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AMC10 2007 A

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AMC10 · 2007 (A)

Q1
One ticket to a show costs $20$ at full price. Susan buys $4$ tickets using a coupon that gives her a $25\%$ discount. Pam buys $5$ tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan?
一张演出票全价为$20$美元。Susan 使用一张提供$25\%$折扣的优惠券买了$4$张票。Pam 使用一张提供$30\%$折扣的优惠券买了$5$张票。Pam 比 Susan 多付了多少美元？
Q2
Define $a@b = ab - b^2$ and $a\#b = a + b - ab^2$. What is $6@2 \div 6\#2$?
定义 $a@b = ab - b^2$ 和 $a\#b = a + b - ab^2$。求 $6@2 \div 6\#2$ 的值？
Q3
An aquarium has a rectangular base that measures $100$ cm by $40$ cm and has a height of $50$ cm. It is filled with water to a height of $40$ cm. A brick with a rectangular base that measures $40$ cm by $20$ cm and a height of $10$ cm is placed in the aquarium. By how many centimeters does the water rise?
一个水族箱底部是长 $100$ 厘米、宽 $40$ 厘米的矩形，高 $50$ 厘米。水族箱中注水至 $40$ 厘米高。将一块底部是 $40$ 厘米 $\times$ $20$ 厘米、高 $10$ 厘米的长方体砖放入水族箱中。水位上升多少厘米？
Q4
The larger of two consecutive odd integers is three times the smaller. What is their sum?
两个连续奇整数中较大的一个是较小的一个的三倍。它们的和是多少？
Q5
A school store sells $7$ pencils and $8$ notebooks for $\$4.15$. It also sells $5$ pencils and $3$ notebooks for $\$1.77$. How much do $16$ pencils and $10$ notebooks cost?
$7$支铅笔和 $8$本笔记本售价 $\$4.15$。$5$支铅笔和 $3$本笔记本售价 $\$1.77$。$16$支铅笔和 $10$本笔记本售价多少？
Q6
At Euclid High School, the number of students taking the AMC10 was $60$ in $2002$, $66$ in $2003$, $70$ in $2004$, $76$ in $2005$, and $78$ in $2006$, and is $85$ in $2007$. Between what two consecutive years was there the largest percentage increase?
在Euclid高中，参加AMC10的学生人数2002年为$60$人，2003年为$66$人，2004年为$70$人，2005年为$76$人，2006年为$78$人，2007年为$85$人。哪两个连续年份之间的百分比增长最大？
Q7
Last year Mr. John Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\$10,500$ for both taxes. How many dollars was the inheritance?
去年，John Q. Public先生收到一笔遗产。他支付了20\%的联邦税，然后对他剩下的钱支付了10\%的州税。两项税款总共支付了$\$$10,500。他继承的遗产有多少美元？
Q8
Triangles $ABC$ and $ADC$ are isosceles with $AB = BC$ and $AD = DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 40^\circ$, and $\angle ADC = 140^\circ$. What is the degree measure of $\angle BAD$?
等腰三角形$ABC$和$ADC$满足$AB = BC$和$AD = DC$。点$D$在$\triangle ABC$内部，$\angle ABC = 40^\circ$，$\angle ADC = 140^\circ$。$\angle BAD$的度数是多少？
Q9
Real numbers $a$ and $b$ satisfy the equations $3^a = 81^{b+2}$ and $125^b = 5^{a-3}$. What is $ab$?
实数$a$和$b$满足方程$3^a = 81^{b+2}$和$125^b = 5^{a-3}$。$ab$等于多少？
Q10
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family?
Dunbar一家有母亲、父亲和一些孩子。全家成员的平均年龄是$20$岁，父亲$48$岁，母亲和孩子们的平均年龄是$16$岁。家里有多少孩子？
Q11
The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
将数字 $1$ 到 $8$ 放置在立方体的顶点上，使得每个面上的四个数字之和相同。这个公共和是多少？
Q12
Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?
有两个导游带领六名游客。导游决定分开。每个游客必须选择一个导游，但规定每个导游至少带一个游客。有多少种不同的导游和游客分组方式？
Q13
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
Yan 在家和体育场之间某处。为了去体育场，他可以直接走路去体育场，或者先走回家然后骑自行车去体育场。他骑车速度是他走路速度的 $7$ 倍，且两种选择所需时间相同。Yan 离家的距离与他离体育场的距离之比是多少？
Q14
A triangle with side lengths in the ratio $3:4:5$ is inscribed in a circle of radius $3$. What is the area of the triangle?
一个边长比为 $3:4:5$ 的三角形内接于半径为 $3$ 的圆中。这个三角形的面积是多少？
Q15
Four circles of radius $1$ are each tangent to two sides of a square and externally tangent to a circle of radius $2$, as shown. What is the area of the square?
四个半径为 $1$ 的圆，每个都与正方形的两条边相切，并与半径为 $2$ 的圆外切，如图所示。正方形的面积是多少？
stem
Q16
Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2007$, inclusive. What is the probability that $ad - bc$ is even?
整数 $a, b, c,$ 和 $d$（不一定互异）独立且随机地从 $0$ 到 $2007$（包含端点）中选取。$ad - bc$ 为偶数的概率是多少？
Q17
Suppose that $m$ and $n$ are positive integers such that $75m = n^3$. What is the minimum possible value of $m + n$?
假设 $m$ 和 $n$ 是正整数使得 $75m = n^3$。$m + n$ 的最小可能值是多少？
Q18
Consider the $12$-sided polygon $ABCDEFHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $AG$ and $CH$ meet at $M$. What is the area of quadrilateral $ABCM$?
考虑如图所示的 12 边形 $ABCDEFHIJKL$。它的每条边长均为 $4$，且每两条连续边形成直角。假设 $AG$ 和 $CH$ 相交于 $M$。四边形 $ABCM$ 的面积是多少？
stem
Q19
A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?
画笔沿正方形的两条对角线扫过，产生如图所示的对称涂漆区域。正方形的一半面积被涂漆。求正方形边长与画笔宽度的比值？
stem
Q20
Suppose that the number $a$ satisfies the equation $4 = a + a^{-1}$. What is the value of $a^4 + a^{-4}$?
假设数 $a$ 满足方程 $4 = a + a^{-1}$。$a^4 + a^{-4}$ 的值是多少？
Q21
A sphere is inscribed in a cube that has a surface area of $24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?
一个球体内切于一个表面积为$24$平方米的立方体。然后，一个第二立方体内切于该球体。内立方体的表面积是多少平方米？
Q22
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. ... Let $S$ be the sum of all the terms in the sequence. What is the largest prime number that always divides $S$?
一个有限的三位整数序列具有如下性质：每个项的十位和个位数字分别作为下一项的百位和十位数字，最后一项的十位和个位数字分别作为第一项的百位和十位数字。……令$S$为序列中所有项之和。总是整除$S$的最大素数是多少？
Q23
How many ordered pairs $(m, n)$ of positive integers, with $m > n$, have the property that their squares differ by $96$?
有多少对正整数有序对$(m, n)$，满足$m > n$，且它们的平方差为$96$？
Q24
Circles centered at $A$ and $B$ each have radius $2$, as shown. Point $O$ is the midpoint of $AB$, and $OA = 2\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$?
以$A$和$B$为圆心、半径均为$2$的圆，如图所示。点$O$是$AB$的中点，且$OA = 2\sqrt{2}$。线段$OC$和$OD$分别切于以$A$和$B$为圆心的圆，且$EF$是公共切线。阴影区域$ECODF$的面积是多少？
stem
Q25
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n + S(n) + S(S(n)) = 2007$?
对于每个正整数$n$，令$S(n)$表示$n$的各位数字之和。有多少个$n$满足$n + S(n) + S(S(n)) = 2007$？
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