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AMC10 2016 A

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AMC10 · 2016 (A)

Q1
What is the value of $\dfrac{11!-10!}{9!}$?
$\dfrac{11!-10!}{9!}$ 的值是多少？
Q2
For what value of $x$ does $10^x \cdot 100^{2x} = 1000^5$?
$10^x \cdot 100^{2x} = 1000^5$ 成立的 $x$ 的值为多少？
Q3
For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid $12.50 more than David. How much did they spend in the bagel store together?
Ben 每花一美元买百吉饼，David 就少花 25 美分。Ben 比 David 多付了 12.50 美元。他们在百吉饼店总共花了多少钱？
Q4
The remainder function can be defined for all real numbers $x$ and $y$ with $y\ne 0$ by $$\mathrm{rem}(x,y)=x-y\left\lfloor \frac{x}{y}\right\rfloor,$$ where $\left\lfloor \frac{x}{y}\right\rfloor$ denotes the greatest integer less than or equal to $\frac{x}{y}$. What is the value of $\mathrm{rem}\left(\frac{3}{8},-\frac{2}{5}\right)$?
余数函数对所有实数 $x$ 和 $y$（其中 $y\ne 0$）定义为 $$\mathrm{rem}(x,y)=x-y\left\lfloor \frac{x}{y}\right\rfloor,$$ 其中 $\left\lfloor \frac{x}{y}\right\rfloor$ 表示不超过 $\frac{x}{y}$ 的最大整数。求 $\mathrm{rem}\left(\frac{3}{8},-\frac{2}{5}\right)$ 的值。
Q5
A rectangular box has integer side lengths in the ratio $1:3:4$. Which of the following could be the volume of the box?
一个长方体的三条棱长为整数，且它们的比为$1:3:4$。下列哪一个可能是该长方体的体积？
Q6
Ximena lists the whole numbers 1 through 30 once. Emilio copies Ximena’s numbers, replacing each occurrence of the digit 2 by the digit 1. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena’s sum than Emilio’s?
Ximena 把从 1 到 30 的所有整数各写一次。Emilio 抄写 Ximena 的这些数，并把每一次出现的数字 2 都替换成数字 1。Ximena 计算她这些数的总和，Emilio 计算他这些数的总和。Ximena 的总和比 Emilio 的总和大多少？
Q7
For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
对于某个正整数 $n$，数 $110n^3$ 有 $110$ 个正整数因数（包括 $1$ 和 $110n^3$ 本身）。那么数 $81n^4$ 有多少个正整数因数？
Q8
Trickster Rabbit agrees with Foolish Fox to double Fox’s money every time Fox crosses the bridge by Rabbit’s house, as long as Fox pays 40 coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
狡猾的兔子与愚蠢的狐狸约定：狐狸每次经过兔子家旁的桥时，兔子都会把狐狸的钱翻倍，但条件是狐狸在每次过桥后要付给兔子 40 枚金币作为过桥费。付款发生在翻倍之后。狐狸为自己的好运而兴奋，直到他发现自己过桥三次后钱全部没了。狐狸最开始有多少枚金币？
Q9
A triangular array of 2016 coins has 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?
由2016枚硬币组成的三角形排列：第一行有1枚硬币，第二行有2枚，第三行有3枚，依此类推，直到第$N$行有$N$枚硬币。问$N$的各位数字之和是多少？
Q10
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle?
如图所示，一块地毯由三种不同颜色组成。三块不同颜色区域的面积成等差数列。内层矩形的宽为 1 英尺，且两层阴影区域在四个方向的宽度都为 1 英尺。问内层矩形的长是多少英尺？
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Q11
What is the area of the shaded region of the given $8\times 5$ rectangle?
给定的 $8\times 5$ 的长方形中，阴影部分的面积是多少？
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Q12
Three distinct integers are selected at random between 1 and 2016, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?
从 1 到 2016（含）之间随机选取三个互不相同的整数。以下哪一项关于这三个整数的乘积为奇数的概率 $p$ 的说法是正确的？
Q13
Five friends sat in a movie theater in a row containing 5 seats, numbered 1 to 5 from left to right. (The directions “left” and “right” are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
五位朋友在电影院同一排的 5 个座位上就坐，座位从左到右编号为 1 到 5。（“左”和“右”的方向以坐在座位上的人的视角为准。）电影进行中，Ada 去大厅买爆米花。她回来时发现：Bea 向右移动了两个座位，Ceci 向左移动了一个座位，Dee 和 Edie 交换了座位，从而给 Ada 留出了一个最边上的座位。问：Ada 起身之前坐在第几个座位上？
Q14
How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2+0\cdot 3$ and $402\cdot 2+404\cdot 3$ are two such ways.)
有多少种方法可以把 $2016$ 写成若干个 $2$ 和若干个 $3$ 的和（不考虑顺序）？（例如，$1008\cdot 2+0\cdot 3$ 和 $402\cdot 2+404\cdot 3$ 就是两种这样的表示。）
Q15
Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
如图所示，从一块圆形的饼干面团中切出 7 块半径为 1 英寸的圆形饼干。相邻的饼干互相相切，且除中心那块饼干外，其余饼干都与面团的外边缘相切。剩余的边角料被重新塑形，做成另一块厚度相同的饼干。问这块“边角料饼干”的半径是多少英寸？
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Q16
A triangle with vertices $A(0,2)$, $B(-3,2)$, and $C(-3,0)$ is reflected about the $x$-axis; then the image $\triangle A'B'C'$ is rotated counterclockwise around the origin by $90^\circ$ to produce $\triangle A''B''C''$. Which of the following transformations will return $\triangle A''B''C''$ to $\triangle ABC$?
一个三角形的顶点为 $A(0,2)$、$B(-3,2)$、$C(-3,0)$，先关于 $x$ 轴对称；然后将所得图形 $\triangle A'B'C'$ 围绕原点逆时针旋转 $90^\circ$，得到 $\triangle A''B''C''$。以下哪一种变换可以将 $\triangle A''B''C''$ 还原为 $\triangle ABC$？
Q17
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N)<\frac{321}{400}$?
设$N$为$5$的正倍数。将$1$个红球和$N$个绿球以随机顺序排成一列。令$P(N)$表示：至少有$\frac{3}{5}$的绿球位于红球同一侧的概率。注意到$P(5)=1$，并且当$N$趋于很大时，$P(N)$趋近于$\frac{4}{5}$。求使得$P(N)<\frac{321}{400}$的最小$N$的各位数字之和。
Q18
Each vertex of a cube is to be labeled with an integer from 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
立方体的每个顶点要标上从 1 到 8 的整数，每个整数恰好使用一次，并且要求立方体每个面的四个顶点上的数字之和对所有面都相同。通过旋转立方体可以互相得到的标号方式视为同一种。问一共有多少种不同的标号方式？
Q19
In rectangle $ABCD$, $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $AE$ and $AF$ intersect $BD$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r$, $s$, and $t$ is $1$. What is $r+s+t$?
在矩形 $ABCD$ 中，$AB=6$，$BC=3$。点 $E$ 在线段 $BC$ 上，点 $F$ 在线段 $EC$ 上，且满足 $BE=EF=FC$。线段 $AE$ 与 $AF$ 分别与对角线 $BD$ 相交于 $P$ 和 $Q$。比值 $BP:PQ:QD$ 可写为 $r:s:t$，其中 $r,s,t$ 的最大公因数为 $1$。求 $r+s+t$。
Q20
For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables, $a$, $b$, $c$, and $d$, each to some positive power. What is $N$?
对于某个特定的 $N$，当 $(a+b+c+d+1)^N$ 展开并合并同类项后，所得表达式恰好包含 $1001$ 项，这些项都同时含有四个变量 $a,b,c,d$，且它们各自的指数都为正整数。求 $N$。
Q21
Circles with centers $P$, $Q$, and $R$, having radii $1$, $2$, and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P'$, $Q'$, and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of $\triangle PQR$?
半径分别为 $1$、$2$、$3$ 的三个圆，其圆心分别为 $P$、$Q$、$R$，它们位于直线 $l$ 的同一侧，并分别在 $P'$、$Q'$、$R'$ 处与直线 $l$ 相切，其中 $Q'$ 位于 $P'$ 与 $R'$ 之间。以 $Q$ 为圆心的圆与另外两个圆都外切。求 $\triangle PQR$ 的面积。
Q22
For some positive integer $n$, the number $110n^3$ has 110 positive integer divisors, including 1 and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?
对于某个正整数 $n$，数 $110n^3$ 有 110 个正整数因数（包括 1 和 $110n^3$ 本身）。那么数 $81n^4$ 有多少个正整数因数？
Q23
A binary operation $\diamond$ has the properties that $a\diamond(b\diamond c)=(a\diamond b)\cdot c$ and that $a\diamond a=1$ for all nonzero real numbers $a$, $b$, and $c$. (Here the dot $\cdot$ represents the usual multiplication operation.) The solution to the equation $2016\diamond(6\diamond x)=100$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?
一种二元运算 $\diamond$ 满足性质：对所有非零实数 $a,b,c$，有 $a\diamond(b\diamond c)=(a\diamond b)\cdot c$，且 $a\diamond a=1$。（这里的点号 $\cdot$ 表示通常的乘法运算。）方程 $2016\diamond(6\diamond x)=100$ 的解可写成 $\frac{p}{q}$，其中 $p$ 和 $q$ 是互素的正整数。求 $p+q$。
Q24
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of its fourth side?
一个四边形内接于半径为 $200\sqrt{2}$ 的圆中。该四边形的三条边长为 $200$。它的第四条边长是多少？
Q25
How many ordered triples $(x,y,z)$ of positive integers satisfy $\mathrm{lcm}(x,y)=72$, $\mathrm{lcm}(x,z)=600$, and $\mathrm{lcm}(y,z)=900$?
有多少个正整数有序三元组 $(x,y,z)$ 满足 $\mathrm{lcm}(x,y)=72$，$\mathrm{lcm}(x,z)=600$，以及 $\mathrm{lcm}(y,z)=900$？
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