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AMC12 2001 A

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AMC12 · 2001 (A)

Q1
The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
两个数的和是 $S$。假设每个数都加上 $3$，然后将得到的结果每个都乘以 $2$。最后两个数的和是多少？
Q2
Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N)+S(N)$. What is the units digit of $N$?
设 $P(n)$ 和 $S(n)$ 分别表示整数 $n$ 的各位数字的积与和。例如，$P(23) = 6$ 且 $S(23) = 5$。假设 $N$ 是一个两位数，满足 $N = P(N)+S(N)$。$N$ 的个位数字是多少？
Q3
The state income tax where Kristin lives is charged at the rate of $p\%$ of the first $\$28000$ of annual income plus $(p + 2)\%$ of any amount above $\$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?
Kristin 所在州的所得税按如下方式征收：年收入的前 $\$28000$ 部分按 $p\%$ 征税，超过 $\$28000$ 的部分按 $(p + 2)\%$ 征税。Kristin 注意到她缴纳的州所得税相当于她年收入的 $(p + 0.25)\%$。她的年收入是多少？
Q4
The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum?
三个数的平均数比其中最小的数大 $10$，且比其中最大的数小 $15$。这三个数的中位数是 $5$。它们的和是多少？
Q5
What is the product of all positive odd integers less than $10000$?
所有小于 $10000$ 的正奇整数的乘积是多少？
Q6
A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$. Find $A$.
电话号码的形式是 $\text{ABC-DEF-GHIJ}$，其中每个字母代表不同的数字。号码每个部分的数字是递减的，即 $A > B > C$，$D > E > F$，且 $G > H > I > J$。此外，$D$、$E$、$F$ 是连续的偶数数字；$G$、$H$、$I$、$J$ 是连续的奇数数字；并且 $A + B + C = 9$。求 $A$ 的值。
Q7
A charity sells 140 benefit tickets for a total of \$2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?
一个慈善机构出售了140张募捐票，总收入为\$2001。有些票按全价出售（票价为整数美元），其余的票按半价出售。全价票筹集了多少钱？
Q8
Which of the cones listed below can be formed from a $252^\circ$ sector of a circle of radius $10$ by aligning the two straight sides?
下面哪个圆锥可以由半径为 $10$ 的圆的 $252^\circ$ 扇形通过对齐两条直边形成？
stem
Q9
Let $f$ be a function satisfying $f(xy) = \frac{f(x)}y$ for all positive real numbers $x$ and $y$. If $f(500) =3$, what is the value of $f(600)$?
设 $f$ 是一个满足 $f(xy) = \frac{f(x)}y$ 对于所有正实数 $x$ 和 $y$ 的函数。若 $f(500) =3$，则 $f(600)$ 的值为多少？
Q10
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
平面由全等的正方形和全等的五边形铺满，如图所示。五边形所覆盖的平面百分比最接近
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Q11
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
一个盒子中正好有五个筹码，三个红色、两个白色。随机逐个无放回取出筹码，直到所有红色筹码都被取出或所有白色筹码都被取出为止。最后一个取出的筹码是白色的概率是多少？
Q12
How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$?
不超过 $2001$ 的正整数中，有多少个数是 $3$ 或 $4$ 的倍数但不是 $5$ 的倍数？
Q13
The parabola with equation $p(x) = ax^2+bx+c$ and vertex $(h,k)$ is reflected about the line $y=k$. This results in the parabola with equation $q(x) = dx^2+ex+f$. Which of the following equals $a+b+c+d+e+f$?
方程为 $p(x) = ax^2+bx+c$、顶点为 $(h,k)$ 的抛物线关于直线 $y=k$ 反射，得到方程为 $q(x) = dx^2+ex+f$ 的抛物线。以下哪一项等于 $a+b+c+d+e+f$？
Q14
Given the nine-sided regular polygon $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1,A_2,\dots,A_9\}$?
给定正九边形 $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$，在该多边形所在平面内，有多少个不同的等边三角形至少有两个顶点属于集合 $\{A_1,A_2,\dots,A_9\}$？
Q15
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
一只昆虫生活在棱长为 $1$ 的正四面体表面上。它希望沿四面体表面从一条棱的中点走到与之相对的棱的中点。这样的最短路径长度是多少？（注：四面体的两条棱若没有公共端点，则称为相对棱。）
Q16
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
一只蜘蛛为它的八条腿各有一只袜子和一只鞋。蜘蛛穿上袜子和鞋的不同顺序有多少种，假设每条腿上袜子必须在鞋子之前穿上？
Q17
A point $P$ is selected at random from the interior of the pentagon with vertices $A = (0,2)$, $B = (4,0)$, $C = (2 \pi + 1, 0)$, $D = (2 \pi + 1,4)$, and $E=(0,4)$. What is the probability that $\angle APB$ is obtuse?
从五边形内部随机选取一点 $P$，五边形的顶点为 $A = (0,2)$，$B = (4,0)$，$C = (2 \pi + 1, 0)$，$D = (2 \pi + 1,4)$，以及 $E=(0,4)$。$\angle APB$ 为钝角的概率是多少？
Q18
A circle centered at $A$ with a radius of 1 and a circle centered at $B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle?
以 $A$ 为圆心半径为 1 的圆与以 $B$ 为圆心半径为 4 的圆外切。第三个圆与前两个圆以及它们的一条公共外公切线相切，如图所示。第三个圆的半径是多少？
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Q19
The polynomial $p(x) = x^3+ax^2+bx+c$ has the property that the average of its zeros, the product of its zeros, and the sum of its coefficients are all equal. The $y$-intercept of the graph of $y=p(x)$ is 2. What is $b$?
多项式 $p(x) = x^3+ax^2+bx+c$ 具有如下性质：它的零点的平均值、零点的乘积以及系数的和都相等。函数 $y=p(x)$ 的图像的 $y$ 轴截距为 2。求 $b$。
Q20
Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$?
点 $A = (3,9)$，$B = (1,1)$，$C = (5,3)$，以及 $D=(a,b)$ 位于第一象限，并且是四边形 $ABCD$ 的顶点。连接 $\overline{AB}$、$\overline{BC}$、$\overline{CD}$ 和 $\overline{DA}$ 的中点所形成的四边形是一个正方形。点 $D$ 的坐标之和是多少？
Q21
Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy: \[\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}\] What is $a-d$?
四个正整数 $a$, $b$, $c$, 和 $d$ 的乘积为 $8!$，且满足： \[\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}\] 求 $a-d$。
Q22
In rectangle $ABCD$, points $F$ and $G$ lie on $AB$ so that $AF=FG=GB$ and $E$ is the midpoint of $\overline{DC}$. Also, $\overline{AC}$ intersects $\overline{EF}$ at $H$ and $\overline{EG}$ at $J$. The area of the rectangle $ABCD$ is $70$. Find the area of triangle $EHJ$.
在矩形 $ABCD$ 中，点 $F$ 和 $G$ 在 $AB$ 上，使得 $AF=FG=GB$，且 $E$ 是 $\overline{DC}$ 的中点。另有 $\overline{AC}$ 与 $\overline{EF}$ 交于 $H$，与 $\overline{EG}$ 交于 $J$。矩形 $ABCD$ 的面积为 $70$。求三角形 $EHJ$ 的面积。
Q23
A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial?
一个四次多项式的首项系数为 1，且系数均为整数。它有两个零点，并且这两个零点都是整数。下列哪一个也可能是该多项式的零点？
Q24
In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB.$
在 $\triangle ABC$ 中，$\angle ABC=45^\circ$。点 $D$ 在 $\overline{BC}$ 上，使得 $2\cdot BD=CD$，且 $\angle DAB=15^\circ$。求 $\angle ACB$。
Q25
Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?
考虑形如 $x, 2000, y, \dots$ 的正实数序列，其中从第二项起，每一项都等于其相邻两项的乘积减 1。问：有多少个不同的 $x$ 值，使得数 $2001$ 会在该序列的某一项中出现？
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