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AMC10 2019 A

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AMC10 · 2019 (A)

Q1
What is the value of $2^{(0^{19})} + ((2^9)^{-1})^9$?
$2^{(0^{19})} + ((2^9)^{-1})^9$ 的值是多少？
Q2
What is the hundreds digit of $(20! - 15!)$?
$(20! - 15!)$ 的百位数字是多少？
Q3
Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was 5 times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n$?
Ana 和 Bonita 在不同年份的同一天出生，相差 $n$ 年。去年 Ana 的年龄是 Bonita 的 5 倍。今年 Ana 的年龄是 Bonita 年龄的平方。$n$ 是多少？
Q4
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn?
一个盒子中有 28 个红球，20 个绿球，19 个黄球，13 个蓝球，11 个白球，9 个黑球。从盒中不放回地抽取最少多少个球，才能保证至少有一种颜色的球抽到 15 个？
Q5
What is the greatest number of consecutive integers whose sum is 45?
和为 45 的连续整数最多有几个？
Q6
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral? * a square * a rectangle that is not a square * a rhombus that is not a square * a parallelogram that is not a rectangle or a rhombus * an isosceles trapezoid that is not a parallelogram
在以下哪几种四边形中，存在一个在四边形平面内的点，该点到四边形四个顶点的距离相等？ * 正方形 * 非正方形的矩形 * 非正方形的菱形 * 既非矩形也非菱形的平行四边形 * 非平行四边形的等腰梯形
Q7
Two lines with slopes $\frac{1}{2}$ and 2 intersect at $(2, 2)$. What is the area of the triangle enclosed by these two lines and the line $x + y = 10$?
两条斜率分别为 $\frac{1}{2}$ 和 2 的直线相交于点 $(2, 2)$。这两条直线与直线 $x + y = 10$ 围成的三角形的面积是多少？
Q8
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? * some rotation around a point on line $\ell$ * some translation in the direction parallel to line $\ell$ * the reflection across line $\ell$ * some reflection across a line perpendicular to line $\ell$
下图显示直线 $\ell$ 上有一个规则、无限、周期重复的正方形和线段图案。在这个图形所在的平面中，以下四种刚体运动变换（除了恒等变换外），有多少种会将这个图形映射到自身？ * 绕直线 $\ell$ 上某点的某些旋转 * 平行于直线 $\ell$ 方向的某些平移 * 关于直线 $\ell$ 的反射 * 关于垂直于直线 $\ell$ 的直线的某些反射
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Q9
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers?
对于三位正整数 $n$，前 $n$ 个正整数的和不整除前 $n$ 个正整数的乘积的最大 $n$ 是多少？
Q10
A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and last tile, how many tiles does the bug visit?
一个宽 10 英尺、长 17 英尺的矩形地板铺了 170 张一英尺见方的瓷砖。一只虫子从一个角直线走到对角。包括首尾瓷砖在内，虫子经过多少张瓷砖？
Q11
How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?
$201^9$ 有多少个正整数除数是完全平方数或完全立方数（或两者皆是）？
Q12
Melanie computes the mean $\mu$, the median $M$, and modes of the 365 values that are the dates in the months of 2019. Thus her data consist of 12 1s, 12 2s, ..., 12 28s, 11 29s, 11 30s, and 7 31s. Let $d$ be the median of the modes. Which of the following statements is true?
Melanie 计算了 2019 年月份日期的 365 个值的均值 $\mu$、中位数 $M$ 和众数。因此她的数据包括 12 个 1、12 个 2、...、12 个 28、11 个 29、11 个 30 和 7 个 31。让 $d$ 为众数的中位数。以下哪个陈述是正确的？
Q13
Let $\triangle ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^\circ$. Construct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC$?
设 $\triangle ABC$ 是等腰三角形，$BC = AC$，且 $\angle ACB = 40^\circ$。构造以 $\overline{BC}$ 为直径的圆，让 $D$ 和 $E$ 分别为该圆与边 $\overline{AC}$ 和 $\overline{AB}$ 的其他交点。让 $F$ 为四边形 $BCDE$ 对角线的交点。$\angle BFC$ 的度数是多少？
Q14
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?
对于平面上一组四条不同的直线，恰有 $N$ 个不同的点位于两条或多条直线上。所有可能的 $N$ 值之和是多少？
Q15
A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and $a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$ for all $n \ge 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
一个数列由 $a_1 = 1$，$a_2 = \frac{3}{7}$，且对于所有 $n \ge 3$，$a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$ 递推定义。那么 $a_{2019}$ 可以写成 $\frac{p}{q}$，其中 $p$ 和 $q$ 是互质的正整数。$p + q$ 是多少？
Q16
The figure below shows 13 circles of radius 1 within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all of the circles of radius 1?
下图显示了一个大圆内有13个半径为1的小圆。所有相交点均为切点。图中阴影区域是大圆内部但所有半径为1的小圆外部的区域面积是多少？
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Q17
A child builds towers using identically shaped cubes of different colors. How many different towers with a height of 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.)
一个孩子使用相同形状但不同颜色的立方体搭建塔形。有2个红色、3个蓝色和4个绿色立方体，能搭建多少种高度为8个立方体的不同塔？（会剩下一个立方体。）
Q18
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.23232323..._k$. What is $k$?
对于某个正整数 $k$，分数 $\frac{7}{51}$ 的基数-$k$ 循环表示为 $0.\overline{23}_k = 0.23232323..._k$。$k$ 是多少？
Q19
What is the least possible value of $(x+1)(x+2)(x+3)(x+4) + 2019$, where $x$ is a real number?
$(x+1)(x+2)(x+3)(x+4) + 2019$ 的最小可能值为多少，其中 $x$ 为实数？
Q20
The numbers 1, 2, ..., 9 are randomly placed into the 9 squares of a 3 × 3 grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
数字 1, 2, ..., 9 随机放置到 3 × 3 网格的 9 个方格中。每个方格放一个数字，每个数字使用一次。每行和每列数字之和均为奇数的概率是多少？
Q21
A sphere with center $O$ has radius 6. A triangle with sides of length 15, 15, and 24 is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
一个以 $O$ 为球心的球半径为 6。一个边长为 15、15 和 24 的三角形位于空间中，使得它的每条边都与该球相切。求 $O$ 与该三角形确定的平面之间的距离。
Q22
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0, 1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x - y| > \frac{1}{2}$?
实数在 0 和 1（包含端点）之间按以下方式选择。先抛一枚公平硬币。如果正面，再抛一次，第二次正面则选 0，反面则选 1。如果第一次反面，则从闭区间 $[0, 1]$ 中均匀随机选择一个数。独立选择两个随机数 $x$ 和 $y$。求 $|x - y| > \frac{1}{2}$ 的概率。
Q23
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number 1, then Todd must say the next two numbers (2 and 3), then Tucker must say the next three numbers (4, 5, 6), then Tadd must say the next four numbers (7, 8, 9, 10), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number 10,000 is reached. What is the 2019th number said by Tadd?
Travis 要照看可怕的 Thompson 三胞胎。知道他们喜欢大数字，Travis 为他们设计了一个计数游戏。先 Tadd 说数字 1，然后 Todd 说接下来的两个数字（2 和 3），然后 Tucker 说接下来的三个数字（4, 5, 6），然后 Tadd 说接下来的四个数字（7, 8, 9, 10），过程继续按三个孩子顺序轮流，每人说的数字比前一人多一个，直到达到数字 10,000。Tadd 说的第 2019 个数字是多少？
Q24
Let $p, q, r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. There exist real numbers $A, B, C$ such that $$\frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}$$ for all real numbers $s$ with $s \notin \{p, q, r\}$. What is $\frac{1}{A} + \frac{1}{B} + \frac{1}{C}$?
设 $p, q, r$ 是多项式 $x^3 - 22x^2 + 80x - 67$ 的不同根。存在实数 $A, B, C$ 使得 $$\frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}$$ 对所有 $s \notin \{p, q, r\}$ 的实数 $s$ 成立。求 $\frac{1}{A} + \frac{1}{B} + \frac{1}{C}$。
Q25
For how many integers $n$ between 1 and 50, inclusive, is $$\frac{(n^2 - 1)!}{(n!)^n}$$ an integer? (Recall that $0! = 1$.)
在 1 到 50（包含）之间的整数 $n$ 中，有多少个使 $$\frac{(n^2 - 1)!}{(n!)^n}$$ 为整数？（回想 $0! = 1$）。
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