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AMC12 2019 A

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AMC12 · 2019 (A)

Q1
The area of a pizza with radius 4 inches is $N$ percent larger than the area of a pizza with radius 3 inches. What is the integer closest to $N$?
半径为 4 英寸的比萨的面积比半径为 3 英寸的比萨的面积大 $N$ 百分比。最接近 $N$ 的整数是多少？
Q2
Suppose $a$ is 150% of $b$. What percent of $a$ is $3b$?
假设 $a$ 是 $b$ 的 150%。那么 $3b$ 是 $a$ 的百分之多少？
Q3
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?
盒子中有 28 个红球、20 个绿球、19 个黄球、13 个蓝球、11 个白球和 9 个黑球。从盒子中不放回地抽取球，至少需要抽取多少个球才能保证抽到至少 15 个同一颜色的球？
Q4
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？
Q5
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？
Q6
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 $\ell$
下面的图形显示直线 $\ell$ 上有一个规则的、无限的、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变换成自身？ • 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
stem
Q7
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 年各月日期的 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？
Q8
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 $N$?
对于平面内四条不同的直线集合，有恰好 $N$ 个不同的点位于两条或更多条直线上。所有可能的 $N$ 值之和是多少？
Q9
A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = 3/7$, and $a_n = (a_{n-2} \cdot a_{n-1})/(2a_{n-2} - a_{n-1})$ for all $n \geq 3$. Then $a_{2019}$ can be written as $p/q$, where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
一个数列由 $a_1 = 1$、$a_2 = 3/7$ 和对于所有 $n \geq 3$，$a_n = (a_{n-2} \cdot a_{n-1})/(2a_{n-2} - a_{n-1})$ 递归定义。那么 $a_{2019}$ 可以写成 $p/q$，其中 $p$ 和 $q$ 是互质的正整数。$p + q$ 是多少？
Q10
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?
下面的图形显示一个大圆内有 13 个半径为 1 的圆。所有交点均发生在相切点处。图形中阴影区域是大圆内部但所有半径为 1 的圆外部的区域面积是多少？
stem
Q11
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.23_k = 0.23232323..._k$. What is $k$?
对于某个正整数$k$，分数$\frac{7}{51}$（十进制）的重复$k$进制表示为$0.23_k = 0.23232323..._k$。$k$是多少？
Q12
Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2 x = \log_y 16$ and $xy = 64$. What is $\left(\log_2 \frac{x}{y}\right)^2$?
正实数$x \neq 1$和$y \neq 1$满足$\log_2 x = \log_y 16$且$xy = 64$。求$\left(\log_2 \frac{x}{y}\right)^2$的值？
Q13
How many ways are there to paint each of the integers 2, 3, ..., 9 either red, green, or blue so that each number has a different color from each of its proper divisors?
有几种方法可以将整数2, 3, ..., 9每个涂成红色、绿色或蓝色，使得每个数与其每个真约数的颜色都不同？
Q14
For a certain complex number $c$, the polynomial $P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)$ has exactly 4 distinct roots. What is $|c|$?
对于某个复数$c$，多项式$P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)$恰有4个不同根。求$|c|$？
Q15
Positive real numbers $a$ and $b$ have the property that $\sqrt{\log a} + \sqrt{\log b} + \log \sqrt{a} + \log \sqrt{b} = 100$, and all four terms on the left are positive integers, where $\log$ denotes the base 10 logarithm. What is $ab$?
正实数$a$和$b$满足$\sqrt{\log a} + \sqrt{\log b} + \log \sqrt{a} + \log \sqrt{b} = 100$，且左边四个项均为正整数，其中$\log$表示以10为底的对数。求$ab$？
Q16
The numbers 1, 2, ..., 9 are randomly placed into the 9 squares of a $3 \times 3$ 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被随机放置到一个$3 \times 3$网格的9个方格中。每个方格得到一个数字，且每个数字只使用一次。每个行和每个列的数字之和为奇数的概率是多少？
Q17
Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_0 = 3$, $s_1 = 5$, and $s_2 = 9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$ for $k = 2, 3, \dots$. What is $a + b + c$?
设$s_k$表示多项式$x^3 - 5x^2 + 8x - 13$的根的$k$次幂之和。特别地，$s_0 = 3$，$s_1 = 5$，且$s_2 = 9$。设$a$，$b$，$c$为实数使得$s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$对$k = 2, 3, \dots$成立。求$a + b + c$？
Q18
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, 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。
Q19
In $\triangle ABC$ with integer side lengths, $$\cos A = \frac{11}{16}, \quad \cos B = \frac{7}{8}, \quad \text{and} \quad \cos C = -\frac{1}{4}.$$ What is the least possible perimeter for $\triangle ABC$?
在具有整数边长的$\triangle ABC$中，$$\cos A = \frac{11}{16}, \quad \cos B = \frac{7}{8}, \quad \text{and} \quad \cos C = -\frac{1}{4}.$$$\triangle ABC$的最小可能周长是多少？
Q20
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到1之间（包含端点）以以下方式选择。抛一次公平硬币。如果正面，则再抛一次，第二抛为正面则选0，反面则选1。另一方面，如果第一抛为反面，则从闭区间$[0, 1]$中均匀随机选择该数。独立选择两个随机数$x$和$y$。求$|x - y| > \frac{1}{2}$的概率？
Q21
Let $$ z = \frac{1+i}{\sqrt{2}}. $$ What is $$ (z^{12} + z^{22} + z^{32} + \dots + z^{122}) \cdot \left( \frac{1}{z^{12}} + \frac{1}{z^{22}} + \frac{1}{z^{32}} + \dots + \frac{1}{z^{122}} \right) ? $$
设 $$ z = \frac{1+i}{\sqrt{2}}. $$ 何为 $$ (z^{12} + z^{22} + z^{32} + \dots + z^{122}) \cdot \left( \frac{1}{z^{12}} + \frac{1}{z^{22}} + \frac{1}{z^{32}} + \dots + \frac{1}{z^{122}} \right) ? $$
Q22
Circles $\omega$ and $\gamma$, both centered at $O$, have radii 20 and 17, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$. Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\frac{BP}{CP} = 3$. Then $AB$ can be written in the form $\frac{m}{\sqrt{n}} - \frac{p}{\sqrt{q}}$ for positive integers $m, n, p, q$ with $\gcd(m, n) = \gcd(p, q) = 1$. What is $m + n + p + q$?
圆 $\omega$ 和 $\gamma$ 都以 $O$ 为圆心，半径分别为 20 和 17。正三角形 $ABC$ 的内部位于 $\omega$ 的内部但位于 $\gamma$ 的外部，顶点 $A$ 在 $\omega$ 上，边 $\overline{BC}$ 所在直线与 $\gamma$ 相切。线段 $\overline{AO}$ 和 $\overline{BC}$ 相交于 $P$，且 $\frac{BP}{CP} = 3$。则 $AB$ 可写成 $\frac{m}{\sqrt{n}} - \frac{p}{\sqrt{q}}$ 的形式，其中 $m, n, p, q$ 为正整数且 $\gcd(m, n) = \gcd(p, q) = 1$。求 $m + n + p + q$。
Q23
Define binary operations $\diamond$ and $\diamond$ by $a \diamond b = a^{\log_7(b)}$ and $a \diamond b = a^{\frac{1}{\log_7(b)}}$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3 \diamond 2$ and $a_n = (n \diamond (n-1)) \diamond a_{n-1}$ for all integers $n \ge 4$. To the nearest integer, what is $\log_7(a_{2019})$?
定义二元运算 $\diamond$ 和 $\diamond$ 如下： $$ a \diamond b = a^{\log_7(b)} $$ 和 $$ a \diamond b = a^{\frac{1}{\log_7(b)}} $$ 对所有这些表达式有定义的实数 $a$ 和 $b$。序列 $(a_n)$ 由 $a_3 = 3 \diamond 2$ 和 $$ a_n = (n \diamond (n-1)) \diamond a_{n-1} $$ 对所有整数 $n \ge 4$ 递归定义。四舍五入到最近整数，$\log_7(a_{2019})$ 是多少？
Q24
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，有多少个使得 $$ \frac{(n^2 - 1)!}{(n!)^n} $$ 是整数？（回想 $0! = 1$）。
Q25
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly 59.999°, 60°, and 60.001°. For each positive integer $n$ define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse?
设 $\triangle A_0B_0C_0$ 是一个角的度数恰好为 59.999°、60° 和 60.001° 的三角形。对每个正整数 $n$，定义 $A_n$ 为从 $A_{n-1}$ 到直线 $B_{n-1}C_{n-1}$ 的高足点。同样地，定义 $B_n$ 为从 $B_{n-1}$ 到直线 $A_{n-1}C_{n-1}$ 的高足点，$C_n$ 为从 $C_{n-1}$ 到直线 $A_{n-1}B_{n-1}$ 的高足点。求最小的正整数 $n$ 使得 $\triangle A_nB_nC_n$ 是钝角三角形。
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