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AMC10 2015 B

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AMC10 · 2015 (B)

Q1
What is the value of $2 - (-2)^{-2}$?
$2 - (-2)^{-2}$ 的值是多少?
Q2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
Marie 连续做了三个耗时相等的任务,没有休息。她在下午 1:00 开始第一个任务,并在下午 2:40 完成第二个任务。她何时完成第三个任务?
Q3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
Isaac 写下了一个整数两次,另一个整数三次。这五个数的和是 100,其中一个数是 28。另一个数是多少?
Q4
Four siblings ordered an extra large pizza. Alex ate $\frac{1}{5}$, Beth $\frac{1}{3}$, and Cyril $\frac{1}{4}$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of the pizza they consumed?
四个兄弟姐妹点了一份超大披萨。Alex 吃了 $\frac{1}{5}$,Beth 吃了 $\frac{1}{3}$,Cyril 吃了 $\frac{1}{4}$。Dan 吃了剩下的部分。他们按吃披萨份额从多到少的顺序是?
Q5
David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?
David、Hikmet、Jack、Marta、Rand 和 Todd 参加了一个 12 人比赛,还有 6 个人。Rand 比 Hikmet 早 6 名完赛。Marta 比 Jack 晚 1 名完赛。David 比 Hikmet 晚 2 名完赛。Jack 比 Todd 晚 2 名完赛。Todd 比 Rand 晚 1 名完赛。Marta 第 6 名完赛。谁第 8 名完赛?
Q6
Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?
Marley 每周每天练习一种运动。她每周跑步三天,但从不连续两天跑步。周一她打篮球,两天后打高尔夫。她游泳和打网球,但从不连续跑步或游泳后打网球。Marley 每周哪天游泳?
Q7
Consider the operation “minus the reciprocal of,” defined by $a \diamond b = a - \frac{1}{b}$. What is $((1 \diamond 2) \diamond 3) - (1 \diamond (2 \diamond 3))$?
考虑操作“减去倒数”,定义为 $a \diamond b = a - \frac{1}{b}$。求 $((1 \diamond 2) \diamond 3) - (1 \diamond (2 \diamond 3))$ 的值?
Q8
The letter F shown below is rotated 90° clockwise around the origin, then reflected in the y-axis, and then rotated a half turn around the origin. What is the final image?
下面的字母 F 绕原点顺时针旋转 90°,然后关于 y 轴反射,然后绕原点旋转半圈。最终图像是什么?
stem
Q9
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius 3 and center (0,0) that lies in the first quadrant, the portion of the circle of radius $\frac{3}{2}$ and center $(0, \frac{3}{2})$ that lies in the first quadrant, and the line segment from (0,0) to (3,0). What is the area of the shark's fin falcata?
下面的阴影区域称为鲨鱼鳍弯刀,是达·芬奇研究过的图形。它被半径为 3、圆心为 (0,0) 的圆的第一象限部分、半径为 $\frac{3}{2}$、圆心为 $(0, \frac{3}{2})$ 的圆的第一象限部分,以及从 (0,0) 到 (3,0) 的线段所包围。鲨鱼鳍弯刀的面积是多少?
stem
Q10
What are the sign and units digit of the product of all the odd negative integers strictly greater than −2015?
所有严格大于 −2015 的奇负整数的乘积的符号和个位数是什么?
Q11
Among the positive integers less than 100, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?
在小于100的正整数中,其每个数字都是素数,从中随机选取一个。选中的数字是素数的概率是多少?
Q12
For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius 10 centered at (5,5)?
有整数$x$使得点$(x, -x)$在以(5,5)为圆心、半径为10的圆内或圆周上?
Q13
The line $12x + 5y = 60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
直线$12x + 5y = 60$与坐标轴围成一个三角形。这个三角形的高的长度之和是多少?
Q14
Let $a, b,$ and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?
设$a, b,$和$c$是三个不同的个位数。方程$(x-a)(x-b)+(x-b)(x-c)=0$的根之和的最大值是多少?
Q15
The town of Hamlet has 3 people for each horse, 4 sheep for each cow, and 3 ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?
哈姆雷特镇每匹马有3个人,每头牛有4只羊,每个人有3只鸭。以下哪个不可能是哈姆雷特镇的总人数、马、羊、牛和鸭的数量?
Q16
Al, Bill, and Cal will each randomly be assigned a whole number from 1 to 10, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
Al、Bill 和 Cal 每个人将被随机分配一个从 1 到 10(包含)的整数,且三人得到的数互不相同。Al 的数是 Bill 的数的整数倍且 Bill 的数是 Cal 的数的整数倍的概率是多少?
Q17
The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of the octahedron?
将下图所示直角长方体各面的中心点连接起来形成一个八面体。这个八面体的体积是多少?
stem
Q18
Johann has 64 fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?
Johann 有 64 个公平硬币。他翻转所有硬币。凡是落在反面的硬币再抛一次。第二抛落在反面的硬币第三次抛掷。现在正面朝上的硬币的期望数量是多少?
Q19
In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
在 $\triangle ABC$ 中,$\angle C = 90^\circ$ 且 $AB = 12$。在三角形外构造正方形 $ABXY$ 和 $ACWZ$。点 $X,Y,Z,W$ 位于同一个圆上。三角形的周长是多少?
Q20
Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
蚂蚁 Erin 从立方体的一个顶点开始,沿着恰好 7 条边爬行,这样她恰好访问每个顶点一次,然后发现无法沿着一条边返回起点。有多少条满足条件的路径?
Q21
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose that Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
小猫 Cozy 和小狗 Dash 要爬上一段有若干级台阶的楼梯。不过他们不是一级一级走,而是用跳的。Cozy 每次跳上 2 级(如果需要,他会只跳最后 1 级)。Dash 每次跳上 5 级(如果需要,若剩余台阶少于 5 级,他会把剩下的台阶一次跳完)。已知 Dash 到达楼梯顶部所用的跳跃次数比 Cozy 少 19 次。设 $s$ 为满足条件的所有可能台阶总数之和。问 $s$ 的各位数字之和是多少?
Q22
In the figure shown below, $ABCDE$ is a regular pentagon and $AG = 1$. What is $FG + JH + CD$?
如图所示,$ABCDE$是一个正五边形,且$AG=1$。求$FG+JH+CD$的值。
stem
Q23
Let n be a positive integer greater than 4 such that the decimal representation of n! ends in k zeros and the decimal representation of (2n)! ends in 3k zeros. Let s denote the sum of the four least possible values of n. What is the sum of the digits of s ?
设$n$是一个大于4的正整数,使得$n!$的小数表示末尾有$k$个零,而$(2n)!$的小数表示末尾有$3k$个零。让$s$表示四个最小可能$n$值的总和。$s$的各位数字之和是多少?
Q24
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\ldots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1)$, $p_3=(0,1)$, $p_4=(-1,1)$, $p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?
蚂蚁 Aaron 按照如下规则在坐标平面上行走。他从原点 $p_0=(0,0)$ 出发,面朝东走 1 个单位,到达 $p_1=(1,0)$。对 $n=1,2,3,\ldots$,当 Aaron 刚到达点 $p_n$ 后,如果他可以向左转 $90^\circ$ 并走 1 个单位到一个未到访过的点 $p_{n+1}$,他就这样做;否则,他就沿着当前方向直走 1 个单位到达 $p_{n+1}$。因此点列继续为 $p_2=(1,1)$,$p_3=(0,1)$,$p_4=(-1,1)$,$p_5=(-1,0)$,以此类推,形成一个逆时针的螺旋路径。求 $p_{2015}$。
Q25
A rectangular box measures a × b × c, where a, b, and c are integers and 1 ≤ a ≤ b ≤ c. The volume and the surface area of the box are numerically equal. How many ordered triples (a, b, c) are possible?
一个长方体盒子尺寸为$a\times b\times c$,其中$a$、$b$、$c$是整数且$1\leq a\leq b\leq c$。盒子的体积与其表面积数值相等。可能的有序三元组$(a,b,c)$有多少个?
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