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AMC12 2015 B

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AMC12 · 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连续完成三项耗时相等的任务，没有休息。她在下午1:00开始第一项任务，并在下午2:40完成第二项任务。她何时完成第三项任务？
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
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名结束。Marta比Jack晚1名结束。David比Hikmet晚2名结束。Jack比Todd晚2名结束。Todd比Rand晚1名结束。Marta是第6名。谁是第8名？
Q5
The Tigers beat the Sharks 2 out of the first 3 times they played. They then played N more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for N ?
Tigers在头三次比赛中2胜Sharks。然后又进行了N场比赛，Sharks最终赢得了所有比赛的至少95%。N的最小可能值为多少？
Q6
Back in 1930, Tillie had to memorize her multiplication facts from $0\times0$ through $12\times12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?
1930年，Tillie需要记忆从$0\times0$到$12\times12$的乘法事实。她得到的乘法表有标记因数的行和列，积形成了表格的主体部分。主体表格中的数字有多少比例是奇数？（结果保留到百分位）
Q7
A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$?
一个正15边形有$L$条对称轴，其最小的正旋转对称角是$R$度。求$L+R$。
Q8
What is the value of $\left(625^{\log_5 2015}\right)^{\frac{1}{4}}$?
$\left(625^{\log_5 2015}\right)^{\frac{1}{4}}$ 的值是多少？
Q9
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?
Larry和Julius在玩一个游戏，轮流向搁架上瓶子投球。Larry先投。获胜者是第一个把瓶子打下搁架的人。每次投球将瓶子打下搁架的概率为$\frac{1}{2}$，且独立于之前发生的事。求Larry获胜的概率。
Q10
How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?
有多少个不全等的、正面积的、周长小于15的整数边三角形既不是等边、等腰，也不是直角三角形？
Q11
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？
Q12
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 $(x - a)(x - b) + (x - b)(x - c) = 0$ 的根的和的最大值是多少？
Q13
Quadrilateral $ABCD$ is inscribed in a circle with $\angle BAC = 70^\circ$, $\angle ADB = 40^\circ$, $AD = 4$, and $BC = 6$. What is $AC$?
四边形 $ABCD$ 内接于一个圆，$\angle BAC = 70^\circ$，$\angle ADB = 40^\circ$，$AD = 4$，$BC = 6$。$AC$ 等于多少？
Q14
A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?
半径为 2 的圆以 $A$ 为圆心。一个边长为 4 的正三角形有一个顶点在 $A$。位于圆内但三角形外的区域面积与位于三角形内但圆外的区域面积之差是多少？
Q15
At Rachelle’s school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She thinks she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5?
在 Rachelle 的学校，A 计 4 分，B 计 3 分，C 计 2 分，D 计 1 分。她四门课的 GPA 是总分除以 4。她确定数学和科学会得 A，英语和历史每门至少 C。她认为英语得 A 的概率是 $\frac{1}{6}$，得 B 的概率是 $\frac{1}{4}$。历史得 A 的概率是 $\frac{1}{4}$，得 B 的概率是 $\frac{1}{3}$，与英语独立。Rachelle 获得 GPA 至少 3.5 的概率是多少？
Q16
A regular hexagon with sides of length 6 has an isosceles triangle attached to each side. Each of these triangles has two sides of length 8. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?
一个边长为6的正六边形，每条边上都附着一个等腰三角形，每个等腰三角形有两条边长为8。这些等腰三角形被折叠起来，形成一个以六边形为底面的金字塔。这个金字塔的体积是多少？
Q17
An unfair coin lands on heads with a probability of $\frac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$?
一枚不公平的硬币正面朝上的概率为 $\frac{1}{4}$。抛掷 $n$ 次时，正好两次正面的概率等于正好三次正面的概率。$n$ 的值为多少？
Q18
For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of 50 is $2\cdot5^{2}$, and $2+5+5=12$. What is the range of the function $r$, $\left\{r(n):n\text{ is a composite positive integer}\right\}$?
对于每个合数 $n$，定义 $r(n)$ 为 $n$ 的质因数分解中各因数的和。例如，$r(50)=12$，因为50的质因数分解为 $2\cdot5^{2}$，且 $2+5+5=12$。函数 $r$ 的值域 $\left\{r(n):n\text{ 是合数}\right\}$ 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 $AB=12$。在三角形外部构造正方形 $ABXY$ 和 $ACWZ$。点 $X,Y,Z,W$ 位于一个圆上。三角形的周长是多少？
Q20
For every positive integer $n$, let $\bmod_{5}(n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f:\{0,1,2,3,\dots\}\times\{0,1,2,3,4\}\to\{0,1,2,3,4\}$ recursively as follows: $$f(i,j)=\begin{cases}\bmod_{5}(j+1) & \text{if $i=0$ and $0\leq j\leq4$,}\\f(i-1,1) & \text{if $i\geq1$ and $j=0$,}\\f(i-1,f(i,j-1)) & \text{if $i\geq1$ and $1\leq j\leq4$.}\end{cases}$$ What is $f(2015,2)$?
对于每个正整数 $n$，令 $\bmod_{5}(n)$ 为 $n$ 除以5的余数。递归定义函数 $f:\{0,1,2,3,\dots\}\times\{0,1,2,3,4\}\to\{0,1,2,3,4\}$ 如下： $$f(i,j)=\begin{cases}\bmod_{5}(j+1) & \text{if $i=0$ and $0\leq j\leq4$,}\\f(i-1,1) & \text{if $i\geq1$ and $j=0$,}\\f(i-1,f(i,j-1)) & \text{if $i\geq1$ and $1\leq j\leq4$.}\end{cases}$$ $f(2015,2)$ 是多少？
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 $s$?
猫咪Cozy和狗狗Dash正在爬一个有一定数量台阶的楼梯。然而，他们不是一步一步走台阶，而是跳跃。Cozy每次跳跃上升两级台阶（不过如果必要，他会只跳最后一级）。Dash每次跳跃上升五级台阶（不过如果必要，他会跳剩下的不足5级的台阶）。假设Dash比Cozy少用了19次跳跃到达楼梯顶部。让$s$表示所有可能的台阶数的和。这个楼梯可能有的台阶数的和$s$的各位数字之和是多少？
Q22
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
六把椅子均匀地围绕一张圆桌摆放。每把椅子上坐着一个人。每人都站起来，坐到不是自己原来椅子且不与原来椅子相邻的椅子上，使得每把椅子又坐着一个人。这样可以有多少种方式？
Q23
A rectangular box measures $a \times b \times c$, where $a, b,$ and $c$ are integers and $1 \le a \le b \le c$. 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$a, b,$和$c$是整数且$1 \le a \le b \le c$。盒子的体积与表面积数值相等。有多少个可能的有序三元组$(a, b, c)$？
Q24
Four circles, no two of which are congruent, have centers at $A, B, C,$ and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $PQ$. What is $AR + BR + CR + DR$?
四个圆，不两两全等的，圆心在$A, B, C,$和$D$，点$P$和$Q$在所有四个圆上。圆$A$的半径是圆$B$半径的$\frac{5}{8}$，圆$C$的半径是圆$D$半径的$\frac{5}{8}$。此外，$AB = CD = 39$且$PQ = 48$。让$R$为$PQ$的中点。$AR + BR + CR + DR$是多少？
Q25
A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^\circ$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a, b, c,$ and $d$ are positive integers and $b$ and $d$ are square-free. What is $a + b + c + d$?
一只蜜蜂从点$P_0$开始飞行。她向正东飞1英寸到点$P_1$。对于$j \ge 1$，到达点$P_j$后，她逆时针转$30^\circ$，然后直飞$j+1$英寸到点$P_{j+1}$。当蜜蜂到达$P_{2015}$时，她离$P_0$正好$a\sqrt{b} + c\sqrt{d}$英寸，其中$a, b, c,$和$d$是正整数且$b$和$d$是无平方因子。$a + b + c + d$是多少？
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