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AMC12 2007 A

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AMC12 · 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?
一场演出的全价门票价格为$\$20$。Susan使用一张可享受25%折扣的优惠券购买了4张票。Pam使用一张可享受30%折扣的优惠券购买了5张票。Pam比Susan多付了多少美元？
Q2
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?
一个水族箱的底面是$100\text{ cm}\times 40\text{ cm}$的矩形，高$50\text{ cm}$。其中注水至$40\text{ cm}$高。将一块底面是$40\text{ cm}\times 20\text{ cm}$的矩形、高$10\text{ cm}$的砖块放入水族箱中。水位上升了多少厘米？
Q3
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？
Q4
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?
Kate先以$16$ mph的速度骑自行车$30$分钟，然后以$4$ mph的速度步行$90$分钟。她的总体平均速度是多少（单位：英里/小时）？
Q5
Last year Mr. Jon 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 $\$10500$ for both taxes. How many dollars was his inheritance?
去年，Mr. Jon Q. Public收到一笔遗产。他为这笔遗产缴纳了$20\%$的联邦税，并对剩余金额缴纳了$10\%$的州税。他两项税共缴纳了 $\$10500$。他的遗产是多少美元？
Q6
Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside triangle $ABC$, angle $ABC$ measures $40$ degrees, and angle $ADC$ measures $140$ degrees. What is the degree measure of angle $BAD$?
等腰三角形 $ABC$ 和 $ADC$ 满足 $AB=BC$ 且 $AD=DC$。点 $D$ 在三角形 $ABC$ 内部，$\angle ABC$ 的度数为 $40$，且 $\angle ADC$ 的度数为 $140$。$\angle BAD$ 的度数是多少？
Q7
Let $a, b, c, d$, and $e$ be five consecutive terms in an arithmetic sequence, and suppose that $a+b+c+d+e=30$. Which of $a, b, c, d,$ or $e$ can be found?
设 $a, b, c, d,$ 和 $e$ 为等差数列中连续的五项，并且 $a+b+c+d+e=30$。在 $a, b, c, d,$ 或 $e$ 中，哪一个可以被确定？
Q8
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?
在钟面上通过从每个数字向顺时针方向数的第五个数字画弦来绘制一个星形多边形。也就是说，画从 12 到 5 的弦，从 5 到 10 的弦，从 10 到 3 的弦，依此类推，最后回到 12。该星形多边形每个顶点处的角的度数是多少？
Q9
Yan 在家和体育场之间的某处。要去体育场，他可以直接步行到体育场，或者先步行回家再骑自行车去体育场。他骑车的速度是他步行速度的 7 倍，并且两种选择所用时间相同。Yan 离家的距离与他离体育场的距离之比是多少？
Q10
A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with 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。该三角形的面积是多少？
Q11
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. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?
一个有限的三位整数序列具有如下性质：每一项的十位和个位数字分别是下一项的百位和十位数字，而最后一项的十位和个位数字分别是第一项的百位和十位数字。例如，这样的序列可能以 247、475、756 开头，并以 824 结尾。设 $S$ 为该序列所有项之和。问：总能整除 $S$ 的最大素因数是多少？
Q12
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?
从 0 到 2007（含端点）中独立且随机选取整数 $a, b, c,$ 和 $d$（不一定互不相同）。$ad-bc$ 为偶数的概率是多少？
Q13
A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a+b$?
坐标平面上，一块奶酪位于 $(12,10)$。一只老鼠在 $(4,-2)$，沿直线 $y=-5x+18$ 向上跑。在点 $(a,b)$ 处，老鼠开始离奶酪越来越远而不是越来越近。求 $a+b$。
Q14
Let $a$, $b$, $c$, $d$, and $e$ be distinct integers such that $(6-a)(6-b)(6-c)(6-d)(6-e)=45$ What is $a+b+c+d+e$?
设 $a$, $b$, $c$, $d$, 和 $e$ 为互不相同的整数，使得 $(6-a)(6-b)(6-c)(6-d)(6-e)=45$ 求 $a+b+c+d+e$。
Q15
The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$?
集合 $\{3,6,9,10\}$ 增加第五个元素 $n$，且 $n$ 不等于另外四个数。所得集合的中位数等于其平均数。求所有可能的 $n$ 的值之和。
Q16
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
有多少个三位数由三个不同数字组成，使得其中一个数字是其他两个的平均数？
Q17
Suppose that $\sin a + \sin b = \sqrt{\frac{5}{3}}$ and $\cos a + \cos b = 1$. What is $\cos (a - b)$?
假设 $\sin a + \sin b = \sqrt{\frac{5}{3}}$ 且 $\cos a + \cos b = 1$。求 $\cos (a - b)$？
Q18
The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ What is $a + b + c + d?$
多项式 $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ 有实系数，且 $f(2i) = f(2 + i) = 0.$ 求 $a + b + c + d$？
Q19
Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B = (0,0),$ $C = (223,0),$ $D = (680,380),$ and $E = (689,389).$ What is the sum of all possible x coordinates of $A$?
三角形 $ABC$ 和 $ADE$ 的面积分别为 $2007$ 和 $7002,$ 其中 $B = (0,0),$ $C = (223,0),$ $D = (680,380),$ 且 $E = (689,389).$ 求点 $A$ 所有可能的 $x$ 坐标之和。
Q20
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
从一个单位立方体的每个顶点切掉小角，使得六个面都变成正八边形。被切掉的四面体总体积是多少？
Q21
The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $f(x)=ax^{2}+bx+c$ are equal. Their common value must also be which of the following?
函数 $f(x)=ax^{2}+bx+c$ 的零点的和、零点的乘积以及系数的和相等。它们的共同值还必须是下列哪一项？
Q22
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 $n + S(n) + S(S(n)) = 2007?$
Q23
Square $ABCD$ has area $36,$ and $\overline{AB}$ is parallel to the x-axis. Vertices $A,$ $B$, and $C$ are on the graphs of $y = \log_{a}x,$ $y = 2\log_{a}x,$ and $y = 3\log_{a}x,$ respectively. What is $a?$
正方形 $ABCD$ 的面积为 $36,$ 且 $\overline{AB}$ 平行于 $x$ 轴。顶点 $A,$ $B$, 和 $C$ 分别在 $y = \log_{a}x,$ $y = 2\log_{a}x,$ 和 $y = 3\log_{a}x,$ 的图像上。求 $a$ 的值。
Q24
For each integer $n>1$, let $F(n)$ be the number of solutions to the equation $\sin{x}=\sin{(nx)}$ on the interval $[0,\pi]$. What is $\sum_{n=2}^{2007} F(n)$?
对于每个整数 $n>1$，令 $F(n)$ 表示方程 $\sin{x}=\sin{(nx)}$ 在区间 $[0,\pi]$ 上的解的个数。求 $\sum_{n=2}^{2007} F(n)$。
Q25
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\ldots,12\},$ including the empty set, are spacy?
称一个整数集合为 spacy，如果它在任意三个连续整数中至多包含一个整数。$\{1,2,3,\ldots,12\}$ 的子集中（包括空集）有多少个是 spacy 的？
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