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AMC10 2020 B

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AMC10 · 2020 (B)

Q1
What is the value of $1 -(-2) -3 -(-4) -5 -(-6)$?
计算 $1 -(-2) -3 -(-4) -5 -(-6)$ 的值？
Q2
Carl has 5 cubes each having side length 1, and Kate has 5 cubes each having side length 2. What is the total volume of these 10 cubes?
Carl 有 5 个边长为 1 的立方体，Kate 有 5 个边长为 2 的立方体。这些 10 个立方体的总体积是多少？
Q3
The ratio of $w$ to $x$ is $4 : 3$, the ratio of $y$ to $z$ is $3 : 2$, and the ratio of $z$ to $x$ is $1 : 6$. What is the ratio of $w$ to $y$?
$w$ 与 $x$ 的比是 $4 : 3$，$y$ 与 $z$ 的比是 $3 : 2$，$z$ 与 $x$ 的比是 $1 : 6$。$w$ 与 $y$ 的比是多少？
Q4
The acute angles of a right triangle are $a^\circ$ and $b^\circ$, where $a > b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?
一个直角三角形的两个锐角为 $a^\circ$ 和 $b^\circ$，其中 $a > b$，且 $a$ 和 $b$ 均为素数。$b$ 的最小可能值是多少？
Q5
How many distinguishable arrangements are there of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)
有 1 块棕色瓷砖、1 块紫色瓷砖、2 块绿色瓷砖和 3 块黄色瓷砖，从左到右排成一行，有多少种不同的排列方式？（相同颜色的瓷砖不可区分。）
Q6
Driving along a highway, Megan noticed that her odometer showed 15951 (miles). This number is a palindrome—it reads the same forward and backward. Then 2 hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this 2-hour period?
梅根在高速公路上行驶时，注意到她的里程表显示15951（英里）。这个数字是一个回文数——正读反读都相同。然后2小时后，里程表显示了下一个更大的回文数。这2小时期间她的平均速度是多少英里每小时？
Q7
How many positive even multiples of 3 less than 2020 are perfect squares?
小于2020的有多少个正偶数3的倍数的完全平方数？
Q8
Points $P$ and $Q$ lie in a plane with $PQ = 8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with area 12 square units?
点$P$和$Q$在平面内，$PQ = 8$。在这个平面中有多少个点$R$的位置使得顶点为$P$、$Q$和$R$的三角形是直角三角形且面积为12平方单位？
Q9
How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$?
有多少个整数有序对$(x, y)$满足方程$x^{2020} + y^2 = 2y$？
Q10
A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
一个半径为4英寸的圆的三刻度扇形连同其内部，可以沿着所示的两条半径粘合卷起形成直圆锥的侧面。锥体的体积是多少立方英寸？
stem
Q11
Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select?
Carr 女士要求她的学生从阅读列表中的 10 本书中阅读任意 5 本。Harold 随机从这个列表中选了 5 本书，Betty 也这样做。他们两人恰好都选了恰好 2 本相同书籍的概率是多少？
Q12
The decimal representation of $\frac{1}{20^{20}}$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
小数表示 $\frac{1}{20^{20}}$ 在小数点后有一串零，然后是一个 9，后面还有几个数字。小数点后那串初始零有多少个？
Q13
Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves 1 unit and then turns $90^\circ$ left. From there, Andy moves 2 units (north) and then turns $90^\circ$ left. He then moves 3 units (west) and again turns $90^\circ$ left. Andy continues this process, increasing his distance each time by 1 unit and always turning left. What is the location of the point at which Andy makes the 2020th left turn?
蚂蚁 Andy 生活在一个坐标平面上，目前位于 $(-20, 20)$，面朝东（即正 $x$ 方向）。Andy 移动 1 个单位，然后左转 $90^\circ$。然后，他移动 2 个单位（向北）并左转 $90^\circ$。接着移动 3 个单位（向西）并再次左转 $90^\circ$。Andy 继续这个过程，每次移动距离增加 1 个单位，并且总是左转。Andy 进行第 2020 次左转时的位置是哪里？
Q14
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region---inside the hexagon but outside all of the semicircles?
如图所示，六条半圆位于边长为 2 的正六边形内部，其半圆的直径与六边形的边重合。阴影区域（在六边形内但在所有半圆外）的面积是多少？
stem
Q15
Steve wrote the digits 1, 2, 3, 4, and 5 in order repeatedly from left to right, forming a list of 10,000 digits, beginning 123451234512 . . . . He then erased every third digit from his list (that is, the 3rd, 6th, 9th, . . . digits from the left), then erased every fourth digit from the resulting list (that is, the 4th, 8th, 12th, . . . digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in positions 2019, 2020, and 2021?
Steve 将数字 1, 2, 3, 4, 5 按顺序反复从左到右写成一个 10,000 个数字的列表，开头是 123451234512 ... 。然后他从列表中删除每个第 3 个数字（即从左数的第 3、6、9... 个数字），接着从剩余列表中删除每个第 4 个数字（即剩余列表从左数的第 4、8、12... 个数字），然后从那时剩下的中删除每个第 5 个数字。当时位置 2019、2020 和 2021 的三个数字之和是多少？
Q16
Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than 4. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
Bela 和 Jenn 在实数轴上的闭区间 $[0, n]$ 上玩以下游戏，其中 $n$ 是大于 4 的固定整数。他们轮流玩，Bela 先手。在他的第一回合，Bela 选择区间 $[0, n]$ 中的任意实数。此后，轮到的一方选择一个与之前双方选择的所有数都距离超过 1 个单位的实数。无法选择这样数的玩家输。使用最优策略，谁会赢？
Q17
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
有 10 个人等间距站在一个圆周上。每人确切认识其他 9 人中的 3 人：站在他/她旁边 2 人和圆周正对面的人。有多少种方法让这 10 人分成 5 对，使得每对成员互相认识？
Q18
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
一个瓮中有一个红球和一个蓝球。旁边有一个额外红蓝球的盒子。George 执行以下操作 4 次：他随机从瓮中抽一个球，然后从盒子取一个同色球，并将这两个同色球放回瓮中。4 次迭代后，瓮中有 6 个球。瓮中包含每种颜色 3 个球的概率是多少？
Q19
In a certain card game, a player is dealt a hand of 10 cards from a deck of 52 distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as 158A00A4AA0. What is the digit A ?
在某种纸牌游戏中，玩家从 52 张不同牌的牌堆中分到 10 张牌。可以分给玩家的不同（无序）牌型数量可以写成 158A00A4AA0。A 是多少？
Q20
Let $B$ be a right rectangular prism (box) with edge lengths 1, 3, and 4, together with its interior. For real $r \ge 0$, let $S(r)$ be the set of points in 3-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^3 + br^2 + cr + d$, where $a$, $b$, $c$, and $d$ are positive real numbers. What is $\frac{bc}{ad}$?
设 $B$ 是一个边长为 1、3 和 4 的直角长方体（盒子）连同其内部。对于实数 $r \ge 0$，设 $S(r)$ 为三维空间中距离 $B$ 中某点的距离不超过 $r$ 的点的集合。$S(r)$ 的体积可以表示为 $ar^3 + br^2 + cr + d$，其中 $a$、$b$、$c$ 和 $d$ 是正实数。$\frac{bc}{ad}$ 是多少？
Q21
In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE = AH$. Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1$. What is $FI^2$?
在正方形 $ABCD$ 中，点 $E$ 和 $H$ 分别位于 $\overline{AB}$ 和 $\overline{DA}$ 上，使得 $AE = AH$。点 $F$ 和 $G$ 分别位于 $\overline{BC}$ 和 $\overline{CD}$ 上，点 $I$ 和 $J$ 位于 $\overline{EH}$ 上，使得 $\overline{FI} \perp \overline{EH}$ 和 $\overline{GJ} \perp \overline{EH}$。见下图。三角形 $AEH$、四边形 $BFIE$、四边形 $DHJG$ 和五边形 $FCGJI$ 的面积均为 $1$。求 $FI^2$。
stem
Q22
What is the remainder when $2^{202} + 202$ is divided by $2^{101} + 2^{51} + 1$?
求 $2^{202} + 202$ 除以 $2^{101} + 2^{51} + 1$ 的余数。
Q23
Square $ABCD$ in the coordinate plane has vertices at the points $A(1, 1)$, $B(-1, 1)$, $C(-1, -1)$, and $D(1, -1)$. Consider the following four transformations: $\bullet$ $L$, a rotation of $90^\circ$ counterclockwise around the origin; $\bullet$ $R$, a rotation of $90^\circ$ clockwise around the origin; $\bullet$ $H$, a reflection across the $x$-axis; and $\bullet$ $V$, a reflection across the $y$-axis. Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. How many sequences of 20 transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions?
坐标平面上的正方形 $ABCD$ 的顶点为 $A(1, 1)$、$B(-1, 1)$、$C(-1, -1)$ 和 $D(1, -1)$。考虑以下四个变换：\bullet$ $L$，绕原点逆时针旋转 $90^\circ$；\bullet$ $R$，绕原点顺时针旋转 $90^\circ$；\bullet$ $H$，关于 $x$ 轴反射；\bullet$ $V$，关于 $y$ 轴反射。这些变换都将正方形映射到自身，但标记顶点的位置会改变。从 $\{L, R, H, V\}$ 中选择 20 个变换的序列有多少个能使所有标记顶点回到原始位置？
Q24
How many positive integers $n$ satisfy $\frac{n + 1000}{70} = \lfloor \sqrt{n} \rfloor$? (Recall that $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)
有几个正整数 $n$ 满足 $\frac{n + 1000}{70} = \lfloor \sqrt{n} \rfloor$？（回想 $\lfloor x \rfloor$ 是不超过 $x$ 的最大整数。）
Q25
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product $n = f_1 \cdot f_2 \cdots f_k$, where $k \ge 1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2 \cdot 3$, and $3 \cdot 2$, so $D(6) = 3$. What is $D(96)$?
设 $D(n)$ 表示将正整数 $n$ 表示为 $n = f_1 \cdot f_2 \cdots f_k$ 的方法数，其中 $k \ge 1$，$f_i$ 均为严格大于 $1$ 的整数，且因子列出的顺序重要（即仅因子顺序不同的两种表示被视为不同）。例如，$6$ 可以写成 $6$、$2 \cdot 3$ 和 $3 \cdot 2$，故 $D(6) = 3$。求 $D(96)$。
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