# amc代考美国数学竞赛代考American Mathematics Competitions代考|AoPS Community

## amc代考美国数学竞赛代考American Mathematics Competitions代考|AoPS Community

2 Consider the statement, “If $n$ is not prime, then $n-2$ is prime.” Which of the following values of $n$ is a counterexample to this statement?
(A) 11
(B) 15
(C) 19
(D) 21
(E) 27
3 Which one of the following rigid transformations (isometries) maps the line segment $\overline{A B}$ onto the line segment $\overline{A^{\prime} B^{\prime}}$ so that the image of $A(-2,1)$ is $A^{\prime}(2,-1)$ and the image of $B(-1,4)$ is $B^{\prime}(1,-4)$ ?
(A) reflection in the $y$-axis (B) counterclockwise rotation around the origin by $90^{\circ}$ (C) translation by 3 units to the right and 5 units down (D) reflection in the $x$-axis (E) clockwise rotation about the origin by $180^{\circ}$
4 A positive integer $n$ satisfies the equation $(n+1) !+(n+2) !=n ! \cdot 440$. What is the sum of the digits of $n$ ?
(A) 2
(B) 5
(C) 10
(D) 12
(E) 15
5 Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$ ?
(A) 18
(B) 21
(C) 24
(D) 25
(E) 28
6 In a given plane, points $A$ and $B$ are 10 units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle A B C$ is 50 units and the area of $\triangle A B C$ is 100 square units?
(A) 0
(B) 2
(C) 4
(D) 8
(E) infinitely many
7 What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17$, and $x$ is equal to the mean of those five numbers?
(A) $-5$
(B) 0
(C) 5
(D) $\frac{15}{4}$
(E) $\frac{35}{4}$

## amc代考美国数学竞赛代考American Mathematics Competitions代考|AoPS Community

16 There are lily pads in a row numbered 0 to 11 , in that order. There are predators on lily pads 3 and 6 , and a morsel of food on lily pad 10 . Fiona the frog starts on pad 0 , and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6 ?
(A) $\frac{15}{256}$
(B) $\frac{1}{16}$
(C) $\frac{15}{128}$
(D) $\frac{1}{8}$
(E) $\frac{1}{4}$
17 How many nonzero complex numbers $z$ have the property that $0, z$, and $z^3$, when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
(A) 0
(B) 1
(C) 2
(D) 4
(E) infinitely many
18 Square pyramid $A B C D E$ has base $A B C D$, which measures $3 \mathrm{~cm}$ on a side, and altitude $\overline{A E}$ perpendicular to the base, which measures $6 \mathrm{~cm}$. Point $P$ lies on $\overline{B E}$, one third of the way from $B$ to $E$; point $Q$ lies on $\overline{D E}$, one third of the way from $D$ to $E$; and point $R$ lies on $\overline{C E}$, two thirds of the way from $C$ to $E$. What is the area, in square centimeters, of $\triangle P Q R$ ?
(A) $\frac{3 \sqrt{2}}{2}$
(B) $\frac{3 \sqrt{3}}{2}$
(C) $2 \sqrt{2}$
(D) $2 \sqrt{3}$
(E) $3 \sqrt{2}$
19 Raashan, Sylvia, and Ted play the following game. Each starts with $\$ 1$. A bell rings every 15 seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives$\$1$ to that player. What is the probability that after the bell has rung 2019 times, each player will have $\$ 1$? (For example, Raashan and Ted may each decide to give$\$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\$ 0$, Sylvia would have$\$2$, and Ted would have $\$ 1$, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their$\$1$ to, and and the holdings will be the same as the end of the second [sic] round.
(A) $\frac{1}{7}$
(B) $\frac{1}{4}$
(C) $\frac{1}{3}$
(D) $\frac{1}{2}$
(E) $\frac{2}{3}$

# 美国数学竞赛代考

## amc代考美国数学竞赛代考American Mathematics Competitions代考|AoPS Community

2 考虑以下陈述：“如果n不是素数，那么n−2是主要的。” 下列哪个值n是这个说法的反例吗？
(A) 11
(B) 15
(C) 19
(D) 21
(E) 27
3 以下哪个刚性变换（等距）映射线段一个乙¯到线段上一个′乙′¯使图像一个(−2,1)是一个′(2,−1)和图像乙(−1,4)是乙′(1,−4)?
(A) 反思是-axis (B) 绕原点逆时针旋转90∘(C) 向右平移 3 个单位，向下平移 5 个单位 (D) 反射X-axis (E) 绕原点顺时针旋转180∘
4 一个正整数n满足方程(n+1)!+(n+2)!=n!⋅440. 数字的总和是多少n?
(A) 2
(B) 5
(C) 10
(D) 12
(E) 15
5 商店中的每一块糖果的价格都是整数美分。Casper 有足够的钱购买 12 颗红色糖果、14 颗绿色糖果、15 颗蓝色糖果，或者n块紫色糖果。一块紫色糖果要 20 美分。的最小可能值是多少n?
(A) 18
(B) 21
(C) 24
(D) 25
(E) 28
6 在给定平面上，点一个和乙相隔10个单位。多少分C是否存在于平面中，使得周长△一个乙C为 50 个单位，面积为△一个乙C100平方单位是多少？
(A) 0
(B) 2
(C) 4
(D) 8
(E) 无穷多
7 所有实数的和是多少X其中数字的中位数4,6,8,17， 和X是否等于这五个数字的平均值？
（一个）−5
(B) 0
(C) 5
(D)154
（和）354

## amc代考美国数学竞赛代考American Mathematics Competitions代考|AoPS Community

16 睡莲按顺序排列，编号为 0 到 11。睡莲 3 和 6 上有捕食者，睡莲 10 上有一点食物。Fiona the frog 从垫子 0 开始，从任何给定的睡莲垫上，都有一个12有机会跳到下一个垫子，跳 2 个垫子的机会均等。Fiona 到达 pad 10 而没有降落在 pad 3 或 pad 6 上的概率是多少？
（一个）15256
(乙)116
（C）15128
(四)18
（和）14
17 有多少个非零复数和拥有的财产0,和， 和和3，当用复平面中的点表示时，是等边三角形的三个不同顶点吗？
(A) 0
(B) 1
(C) 2
(D) 4
(E) 无穷多
18 四棱锥一个乙CD和有基地一个乙CD, 测量3 C米在一边，和高度一个和¯垂直于底座，测量6 C米. 观点磷位于乙和¯，距离的三分之一乙至和; 观点问位于D和¯，距离的三分之一D至和; 并指出R位于C和¯, 三分之二的距离C至和. 面积是多少平方厘米△磷问R?
（一个）322
(乙)332
（C）22
(四)23
（和）32
19 Raashan、Sylvia 和 Ted 玩以下游戏。每个都以$1. 每 15 秒钟响一次，此时每个当前有钱的玩家同时独立随机选择另外两个玩家中的一个，并给出$1给那个玩家。2019年钟声敲响后，每个玩家拥有的概率是多少$1? （例如，Raashan 和 Ted 可能各自决定给$1给 Sylvia，Sylvia 可能决定把她的美元给 Ted，此时 Raashan 将拥有$0, 西尔维娅会$2, 泰德会$1，然后第一轮比赛就结束了。第二轮 Raashan 没有钱给，但 Sylvia 和 Ted 可能会选择对方给他们$1to, and and the holdings 将与第二轮 [原文如此] 结束时相同。
（一个）17
(乙)14
（C）13
(四)12
（和）23

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