/

AMC12 2006 B

You are not logged in. After submit, your report may not be available on other devices. Login

AMC12 · 2006 (B)

Q1
What is $( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$?
$( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$ 的值是多少？
Q2
For real numbers $x$ and $y$, define $x\spadesuit y = (x + y)(x - y)$. What is $3\spadesuit(4\spadesuit 5)$?
对实数 $x$ 和 $y$，定义 $x\spadesuit y = (x + y)(x - y)$。$3\spadesuit(4\spadesuit 5)$ 的值是多少？
Q3
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
一场橄榄球比赛在两支球队 Cougars 和 Panthers 之间进行。两队总共得了 34 分，且 Cougars 以 14 分的优势获胜。Panthers 得了多少分？
Q4
Mary is about to pay for five items at the grocery store. The prices of the items are $7.99$, $4.99$, $2.99$, $1.99$, and $0.99$. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $20.00$ that she will receive in change?
Mary 在杂货店准备购买五件商品。商品价格分别为 $7.99$、$4.99$、$2.99$、$1.99$ 和 $0.99$。Mary 将用一张二十美元钞票付款。下列哪一项最接近她将收到的找零占 $20.00$ 的百分比？
Q5
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
John 以每小时 3 英里的速度向东走，而 Bob 也向东走，但速度为每小时 5 英里。如果 Bob 现在在 John 以西 1 英里处，Bob 追上 John 需要多少分钟？
Q6
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
Francesca 使用 100 克柠檬汁、100 克糖和 400 克水制作柠檬水。100 克柠檬汁含有 25 卡路里，100 克糖含有 386 卡路里。水不含卡路里。她的柠檬水 200 克含有多少卡路里？
Q7
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?
Lopez 先生和 Lopez 太太有两个孩子。当他们坐进家庭用车时，两人坐在前排，另外两人坐在后排。Lopez 先生或 Lopez 太太必须坐在驾驶座。有多少种可能的座位安排？
Q8
The lines $x = \frac 14y + a$ and $y = \frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$?
直线 $x = \frac 14y + a$ 和 $y = \frac 14x + b$ 相交于点 $(1,2)$。$a + b$ 等于多少？
Q9
How many even three-digit integers have the property that their digits, all read from left to right, are in strictly increasing order?
有多少个偶数的三位整数满足其数字从左到右严格递增？
Q10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
在一个具有整数边长的三角形中，一条边是第二条边的三倍，第三条边的长度为 15。这个三角形的最大可能周长是多少？
Q11
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
Joe 和 JoAnn 各买了 12 盎司的咖啡，装在 16 盎司的杯子里。Joe 喝了 2 盎司咖啡，然后加了 2 盎司奶油。JoAnn 加了 2 盎司奶油，充分搅拌咖啡，然后喝了 2 盎司。Joe 的咖啡中奶油量与 JoAnn 的咖啡中奶油量的比值为多少？
Q12
The parabola $y=ax^2+bx+c$ has vertex $(p,p)$ and $y$-intercept $(0,-p)$, where $p\ne 0$. What is $b$?
抛物线 $y=ax^2+bx+c$ 的顶点为 $(p,p)$，$y$-截距为 $(0,-p)$，其中 $p\ne 0$。$b$ 等于多少？
Q13
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is 24, and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$?
菱形 $ABCD$ 与菱形 $BFDE$ 相似。菱形 $ABCD$ 的面积为 24，且 $\angle BAD = 60^\circ$。菱形 $BFDE$ 的面积是多少？
stem
Q14
Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per glob. The cost of the peanut butter and jam to make all the sandwiches is $2.53$. Assume that $B$, $J$ and $N$ are all positive integers with $N>1$. What is the cost of the jam Elmo uses to make the sandwiches?
Elmo 为筹款活动做了 $N$ 个三明治。每个三明治使用 $B$ 团花生酱（每团 4 美分）和 $J$ 团果酱（每团 5 美分）。制作所有三明治的花生酱和果酱总成本为 $2.53$ 美元。假设 $B$、$J$ 和 $N$ 都是正整数，且 $N>1$。Elmo 用于制作三明治的果酱成本是多少？
Q15
Circles with centers $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent. Points $A$ and $B$ are on the circle centered at $O$, and points $C$ and $D$ are on the circle centered at $P$, such that $\overline{AD}$ and $\overline{BC}$ are common external tangents to the circles. What is the area of hexagon $AOBCPD$?
圆心为 $O$ 和 $P$ 的圆半径分别为 2 和 4，且外切。点 $A$ 和 $B$ 在以 $O$ 为圆心的圆上，点 $C$ 和 $D$ 在以 $P$ 为圆心的圆上，使得 $\overline{AD}$ 和 $\overline{BC}$ 是两圆的公外切线。六边形 $AOBCPD$ 的面积是多少？
stem
Q16
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area?
正六边形 $ABCDEF$ 的顶点 $A$ 和 $C$ 分别位于 $(0,0)$ 和 $(7,1)$。它的面积是多少？
Q17
For a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are in the ratio $1:2:3:4:5:6$. What is the probability of rolling a total of $7$ on the two dice?
对于一对特殊的骰子，每个骰子上掷出 $1$、$2$、$3$、$4$、$5$ 和 $6$ 的概率之比为 $1:2:3:4:5:6$。两个骰子点数之和为 $7$ 的概率是多少？
Q18
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
平面上的一个物体从一个格点移动到另一个格点。每一步，物体可以向右移动一单位、向左移动一单位、向上移动一单位或向下移动一单位。如果物体从原点开始，走一条十步路径，可能的终点有多少个不同的点？
Q19
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
琼斯先生有八个不同年龄的孩子。在一次家庭旅行中，他最大的孩子（9 岁）看到一个车牌号是一个 4 位数，其中恰有两个数字各出现两次。“看，爸爸！”她喊道，“这个数能被我们每个孩子的年龄整除！”“没错，”琼斯先生回答，“而且最后两位数字恰好是我的年龄。”以下哪项不是琼斯先生某个孩子的年龄？
Q20
Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.
从区间 $(0,1)$ 中随机选择 $x$。满足 $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$ 的概率是多少？ 这里 $\lfloor x\rfloor$ 表示不超过 $x$ 的最大整数。
Q21
Rectangle $ABCD$ has area $2006$. An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. What is the perimeter of the rectangle? (The area of an ellipse is $ab\pi$ where $2a$ and $2b$ are the lengths of the axes.)
矩形 $ABCD$ 的面积为 $2006$。一个面积为 $2006\pi$ 的椭圆通过 $A$ 和 $C$，且焦点在 $B$ 和 $D$ 处。该矩形的周长是多少？（椭圆的面积为 $ab\pi$，其中 $2a$ 和 $2b$ 是其轴的长度。）
Q22
Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$?
设 $a$、$b$ 和 $c$ 是正整数，且 $a+b+c=2006$，并且 $a!b!c!=m\cdot 10^n$，其中 $m$ 和 $n$ 为整数且 $m$ 不被 $10$ 整除。$n$ 的最小可能值是多少？
Q23
Isosceles $\triangle ABC$ has a right angle at $C$. Point $P$ is inside $\triangle ABC$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?
等腰 $\triangle ABC$ 在 $C$ 处有直角。点 $P$ 在 $\triangle ABC$ 内部，且 $PA=11$，$PB=7$，$PC=6$。直角边 $\overline{AC}$ 和 $\overline{BC}$ 的长度为 $s=\sqrt{a+b\sqrt{2}}$，其中 $a$ 和 $b$ 为正整数。求 $a+b$。
stem
Q24
Let $S$ be the set of all point $(x,y)$ in the coordinate plane such that $0 \le x \le \frac{\pi}{2}$ and $0 \le y \le \frac{\pi}{2}$. What is the area of the subset of $S$ for which \[\sin^2x-\sin x \sin y + \sin^2y \le \frac34?\]
设 $S$ 为坐标平面中所有点 $(x,y)$ 的集合，使得 $0 \le x \le \frac{\pi}{2}$ 且 $0 \le y \le \frac{\pi}{2}$。满足 \[\sin^2x-\sin x \sin y + \sin^2y \le \frac34?\] 的 $S$ 的子集面积是多少？
Q25
A sequence $a_1,a_2,\dots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq 1$. If $a_1=999$, $a_2<999$ and $a_{2006}=1$, how many different values of $a_2$ are possible?
一个非负整数序列 $a_1,a_2,\dots$ 由规则 $a_{n+2}=|a_{n+1}-a_n|$（对 $n\geq 1$）定义。若 $a_1=999$，$a_2<999$ 且 $a_{2006}=1$，则 $a_2$ 可能取多少个不同的值？
Time Left 75:00