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AMC10 2020 A

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AMC10 · 2020 (A)

Q1
What value of $x$ satisfies $x - \frac{3}{4} = \frac{5}{12} - \frac{1}{3}$?
什么值$x$满足方程$x - \frac{3}{4} = \frac{5}{12} - \frac{1}{3}$?
Q2
The numbers 3, 5, 7, $a$, and $b$ have an average (arithmetic mean) of 15. What is the average of $a$ and $b$?
数字3、5、7、$a$和$b$的平均数(算术平均)为15。$a$和$b$的平均数是多少?
Q3
Assuming $a \neq 3$, $b \neq 4$, and $c \neq 5$, what is the value in simplest form of the following expression? $\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$
假设$a \neq 3$,$b \neq 4$,$c \neq 5$,下式的最简形式的值是多少?$\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$
Q4
A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid \$0.50 per mile, and her only expense is gasoline at \$2.00 per gallon. What is her net rate of pay, in dollars per hour, after this expense?
一位司机以60英里/小时的速度行驶2小时,其间她的汽车每加仑汽油行驶30英里。她每英里获薪0.50美元,唯一开支是每加仑2.00美元的汽油。扣除此开支后,她的净时薪是多少美元/小时?
Q5
What is the sum of all real numbers $x$ for which $|x^2 - 12x + 34| = 2$?
所有满足$|x^2 - 12x + 34| = 2$的实数$x$之和是多少?
Q6
How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5?
有多少个仅由偶数位组成的4位正整数(即1000到9999之间的整数,包含两端)能被5整除?
Q7
The 25 integers from −10 to 14, inclusive, can be arranged to form a 5-by-5 square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
从−10到14(包含两端)的25个整数可以排列成一个5×5的方阵,其中每行之和、每列之和以及两条主对角线之和都相同。这个公共和的值是多少?
Q8
What is the value of $1 + 2 + 3 -4 + 5 + 6 + 7 -8 + \cdots + 197 + 198 + 199 -200$?
求$1 + 2 + 3 -4 + 5 + 6 + 7 -8 + \cdots + 197 + 198 + 199 -200$的值。
Q9
A single bench section at a school event can hold either 7 adults or 11 children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N$?
学校活动中的一个长椅段可以容纳7个成人或11个儿童。当$N$个长椅段首尾相连时,相同数量的成人和儿童坐在一起将正好占满所有长椅空间。$N$的最小正整数值是多少?
Q10
Seven cubes, whose volumes are 1, 8, 27, 64, 125, 216, and 343 cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
七个立方体,体积分别为1、8、27、64、125、216和343立方单位,沿垂直方向堆叠成塔,体积从底部到顶部递减。除了最底部的立方体,每个立方体的底面完全位于下方立方体的顶面上。塔的总表面积(包括底部)有多少平方单位?
Q11
What is the median of the following list of 4040 numbers? $1, 2, 3, \dots, 2020, 1^{2}, 2^{2}, 3^{2}, \dots, 2020^{2}$
以下 4040 个数的列表的中位数是多少? $1, 2, 3, \dots, 2020, 1^{2}, 2^{2}, 3^{2}, \dots, 2020^{2}$
Q12
Triangle $AMC$ is isosceles with $AM = AC$. Medians $\overline{MV}$ and $\overline{CU}$ are perpendicular to each other, and $MV = CU = 12$. What is the area of $\triangle AMC$?
三角形 $AMC$ 是等腰三角形,$AM = AC$。中线 $\overline{MV}$ 和 $\overline{CU}$ 相互垂直,且 $MV = CU = 12$。三角形 $AMC$ 的面积是多少?
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Q13
A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices (0, 0), (0, 4), (4, 4), and (4, 0). What is the probability that the sequence of jumps ends on a vertical side of the square?
一只青蛙坐在点 (1, 2),开始一系列跳跃,每跳平行于坐标轴,长度为 1,每次跳的方向(上、下、右、左)独立随机选择。序列在青蛙到达顶点为 (0, 0)、(0, 4)、(4, 4) 和 (4, 0) 的正方形边时结束。序列结束在正方形垂直边上的概率是多少?
Q14
Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of $\frac{x + x^{3}}{y^{2} + y^{3}} \times (x^{2} + y)$?
实数 $x$ 和 $y$ 满足 $x + y = 4$ 和 $x \cdot y = -2$。求 $\frac{x + x^{3}}{y^{2} + y^{3}} \times (x^{2} + y)$ 的值?
Q15
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
随机选择 $12!$ 的一个正整数除数。所选除数是完全平方的概率可以表示为 $\frac{m}{n}$,其中 $m$ 和 $n$ 是互质的正整数。求 $m+n$?
Q16
A point is chosen at random within the square in the coordinate plane whose vertices are $(0,0)$, $(2020,0)$, $(2020,2020)$, and $(0,2020)$. The probability that the point lies within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x,y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth?
在坐标平面内的一个正方形中随机选择一点,该正方形的顶点为$(0,0)$、$(2020,0)$、$(2020,2020)$和$(0,2020)$。该点距离格点$d$单位以内的概率为$\frac{1}{2}$。(点$(x,y)$是格点当且仅当$x$和$y$均为整数。)$d$的最接近的十分位数是多少?
Q17
Define $P(x) = (x - 1^2)(x - 2^2)\cdots(x - 100^2)$. How many integers $n$ are there such that $P(n) \le 0$?
定义$P(x) = (x - 1^2)(x - 2^2)\cdots(x - 100^2)$。有多少个整数$n$使得$P(n) \le 0$?
Q18
Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}$. For how many such quadruples is it true that $a \cdot d - b \cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0 \cdot 1 - 3 \cdot 1 = -3$ is odd.)
让$(a,b,c,d)$为一个有序四元组,其中的元素(不一定不同)均来自集合$\{0,1,2,3\}$。有多少这样的四元组满足$a \cdot d - b \cdot c$为奇数?(例如,$(0,3,1,1)$是一个这样的四元组,因为$0 \cdot 1 - 3 \cdot 1 = -3$为奇数。)
Q19
As shown in the figure below, a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
如图所示,一个正十二面体(由12个全等正五边形面组成的多面体)在空间中漂浮,有两个水平面。注意,顶部面相邻有5个倾斜面组成的环,底部面相邻也有5个倾斜面组成的环。从顶部面到底部面,通过相邻面的序列移动,有多少种方法,使得每个面至多访问一次,且不允许从底部环移动到顶部环?
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Q20
Quadrilateral ABCD satisfies ∠ABC = ∠ACD = 90°, AC = 20, and CD = 30. Diagonals AC and BD intersect at point E, and AE = 5. What is the area of quadrilateral ABCD?
四边形ABCD满足∠ABC = ∠ACD = 90°,AC = 20,CD = 30。对角线AC和BD相交于点E,且AE = 5。四边形ABCD的面积是多少?
Q21
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that $\frac{2^{289} + 1}{2^{17} + 1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}$. What is $k$?
存在唯一的严格递增的非负整数序列 $a_1 < a_2 < \dots < a_k$ 使得 $\frac{2^{289} + 1}{2^{17} + 1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}$。$k$ 是多少?
Q22
For how many positive integers $n \le 1000$ is $\left\lfloor \frac{998}{n} \right\rfloor + \left\lfloor \frac{999}{n} \right\rfloor + \left\lfloor \frac{1000}{n} \right\rfloor$ not divisible by 3? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)
对于多少个正整数 $n \le 1000$,$\left\lfloor \frac{998}{n} \right\rfloor + \left\lfloor \frac{999}{n} \right\rfloor + \left\lfloor \frac{1000}{n} \right\rfloor$ 不能被 3 整除?(回忆 $\lfloor x \rfloor$ 是小于或等于 $x$ 的最大整数。)
Q23
Let $T$ be the triangle in the coordinate plane with vertices $(0, 0)$, $(4, 0)$, and $(0, 3)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^\circ$, $180^\circ$, and $270^\circ$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the 125 sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^\circ$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^\circ$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)
让 $T$ 为坐标平面上的三角形,顶点为 $(0, 0)$、$(4, 0)$ 和 $(0, 3)$。考虑平面的以下五种等距变换(刚性变换):绕原点逆时针旋转 $90^\circ$、$180^\circ$ 和 $270^\circ$,反射过 $x$ 轴,反射过 $y$ 轴。其中 125 种由这些变换组成的三个变换序列(不一定不同)有多少种会使 $T$ 回到原位置?(例如,$180^\circ$ 旋转后反射过 $x$ 轴再反射过 $y$ 轴会使 $T$ 回到原位置,但 $90^\circ$ 旋转后反射过 $x$ 轴再反射过 $x$ 轴不会。)
Q24
Let $n$ be the least positive integer greater than 1000 for which $\gcd(63, n + 120) = 21$ and $\gcd(n + 63, 120) = 60$. What is the sum of the digits of $n$?
让 $n$ 为大于 1000 的最小正整数,使得 $\gcd(63, n + 120) = 21$ 且 $\gcd(n + 63, 120) = 60$。$n$ 的各位数字之和是多少?
Q25
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
Jason 掷三个公平的标准六面骰子。然后他查看结果并选择一部分骰子(可能为空,可能全部三个)重新掷。重新掷后,当且仅当三个骰子上面数字之和恰为 7 时他获胜。Jason 总是为了优化获胜几率而玩。他选择重新掷恰好两个骰子的概率是多少?
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