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AMC12 2010 A

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AMC12 · 2010 (A)

Q1
What is $\left(20-\left(2010-201\right)\right)+\left(2010-\left(201-20\right)\right)$?
$\left(20-\left(2010-201\right)\right)+\left(2010-\left(201-20\right)\right)$ 的值是多少？
Q2
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
一艘渡船从上午 10 点开始每小时运送游客到一个岛上，直到最后一趟在下午 3 点出发。某天船长注意到上午 10 点那趟船上有 100 名游客，并且之后每一趟的游客人数都比前一趟少 1 人。那天渡船一共把多少名游客送到了岛上？
Q3
Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$?
如下图所示，长方形 $ABCD$ 与正方形 $EFGH$ 的重叠部分占长方形面积的 $50\%$。同时，正方形 $EFGH$ 与长方形 $ABCD$ 的重叠部分占正方形面积的 $20\%$。求 $\frac{AB}{AD}$。
stem
Q4
If $x<0$, then which of the following must be positive?
若 $x<0$，则下列哪个一定为正？
Q5
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next $n$ shots are bullseyes she will be guaranteed victory. What is the minimum value for $n$?
在一个 100 箭的射箭比赛进行到一半时，Chelsea 领先 50 分。每箭射中靶心得 10 分，其他可能得分为 8、4、2 和 0 分。Chelsea 每箭至少得 4 分。若 Chelsea 接下来的 $n$ 箭都是靶心，则她将确保获胜。求 $n$ 的最小值。
Q6
A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?
回文数（如 $83438$）是指将其数字倒序后仍保持不变的数。数字 $x$ 和 $x+32$ 分别是三位数和四位数回文数。$x$ 的各位数字之和是多少？
Q7
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
Logan 正在制作他城镇的缩比模型。城市的水塔高 40 米，顶部是一个容纳 100,000 升水的球体。Logan 的迷你水塔容纳 0.1 升水。Logan 应该把他的水塔做多高（米）？
Q8
Triangle $ABC$ has $AB=2 \cdot AC$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$?
三角形 $ABC$ 有 $AB=2 \cdot AC$。点 $D$ 和 $E$ 分别在 $\overline{AB}$ 和 $\overline{BC}$ 上，使得 $\angle BAE = \angle ACD$。$F$ 是线段 $AE$ 和 $CD$ 的交点，且 $\triangle CFE$ 是等边三角形。$\angle ACB$ 是多少？
Q9
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
一个边长 3 英寸的实心立方体。在每个面中心切出一个 $2$ 英寸乘 $2$ 英寸的正方形孔。每个切口的边与立方体的边平行，并且每个孔都贯穿整个立方体。剩余固体的体积是多少（立方英寸）？
Q10
The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^\text{th}$ term of this sequence?
一个等差数列的前四项是 $p$、$9$、$3p-q$ 和 $3p+q$。该数列的第 $2010^\text{th}$ 项是多少？
Q11
The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \log_b 7^7$. What is $b$?
方程 $7^{x+7} = 8^x$ 的解可以表示为 $x = \log_b 7^7$ 的形式。$b$ 的值是多少？
Q12
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements. Brian: "Mike and I are different species." Chris: "LeRoy is a frog." LeRoy: "Chris is a frog." Mike: "Of the four of us, at least two are toads." How many of these amphibians are frogs?
在一个神奇的沼泽中有两种会说话的两栖动物：蟾蜍，它们的陈述总是真实的；青蛙，它们的陈述总是假的。四只两栖动物 Brian、Chris、LeRoy 和 Mike 一起生活在这个沼泽中，它们做出了以下陈述。 Brian：“Mike 和我是不同物种。” Chris：“LeRoy 是青蛙。” LeRoy：“Chris 是青蛙。” Mike：“我们四个中至少有两个是蟾蜍。” 这四只两栖动物中有多少是青蛙？
Q13
For how many integer values of $k$ do the graphs of $x^2+y^2=k^2$ and $xy = k$ not intersect?
对于多少个整数 $k$ 值，$x^2+y^2=k^2$ 和 $xy = k$ 的图像不相交？
Q14
Nondegenerate $\triangle ABC$ has integer side lengths, $\overline{BD}$ is an angle bisector, $AD = 3$, and $DC=8$. What is the smallest possible value of the perimeter?
非退化 $\triangle ABC$ 有整数边长，$\overline{BD}$ 是角平分线，$AD = 3$，$DC=8$。周长的最小可能值是多少？
Q15
A coin is altered so that the probability that it lands on heads is less than $\frac{1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $\frac{1}{6}$. What is the probability that the coin lands on heads?
一枚硬币被修改，使得它正面朝上的概率小于 $\frac{1}{2}$，当翻转四次时，正反面个数相等的概率为 $\frac{1}{6}$。这枚硬币正面朝上的概率是多少？
Q16
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,...,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,...,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
Bernardo 从集合 $\{1,2,3,...,7,8,9\}$ 中随机选取 3 个不同的数，并按降序排列形成一个三位数。Silvia 从集合 $\{1,2,3,...,6,7,8\}$ 中随机选取 3 个不同的数，也按降序排列形成一个三位数。Bernardo 的数大于 Silvia 的数的概率是多少？
Q17
Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$?
等角六边形 $ABCDEF$ 有边长 $AB=CD=EF=1$ 且 $BC=DE=FA=r$。$\triangle ACE$ 的面积是该六边形面积的 $70\%$。所有可能的 $r$ 值之和是多少？
Q18
A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step?
一条 16 步路径从 $(-4,-4)$ 走到 $(4,4)$，每一步使 $x$ 坐标或 $y$ 坐标增加 1。有多少条这样的路径在每一步都保持在正方形 $-2 \le x \le 2$, $-2 \le y \le 2$ 的外部或边界上？
Q19
Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{2010}$?
一排 2010 个盒子，每个盒子含有一个红弹珠，并且对 $1 \le k \le 2010$，第 $k\text{th}$ 个盒子还包含 $k$ 个白弹珠。Isabella 从第一个盒子开始，依次从每个盒子随机抽取一颗弹珠。她在第一次抽到红弹珠时停止。令 $P(n)$ 为 Isabella 恰好抽取 $n$ 颗弹珠后停止的概率。求使得 $P(n) < \frac{1}{2010}$ 的最小 $n$ 值。
Q20
Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$?
等差数列 $\left(a_n\right)$ 和 $\left(b_n\right)$ 的各项均为整数，且 $a_1=b_1=1<a_2 \le b_2$，并存在某个 $n$ 使得 $a_n b_n = 2010$。最大的可能 $n$ 值是多少？
Q21
The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values?
图像 $y=x^6-10x^5+29x^4-4x^3+ax^2$ 位于直线 $y=bx+c$ 上方，除了有三个 $x$ 值处，图像与直线相交。这些值中最大的是多少？
Q22
What is the minimum value of $f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$?
$f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$ 的最小值是多少？
Q23
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?
由 $90!$ 的最后两个非零数字组成的数等于 $n$。$n$ 是多少？
Q24
Let $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n$?
令 $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$。$f(x)$ 的定义域与区间 $[0,1]$ 的交集是 $n$ 个不相交开区间的并。$n$ 是多少？
Q25
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?
若一个四边形可由另一个通过旋转和平移得到，则视作相同。具有整数边长、周长为 32 的不同凸圆内接四边形有多少个？
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