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AMC10 2013 A

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AMC10 · 2013 (A)

Q1
A taxi ride costs \$1.50 plus \$0.25 per mile traveled. How much does a 5-mile taxi ride cost?
出租车行程费用为1.50美元加上每英里0.25美元。5英里出租车行程需要多少钱？
Q2
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?
Alice在做一批饼干，需要$2\frac{1}{2}$杯糖。不幸的是，她的量杯只能装$\frac{1}{4}$杯糖。她需要填充量杯多少次才能得到正确的糖量？
Q3
Square $ABCD$ has side length 10. Point $E$ is on $\overline{BC}$, and the area of $\triangle ABE$ is 40. What is $BE$?
正方形$ABCD$边长为10。点$E$在$\overline{BC}$上，$\triangle ABE$的面积为40。$BE$是多少？
stem
Q4
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?
垒球队打了十场比赛，得分为1, 2, 3, 4, 5, 6, 7, 8, 9和10分。他们恰好在五场比赛中以一分的劣势输掉。在其他比赛中，他们得分的两次是对手得分。他们的对手总共得了多少分？
Q5
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 the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t - d$?
Tom、Dorothy和Sammy去度假，同意平分费用。在旅行中Tom付了105美元，Dorothy付了125美元，Sammy付了175美元。为了平等分担费用，Tom给了Sammy $t$美元，Dorothy给了Sammy $d$美元。$t - d$是多少？
Q6
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?
乔伊和他的五个兄弟的年龄分别是3、5、7、9、11和13岁。一天下午，两个年龄和为16岁的兄弟去看电影，两个小于10岁的兄弟去打棒球，乔伊和5岁的兄弟留在家里。乔伊多大年龄？
Q7
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?
一名学生必须从英语、代数、几何、历史、艺术和拉丁语课程中选择四个课程组成课程计划。该计划必须包含英语，并且至少有一门数学课程。有多少种方式可以选择这个课程计划？
Q8
What is the value of $\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$?
$\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$ 的值是多少？
Q9
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?
在一场最近的篮球比赛中，Shenille只尝试三分球和两分球。她三分球命中率为20%，两分球命中率为30%。Shenille总共尝试了30次投篮。她总共得了多少分？
Q10
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?
一个花束包含粉色玫瑰、红色玫瑰、粉色康乃馨和红色康乃馨。粉色花中的三分之一是玫瑰，红色花中的四分之三是康乃馨，花束中有十分之六是粉色花。花束中有百分之多少是康乃馨？
Q11
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?
学生会需要从其成员中选出一个两人迎接委员会和一个三人规划委员会。有恰好10种方法选出两人迎接委员会。学生可以同时在两个委员会任职。有多少种不同的方法可以选出三人规划委员会？
Q12
In $\triangle ABC$, $AB = AC = 28$ and $BC = 20$. Points $D, E, F$ are on sides $\overline{AB}, \overline{BC},$ and $\overline{AC}$, respectively, such that $DE$ and $EF$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?
在$\triangle ABC$中，$AB = AC = 28$且$BC = 20$。点$D, E, F$分别在边$\overline{AB}, \overline{BC},$和$\overline{AC}$上，使得$DE$平行于$\overline{AC}$，$EF$平行于$\overline{AB}$。平行四边形$ADEF$的周长是多少？
stem
Q13
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?
有多少个三位数不被5整除，数字之和小于20，且首位数字等于末位数字？
Q14
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?
从边长为3的实心立方体每个角上切掉一个边长为1的实心立方体。剩余实心体有多少条边？
Q15
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?
一个三角形的两边长分别为10和15。到第三边的垂线长度是到这两边垂线长度平均值。第三边有多长？
Q16
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?
有一个顶点为$(6, 5)$、$(8, -3)$和$(9, 1)$的三角形，关于直线$x = 8$反射得到第二个三角形。两个三角形的并集面积是多少？
Q17
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?
Daphne 定期被她的三位最好的朋友访问：Alice、Beatrix 和 Claire。Alice 每三天访问一次，Beatrix 每四天访问一次，Claire 每五天访问一次。三位朋友昨天都访问了 Daphne。在接下来的 365 天周期中，有多少天恰好有两位朋友访问她？
Q18
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 $CD$ at point $\left( \frac{p}{q}, \frac{r}{s} \right)$, where these fractions are in lowest terms. What is $p + q + r + s$?
设点$A = (0, 0)$、$B = (1, 2)$、$C = (3, 3)$和$D = (4, 0)$。四边形$ABCD$被一条经过$A$的直线切割成相等面积的部分。该直线与$CD$相交于点$\left( \frac{p}{q}, \frac{r}{s} \right)$，其中这些分数为最简形式。求$p + q + r + s$？
Q19
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 ?
在10进制下，数字2013 以数字3结尾。而在9进制下，同一个数字写成$(2676)_9$，以数字6结尾。有多少个正整数$b$使得2013在$b$进制表示以数字3结尾？
Q20
A unit square is rotated 45° about its center. What is the area of the region swept out by the interior of the square?
一个单位正方形绕其中心旋转45°。正方形内部扫过的区域面积是多少？
Q21
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 12th pirate receive?
一共有12名海盗，他们同意按照如下方式分配一箱金币。第$k$名海盗拿走剩余金币的$\frac{k}{12}$。箱子中最初的金币数是最小使得这种分配方式每个海盗都能得到正整数金币的数。第12名海盗得到多少金币？
Q22
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?
六个半径为1的球体，其中心位于边长为2的正六边形的顶点上。这六个球体都与一个较大的球体内切，较大球的中心是六边形的中心。还有一个第八个球体，与六个小球体外切，并与较大球体内切。这个第八个球体的半径是多少？
Q23
In $\triangle ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $BC$ at points $B$ and $X$. Moreover $BX$ and $CX$ have integer lengths. What is $BC$ ?
在$\triangle ABC$中，$AB=86$，$AC=97$。以$A$为圆心、$AB$为半径的圆与$BC$相交于点$B$和$X$。而且$BX$和$CX$为整数长度。$BC$是多少？
Q24
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?
中央高中与北方高中进行背gammon比赛。每校各有3名选手，比赛规则要求每位选手与对方学校每位选手各打两场比赛。比赛分6轮进行，每轮同时进行3场比赛。比赛可以以多少种不同的方式安排？
Q25
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?
在正八边形中画出全部20条对角线。八边形内部（不在边界上）有多少个不同的点使得两条或多条对角线相交？
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