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AMC12 2016 B

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AMC12 · 2016 (B)

Q1
What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a=\frac{1}{2}$?
当$a=\frac{1}{2}$时，$\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$的值是多少？
Q2
The harmonic mean of two numbers can be computed as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?
两个数的调和平均数可以用“它们的积的两倍除以它们的和”来计算。$1$ 和 $2016$ 的调和平均数最接近哪个整数？
Q3
Let $x=-2016$. What is the value of $\left|\left||x|-x\right|-|x|\right|-x$?
设$x=-2016$。求$\left|\left||x|-x\right|-|x|\right|-x$的值。
Q4
The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
两个锐角的度数之比为$5:4$，并且这两个角中其中一个角的余角是另一个角的余角的两倍。求这两个角的度数之和。
Q5
The War of 1812 started with a declaration of war on Thursday, June 18, 1812. The peace treaty to end the war was signed 919 days later, on December 24, 1814. On what day of the week was the treaty signed?
1812年战争于1812年6月18日（星期四）通过宣战开始。结束战争的和平条约在919天后签署，即1814年12月24日。该条约是在星期几签署的？
Q6
All three vertices of $\triangle ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $BC$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length $BC$?
$\triangle ABC$ 的三个顶点都在抛物线 $y=x^2$ 上，点 $A$ 在原点，且 $BC$ 平行于 $x$ 轴。该三角形的面积为 $64$。求线段 $BC$ 的长度。
Q7
Josh writes the numbers 1, 2, 3, $\ldots$, 99, 100. He marks out 1, skips the next number (2), marks out 3, and continues skipping and marking out the next number to the end of his list. Then he goes back to the start of his list, marks out the first remaining number (2), skips the next number (4), marks out 6, skips 8, marks out 10, and so on to the end. Josh continues in this manner until only one number remains. What is that number?
Josh 写下数字 1, 2, 3, $\ldots$, 99, 100。他划去 1，跳过下一个数（2），划去 3，并继续按“划去一个、跳过一个”的方式直到列表末尾。然后他回到列表开头，划去剩下的第一个数（2），跳过下一个数（4），划去 6，跳过 8，划去 10，如此一直到末尾。Josh 按这种方式不断重复，直到只剩下一个数。这个数是多少？
Q8
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length 5 inches. Which of the following is closest to the weight, in ounces, of the second piece?
一块密度均匀的薄木片，形状为边长为 3 英寸的正三角形，重 12 盎司。第二块同种木材（厚度相同）的薄木片也为正三角形，边长为 5 英寸。下列哪一项最接近第二块木片的重量（单位：盎司）？
Q9
Carl decided to fence in his rectangular garden. He bought 20 fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly 4 yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl's garden?
卡尔决定把他的长方形花园围起来。他买了 20 根栅栏桩，在四个角各放了一根，其余的均匀地沿着花园边缘放置，使相邻两根桩之间的距离恰好为 4 码。花园的长边（包含两端角上的桩）所包含的桩数是短边（包含两端角上的桩）桩数的两倍。卡尔的花园面积是多少（单位：平方码）？
Q10
A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a,-b)$, and $S(-b,-a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?
一个四边形的顶点为 $P(a,b)$、$Q(b,a)$、$R(-a,-b)$ 和 $S(-b,-a)$，其中 $a$ 和 $b$ 是整数且满足 $a>b>0$。四边形 $PQRS$ 的面积为 $16$。求 $a+b$。
Q11
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$, and the line $x=5.1$?
有多少个边与坐标轴平行、且顶点坐标都是整数的正方形，能够完全位于由直线 $y=\pi x$、直线 $y=-0.1$ 以及直线 $x=5.1$ 所围成的区域内？
Q12
All the numbers $1,2,3,4,5,6,7,8,9$ are written in a $3 \times 3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What number is in the center?
把所有数字 $1,2,3,4,5,6,7,8,9$ 写在一个 $3 \times 3$ 的方格阵列中，每个方格里写一个数，并且满足：如果两个数字是相邻的连续整数，那么它们所在的方格必须有一条公共边相邻。四个角上的数字之和为 $18$。问：中心格里是什么数字？
Q13
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice’s position and $60^\circ$ from Bob’s position. Which of the following is closest to the airplane’s altitude, in miles?
爱丽丝和鲍勃相距 10 英里。一天，爱丽丝从家中向正北方向看，看到一架飞机。与此同时，鲍勃从家中向正西方向看，也看到了同一架飞机。从爱丽丝的位置看，飞机的仰角是 $30^\circ$；从鲍勃的位置看，飞机的仰角是 $60^\circ$。下列哪个选项最接近飞机的高度（单位：英里）？
Q14
The sum of an infinite geometric series is a positive number $S$, and the second term in the series is 1. What is the smallest possible value of $S$?
一个无穷等比数列的和是一个正数 $S$，且该数列的第二项为 1。问 $S$ 的最小可能值是多少？
Q15
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
将数字 2、3、4、5、6、7 分别分配到一个立方体的六个面上，每个面一个数字。对于立方体的每一个顶点（共 8 个），计算一个三个数字的乘积，这三个数字分别是包含该顶点的三个面的数字。求这 8 个乘积之和的最大可能值。
Q16
In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?
345 可以用多少种方式表示为两个或更多个连续正整数的递增序列之和？
Q17
In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $AH$ is an altitude. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, so that $BD$ and $CE$ are angle bisectors, intersecting $AH$ at $Q$ and $P$, respectively. What is $PQ$?
在图示的 $\triangle ABC$ 中，$AB=7$，$BC=8$，$CA=9$，且 $AH$ 为高。点 $D$ 和 $E$ 分别位于边 $AC$ 和 $AB$ 上，使得 $BD$ 与 $CE$ 为角平分线，分别与 $AH$ 交于 $Q$ 和 $P$。求 $PQ$ 的长度。
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Q18
What is the area of the region enclosed by the graph of the equation $x^2 + y^2 = |x| + |y|$?
由方程 $x^2 + y^2 = |x| + |y|$ 的图像所围成的区域面积是多少？
Q19
Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?
汤姆、迪克和哈里在玩一个游戏。他们同时开始，每个人反复掷一枚公平硬币，直到第一次掷出正面为止，然后停止。问三个人掷硬币的次数都相同的概率是多少？
Q20
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won 10 games and lost 10 games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$?
一组队伍进行了循环赛，每支队伍与其他每支队伍都恰好比赛一次。每支队伍赢了 10 场、输了 10 场；没有平局。问有多少组三支队伍的集合 $\{A, B, C\}$ 满足：$A$ 战胜 $B$，$B$ 战胜 $C$，且 $C$ 战胜 $A$？
Q21
Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\ldots$, let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is $\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_iP_i$?
设$ABCD$为单位正方形。设$Q_1$为线段$\overline{CD}$的中点。对$i=1,2,\ldots$，令$P_i$为$\overline{AQ_i}$与$\overline{BD}$的交点，并令$Q_{i+1}$为从$P_i$向$\overline{CD}$作垂线的垂足。求 $\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_iP_i$。
Q22
For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?
对于某个小于 $1000$ 的正整数 $n$，$\frac{1}{n}$ 的小数表示为 $0.\overline{abcdef}$，这是一个循环节为 $6$ 的循环小数；而 $\frac{1}{n+6}$ 的小数表示为 $0.\overline{wxyz}$，这是一个循环节为 $4$ 的循环小数。问：$n$ 位于哪个区间内？
Q23
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le 1$ and $|x|+|y|+|z-1|\le 1$?
由不等式 $|x|+|y|+|z|\le 1$ 和 $|x|+|y|+|z-1|\le 1$ 所定义的三维空间区域的体积是多少？
Q24
There are exactly $77{,}000$ ordered quadruples $(a,b,c,d)$ such that $\gcd(a,b,c,d)=77$ and $\operatorname{lcm}(a,b,c,d)=n$. What is the smallest possible value of $n$?
恰好有 $77{,}000$ 个有序四元组 $(a,b,c,d)$ 满足 $\gcd(a,b,c,d)=77$ 且 $\operatorname{lcm}(a,b,c,d)=n$。问 $n$ 的最小可能值是多少？
Q25
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\ge 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer?
数列$(a_n)$按递推方式定义：$a_0=1$，$a_1=\sqrt[19]{2}$，并且当$n\ge 2$时，$a_n=a_{n-1}a_{n-2}^2$。求使得乘积$a_1a_2\cdots a_k$为整数的最小正整数$k$。
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