square?
💡 解题思路
First, we need to see what this looks like. Below is a diagram.
21
第 21 题
概率
A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The k^{th} pirate to take a share takes \frac{k}{12} of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the 12^{th} pirate receive?
💡 解题思路
Let $x$ be the number of coins. After the $k^{\text{th}}$ pirate takes his share, $\frac{12-k}{12}$ of the original amount is left. Thus, we know that
22
第 22 题
立体几何
Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2 . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
💡 解题思路
Set up an isosceles triangle between the center of the $8$ th sphere and two opposite ends of the hexagon. Then set up another triangle between the point of tangency of the $7$ th and $8$ th spheres,
23
第 23 题
几何·面积
In \triangle ABC , AB = 86 , and AC=97 . A circle with center A and radius AB intersects \overline{BC} at points B and X . Moreover \overline{BX} and \overline{CX} have integer lengths. What is BC ?
💡 解题思路
Let $BX = q$ , $CX = p$ , and $AC$ meets the circle at $Y$ and $Z$ , with $Y$ on $AC$ . Then $AZ = AY = 86$ . Using the Power of a Point (Secant-Secant Power Theorem), we get that $p(p+q) = 11(183) =
24
第 24 题
计数
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
💡 解题思路
Let us label the players of the first team $A$ , $B$ , and $C$ , and those of the second team, $X$ , $Y$ , and $Z$ .
25
第 25 题
综合
All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?
💡 解题思路
If you draw a clear diagram like the one below, it is easy to see that there are $\boxed{\textbf{(A) }49}$ points.