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AMC10 2016 B

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AMC10 · 2016 (B)

Q1
What is the value of $\dfrac{2a^{-1}+\dfrac{a^{-1}}{2}}{a}$ when $a=\dfrac{1}{2}$?
当 $a=\dfrac{1}{2}$ 时，$\dfrac{2a^{-1}+\dfrac{a^{-1}}{2}}{a}$ 的值是多少？
Q2
If $n \heartsuit m = n^3 m^2$, what is $\dfrac{2 \heartsuit 4}{4 \heartsuit 2}$?
如果 $n \heartsuit m = n^3 m^2$，那么 $\dfrac{2 \heartsuit 4}{4 \heartsuit 2}$ 的值是多少？
Q3
Let $x=-2016$. What is the value of $||x|-x|-|x|-x$?
设 $x=-2016$。求 $||x|-x|-|x|-x$ 的值。
Q4
Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each book taking her 1 more day to read than the previous book. Zoey finished the first book on a Monday and the second on a Wednesday. On what day of the week did she finish her 15th book?
佐伊一次读一本书，共读了15本。第一本花了她1天读完，第二本花了她2天读完，第三本花了她3天读完，依此类推，每本书比前一本多花1天。佐伊在星期一读完第一本，在星期三读完第二本。她读完第15本是在星期几？
Q5
The mean age of Amanda’s 4 cousins is 8, and their median age is 5. What is the sum of the ages of Amanda’s youngest and oldest cousins?
Amanda 的 4 个表兄弟姐妹的平均年龄是 8，他们的年龄中位数是 5。Amanda 最小和最大的表兄弟姐妹的年龄之和是多少？
Q6
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$?
劳拉把两个三位正整数相加。这两个数的六个数字都互不相同。劳拉得到的和是一个三位数 $S$。求 $S$ 的各位数字之和的最小可能值。
Q7
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 $5:4$，并且其中一个角的余角是另一个角余角的两倍。求这两个角的度数之和。
Q8
What is the tens digit of $2015^{2016}-2017$?
$2015^{2016}-2017$ 的十位数字是多少？
Q9
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。
Q10
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板，形状为边长为 3 英寸的等边三角形，重量为 12 盎司。另一块同种木材且厚度相同的薄木板，同样为等边三角形，边长为 5 英寸。下列选项中，哪一个最接近第二块木板的重量（单位：盎司）？
Q11
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?
Carl 决定把他的长方形花园围起来。他买了 20 根栅栏桩，在四个角各放一根，其余的沿着花园边缘均匀放置，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）上的桩数是短边（包含两端角上的桩）桩数的两倍。Carl 的花园面积是多少（单位：平方码）？
Q12
Two different numbers are selected at random from $\{1,2,3,4,5\}$ and multiplied together. What is the probability that the product is even?
从 $\{1,2,3,4,5\}$ 中随机选取两个不同的数并将它们相乘。乘积为偶数的概率是多少？
Q13
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?
在大都会医院的某一年，多胎出生的统计如下：双胞胎、三胞胎和四胞胎这些组合共占当年出生婴儿中的1000名。三胞胎的组数是四胞胎组数的4倍，双胞胎的组数是三胞胎组数的3倍。在这1000名婴儿中，有多少名属于四胞胎组合？
Q14
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、且顶点坐标均为整数的正方形，能够完全位于由直线 $y=\pi x$、直线 $y=-0.1$ 和直线 $x=5.1$ 所围成的区域内？
Q15
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?
把数字 1、2、3、4、5、6、7、8、9 全部填入一个 $3 \times 3$ 的方格阵列中，每个小方格填一个数，并且要求：如果两个数相邻（差为 1），那么它们所在的方格必须共享一条边。四个角上的数字之和为 18。问：中心格里的数字是多少？
Q16
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？
Q17
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 $2,3,4,5,6,7$ 分别分配到正方体的六个面上，每个面一个数字。对于正方体的 $8$ 个顶点中的每一个，计算一个由三个数字相乘得到的乘积，其中这三个数字是与该顶点相邻的三个面上所分配的数字。问这 $8$ 个乘积之和的最大可能值是多少？
Q18
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构成的递增序列之和？
Q19
Rectangle $ABCD$ has $\overline{AB}=5$ and $\overline{BC}=4$. Point $E$ lies on $\overline{AB}$ so that $\overline{EB}=1$, point $G$ lies on $\overline{BC}$ so that $\overline{CG}=1$, and point $F$ lies on $\overline{CD}$ so that $\overline{DF}=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\dfrac{\overline{PQ}}{\overline{EF}}$?
矩形 $ABCD$ 满足 $\overline{AB}=5$ 且 $\overline{BC}=4$。点 $E$ 在 $\overline{AB}$ 上，且 $\overline{EB}=1$；点 $G$ 在 $\overline{BC}$ 上，且 $\overline{CG}=1$；点 $F$ 在 $\overline{CD}$ 上，且 $\overline{DF}=2$。线段 $\overline{AG}$ 与 $\overline{AC}$ 分别与 $\overline{EF}$ 相交于 $Q$ 和 $P$。求 $\dfrac{\overline{PQ}}{\overline{EF}}$ 的值。
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Q20
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A'(5,6)$. What distance does the origin $O(0,0)$ move under this transformation?
平面上的一次伸缩变换（即缩放因子为正的比例变换）把以 $A(2,2)$ 为圆心、半径为 $2$ 的圆变换为以 $A'(5,6)$ 为圆心、半径为 $3$ 的圆。在该变换下，原点 $O(0,0)$ 移动了多远距离？
Q21
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？
Q22
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 $A$?
若干支队伍进行了一次循环赛（每支队伍与其他每支队伍恰好比赛一次）。每支队伍赢了 10 场、输了 10 场，且没有平局。问：有多少个由三支队伍组成的集合 $\{A,B,C\}$ 满足 $A$ 战胜 $B$，$B$ 战胜 $C$，且 $C$ 战胜 $A$？
Q23
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$, respectively, so that lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?
在正六边形 $ABCDEF$ 中，点 $W$、$X$、$Y$、$Z$ 分别取在边 $\overline{BC}$、$\overline{CD}$、$\overline{EF}$、$\overline{FA}$ 上，使得直线 $AB$、$ZW$、$YX$ 和 $ED$ 互相平行且等距。求六边形 $WCXYFZ$ 的面积与六边形 $ABCDEF$ 的面积之比。
Q24
How many four-digit positive integers $abcd$, with $a\ne 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
有多少个四位正整数 $abcd$（其中 $a\ne 0$）满足：三个两位整数 $ab<bc<cd$ 构成一个递增的等差数列？例如，$4692$ 就满足条件，其中 $a=4$，$b=6$，$c=9$，$d=2$。
Q25
Let $f(x)=\sum_{k=2}^{10}\big(\lfloor kx\rfloor-k\lfloor x\rfloor\big)$, where $\lfloor r\rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x\ge 0$?
设 $f(x)=\sum_{k=2}^{10}\big(\lfloor kx\rfloor-k\lfloor x\rfloor\big)$，其中 $\lfloor r\rfloor$ 表示不超过 $r$ 的最大整数（取整函数）。当 $x\ge 0$ 时，$f(x)$ 能取到多少个不同的值？
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