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2022 AMC 10A Problems 2022 AMC 10A 问题

 

Problem 1 问题 1

What is the value of\[3+\frac{1}{3+\frac{1}{3+\frac13}}?\]$\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{33}{10}\qquad\textbf{(D)}\ \frac{109}{33}\qquad\textbf{(E)}\ \frac{15}{4}$
价值是什么 $\textbf{(A)}\ \frac{31}{10}\qquad\textbf{(B)}\ \frac{49}{15}\qquad\textbf{(C)}\ \frac{33}{10}\qquad\textbf{(D)}\ \frac{109}{33}\qquad\textbf{(E)}\ \frac{15}{4}$


Problem 2 问题 2

Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes?
迈克在几分钟内 $57$ 循环了 $15$ 一圈。假设他全程以恒定的速度骑行。他在最初的 $27$ 几分钟里大约完成了多少圈?

$\textbf{(A) } 5 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 13$


Problem 3 问题 3

The sum of three numbers is $96.$ The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?
三个数字之和为 $96.$ 第一个数字 $6$ 乘以第三个数字,第三个数字 $40$ 小于第二个数字。第一个和第二个数字之差的绝对值是多少?

$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$


Problem 4 问题 4

In some countries, automobile fuel efficiency is measured in liters per $100$ kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals $m$ miles, and $1$ gallon equals $l$ liters. Which of the following gives the fuel efficiency in liters per $100$ kilometers for a car that gets $x$ miles per gallon?
在一些国家/地区,汽车燃油效率以升/ $100$ 公里为单位,而其他国家/地区则使用每加仑英里数。假设 1 公里等于 $m$ 英里, $1$ 加仑等于 $l$ 升。以下哪一项给出了每加仑行驶 $x$ 里程的汽车的燃油效率(以升/ $100$ 公里为单位)?

$\textbf{(A) } \frac{x}{100lm} \qquad \textbf{(B) } \frac{xlm}{100} \qquad \textbf{(C) } \frac{lm}{100x} \qquad \textbf{(D) } \frac{100}{xlm} \qquad \textbf{(E) } \frac{100lm}{x}$


Problem 5 问题 5

Square $ABCD$ has side length $1$. Points $P$$Q$$R$, and $S$ each lie on a side of $ABCD$ such that $APQCRS$ is an equilateral convex hexagon with side length $s$. What is $s$?
方块 $ABCD$ 有边长 $1$ 。点 $P$$Q$$R$ 、 和 $S$ 每个点都位于 $ABCD$$APQCRS$$s$ 为 的等边凸六边形 的一侧。什么是 $s$

$\textbf{(A) } \frac{\sqrt{2}}{3} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } 2 - \sqrt{2} \qquad \textbf{(D) } 1 - \frac{\sqrt{2}}{4} \qquad \textbf{(E) } \frac{2}{3}$


Problem 6 问题 6

Which expression is equal to\[\left|a-2-\sqrt{(a-1)^2}\right|\]for $a<0?$
哪个表达式等于 $a<0?$

$\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$


Problem 7 问题 7

The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?
正整数 $n$$18$ 的最小公倍数是 $180$ ,而 和 $45$ 的最大 $n$ 公约数是 $15$ 。的位数之和是多少 $n$

$\textbf{(A) } 3 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 12$


Problem 8 问题 8

A data set consists of $6$ (not distinct) positive integers: $1$$7$$5$$2$$5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all possible values of $X$?
数据集由 $6$ (不是不同的)正整数组成: $1$ 、、 $7$$5$ $2$ $5$ 、 和 $X$$6$ 数字的平均值(算术平均值)等于数据集中的一个值。的所有可能值之和是多少 $X$

$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 40$


Problem 9 问题 9

A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color – red, orange, yellow, blue, or green – so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
矩形被划分为 $5$ 多个区域,如图所示。每个区域都将被涂成纯色 - 红色、橙色、黄色、蓝色或绿色 - 以便接触的区域被涂上不同的颜色,并且颜色可以多次使用。有多少种不同的颜色可供选择?

[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]

$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$


Problem 10 问题 10

Daniel finds a rectangular index card and measures its diagonal to be $8$ centimeters. Daniel then cuts out equal squares of side $1$ cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card?[asy] // Diagram by MRENTHUSIASM, edited by Djmathman size(200); defaultpen(linewidth(0.6)); draw((489.5,-213) -- (225.5,-213) -- (225.5,-185) -- (199.5,-185) -- (198.5,-62) -- (457.5,-62) -- (457.5,-93) -- (489.5,-93) -- cycle); draw((206.29,-70.89) -- (480.21,-207.11), linetype ("6 6"),Arrows(size=4,arrowhead=HookHead)); draw((237.85,-182.24) -- (448.65,-95.76),linetype ("6 6"),Arrows(size=4,arrowhead=HookHead)); label("$1$",(450,-80)); label("$1$",(475,-106)); label("$8$",(300,-103)); label("$4\sqrt 2$",(300,-173)); [/asy]$\textbf{(A) } 14 \qquad \textbf{(B) } 10\sqrt{2} \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 12\sqrt{2} \qquad \textbf{(E) } 18$
Daniel 找到一张矩形索引卡,测量它的对角线为 $8$ 厘米。然后,Daniel 在索引卡的两个相对角上切出边 $1$ cm 的相等方格,并将这些方格的两个最近顶点之间的距离测量为 $4\sqrt{2}$ 厘米,如下所示。原始索引卡的面积是多少? $\textbf{(A) } 14 \qquad \textbf{(B) } 10\sqrt{2} \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 12\sqrt{2} \qquad \textbf{(E) } 18$


Problem 11 问题 11

Ted mistakenly wrote $2^m\cdot\sqrt{\frac{1}{4096}}$ as $2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ for which these two expressions have the same value?
Ted 错误地写 $2^m\cdot\sqrt{\frac{1}{4096}}$$2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ that both expressions have the same value?

$\textbf{(A) } 5 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$


Problem 12 问题 12

On Halloween $31$ children walked into the principal’s office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
万圣节那天 $31$ ,孩子们走进校长办公室要糖果。他们可以分为三种类型:有些人总是撒谎;有些人总是说真话;有些人交替撒谎和说真话。替代者可以任意选择他们的第一个响应,要么是谎言,要么是真值,但每个后续陈述都具有与其前一个相反的真值。校长按此顺序向每个人提出了相同的三个问题。

“Are you a truth-teller?” The principal gave a piece of candy to each of the $22$ children who answered yes.
“你是讲真话的人吗?”校长给每个回答“是” $22$ 的孩子发了一块糖果。

“Are you an alternater?” The principal gave a piece of candy to each of the $15$ children who answered yes.
“你是候补车手吗?”校长给每个回答“是” $15$ 的孩子发了一块糖果。

“Are you a liar?” The principal gave a piece of candy to each of the $9$ children who answered yes.
“你是骗子吗?”校长给每个回答“是” $9$ 的孩子发了一块糖果。

How many pieces of candy in all did the principal give to the children who always tell the truth?
校长一共给了那些总是说真话的孩子们多少颗糖果呢?

$\textbf{(A) } 7 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 31$


Problem 13 问题 13

Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP=2$ and $PC=3.$ What is $AD?$
$\triangle ABC$ 为斜角三角形。点 $P$ 位于 $\overline{BC}$ 上 因此 $\overline{AP}$ 平分 $\angle BAC.$ 通过 $B$ 垂直于 $\overline{AP}$ 的线与平行于 $\overline{BC}$$D.$ 的线 $A$ 相交 假设 $BP=2$$PC=3.$ 什么是 $AD?$

$\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$


Problem 14 问题 14

How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?
有多少种方法可以将整数 $1$ $14$ 分成 $7$ 对,以便在每对中,较大的数字至少 $2$ 是较小数字的乘以?

$\textbf{(A) } 108 \qquad \textbf{(B) } 120 \qquad \textbf{(C) } 126 \qquad \textbf{(D) } 132 \qquad \textbf{(E) } 144$


Problem 15 问题 15

Quadrilateral $ABCD$ with side lengths $AB=7, BC=24, CD=20, DA=15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi-b}{c},$ where $a,b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c?$
边长的 $AB=7, BC=24, CD=20, DA=15$ 四边 $ABCD$ 形内接在一个圆圈中。圆内部但四边形外部的面积可以写成其中 $a,b,$ 和 是正整数的形式 $\frac{a\pi-b}{c},$ ,使得 $a$$c$ 没有公共质因 $c$ 数。什么 $a+b+c?$

$\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$


Problem 16 问题 16

The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
多项式 $10x^3 - 39x^2 + 29x - 6$ 的根是矩形框(直直角棱柱)的高度、长度和宽度。通过按 $2$ 单位延长原始框的每个边缘来形成一个新的矩形框。新盒子的体积是多少?

$\textbf{(A) } \frac{24}{5} \qquad \textbf{(B) } \frac{42}{5} \qquad \textbf{(C) } \frac{81}{5} \qquad \textbf{(D) } 30 \qquad \textbf{(E) } 48$


Problem 17 问题 17

How many three-digit positive integers $\underline{a} \ \underline{b} \ \underline{c}$ are there whose nonzero digits $a,b,$ and $c$ satisfy\[0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?\](The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ is the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$)
有多少位三位数的正整数 $\underline{a} \ \underline{b} \ \underline{c}$ ,其非零数字 $a,b,$$c$ 满足(条形表示重复,因此 $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ 是无限重复十进制 $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$

$\textbf{(A) } 9 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$


Problem 18 问题 18

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3,...,T_n$ returns the point $(1, 0)$ back to itself?
$T_k$ 为坐标平面的变换,该坐标平面首先围绕原点逆时针旋转 $k$ 度数,然后沿 $y$ -轴反射平面。执行转换序列将 $T_1, T_2, T_3,...,T_n$$(1, 0)$ 返回自身的最小正整数 $n$ 是多少?

$\textbf{(A) } 359 \qquad \textbf{(B) } 360 \qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$


Problem 19 问题 19

Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that\[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{17}=\frac{h}{L_{17}}\]What is the remainder when $h$ is divided by $17$?
定义为 $L_n$ 所有整数 from to (含 to $1$ $n$ ) 的最小公倍数。有一个唯一的整数 $h$ ,使得 What is the remainder when $h$ is divided by $17$

$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$


Problem 20 问题 20

A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$$60$, and $91$. What is the fourth term of this sequence?
通过将正整数的四项算术序列的每一项与正整数的四项几何序列的相应项相加来形成四项序列。生成的四项序列的前三项是 $57$$60$$91$ 。这个序列的第四项是什么?

$\textbf{(A) } 190 \qquad \textbf{(B) } 194 \qquad \textbf{(C) } 198 \qquad \textbf{(D) } 202 \qquad \textbf{(E) } 206$


Problem 21 问题 21

A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$. The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?[asy] import three; size(225); currentprojection=   orthographic(camera=(-5.52541796301147,-2.61548797564715,1.6545450372312),                up=(0.00247902062334861,0.000877141782387748,0.00966536329192992),                target=(0,0,0),                zoom=0.570588560870951); currentpen = black+1.5bp; triple A = O; triple M = (X+Y)/2; triple B = (-1/2,-1/2,1/sqrt(2)); triple C = (-1,0,sqrt(2)); triple D = (0,-1,sqrt(2)); transform3 rho = rotate(90,M,M+Z);  //arrays of vertices for the lower level (the square), the middle level, //and the interleaves vertices of the upper level (the octagon) triple[] lVs = {A}; triple[] mVs = {B}; triple[] uVsl = {C}; triple[] uVsr = {D};  for(int i = 0; i < 3; ++i){   lVs.push(rho*lVs[i]);   mVs.push(rho*mVs[i]);   uVsl.push(rho*uVsl[i]);   uVsr.push(rho*uVsr[i]); }  lVs.cyclic = true; uVsl.cyclic = true;  for(int i : new int[] {0,1,2,3}){   draw(uVsl[i]--uVsr[i]);   draw(uVsr[i]--uVsl[i+1]); } draw(lVs[0]--lVs[1]^^lVs[0]--lVs[3]); for(int i : new int[] {0,1,3}){   draw(lVs[0]--lVs[i]);   draw(lVs[i]--mVs[i]);   draw(mVs[i]--uVsl[i]); } for(int i : new int[] {0,3}){   draw(mVs[i]--uVsr[i]); }  for(int i : new int[] {1,3}) draw(lVs[2]--lVs[i],dashed); draw(lVs[2]--mVs[2],dashed); draw(mVs[2]--uVsl[2]^^mVs[2]--uVsr[2],dashed); draw(mVs[1]--uVsr[1],dashed);  //Comment two lines below to remove red edges //draw(lVs[1]--lVs[3],red+2bp); //draw(uVsl[0]--uVsr[0],red+2bp); [/asy]$\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5+2\sqrt{2} \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$
碗是通过将边 $1$ 的四个正六边形连接到边 $1$ 的正方形 上而形成的。相邻六边形的边重合,如图所示。通过连接位于碗边缘的四个六边形的前八个顶点得到的八边形的面积是多少? $\textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5+2\sqrt{2} \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$


Problem 22 问题 22

Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on the fourth pass, and $11, 12, 13$ on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?
假设编号的 $13$ $1, 2, 3, \ldots, 13$ 卡片排成一行。任务是按数字递增的顺序拾取它们,从左到右重复工作。在下面的示例中,在第一次传递、 $4$ $5$ 第二次传递、 $6$ 第三次传递、 $7, 8, 9, 10$ 第四次传递和 $11, 12, 13$ 第五次传递时拾取卡片 $1, 2, 3$ 。这些 $13$ 牌的 $13!$ 可能顺序中,将恰好两次通过?

[asy] size(11cm); draw((0,0)--(2,0)--(2,3)--(0,3)--cycle); label("7", (1,1.5)); draw((3,0)--(5,0)--(5,3)--(3,3)--cycle); label("11", (4,1.5)); draw((6,0)--(8,0)--(8,3)--(6,3)--cycle); label("8", (7,1.5)); draw((9,0)--(11,0)--(11,3)--(9,3)--cycle); label("6", (10,1.5)); draw((12,0)--(14,0)--(14,3)--(12,3)--cycle); label("4", (13,1.5)); draw((15,0)--(17,0)--(17,3)--(15,3)--cycle); label("5", (16,1.5)); draw((18,0)--(20,0)--(20,3)--(18,3)--cycle); label("9", (19,1.5)); draw((21,0)--(23,0)--(23,3)--(21,3)--cycle); label("12", (22,1.5)); draw((24,0)--(26,0)--(26,3)--(24,3)--cycle); label("1", (25,1.5)); draw((27,0)--(29,0)--(29,3)--(27,3)--cycle); label("13", (28,1.5)); draw((30,0)--(32,0)--(32,3)--(30,3)--cycle); label("10", (31,1.5)); draw((33,0)--(35,0)--(35,3)--(33,3)--cycle); label("2", (34,1.5)); draw((36,0)--(38,0)--(38,3)--(36,3)--cycle); label("3", (37,1.5)); [/asy]$\textbf{(A) } 4082 \qquad \textbf{(B) } 4095 \qquad \textbf{(C) } 4096 \qquad \textbf{(D) } 8178 \qquad \textbf{(E) } 8191$


Problem 23 问题 23

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
等腰梯形 $ABCD$ 具有平行的边 $\overline{AD}$$\overline{BC},$$BC < AD$ $AB = CD.$ 在平面上有一个点 $P$ ,使得 $PA=1, PB=2, PC=3,$$PD=4.$ 什么是 $\tfrac{BC}{AD}?$

$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$


Problem 24 问题 24

How many strings of length $5$ formed from the digits $0$$1$$2$$3$$4$ are there such that for each $j \in \{1,2,3,4\}$, at least $j$ of the digits are less than $j$? (For example, $02214$ satisfies this condition because it contains at least $1$ digit less than $1$, at least $2$ digits less than $2$, at least $3$ digits less than $3$, and at least $4$ digits less than $4$. The string $23404$ does not satisfy the condition because it does not contain at least $2$ digits less than $2$.)
有多少个由数字 $0$$1$$2$ $3$ $4$ 、 形成的长度 $5$ 字符串,使得对于每个 $j \in \{1,2,3,4\}$ ,至少有 $j$ 数字小于 $j$ ?(例如, 满足此条件, $02214$ 因为它至少包含小于 $1$ $1$ 、 至少 $2$ 小于 $2$ 、 至少 $3$ 小于 、 至少小于 $3$ 和 至少 $4$ 小于 $4$ 的数字。字符串 $23404$ 不满足条件,因为它不包含至少 $2$ 小于 $2$ .

$\textbf{(A) }500\qquad\textbf{(B) }625\qquad\textbf{(C) }1089\qquad\textbf{(D) }1199\qquad\textbf{(E) }1296$


Problem 25 问题 25

Let $R$$S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S.$ See the figure (not drawn to scale).[asy] size(8cm); label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow()); draw((-34,0)--(60,0),EndArrow()); [/asy]The fraction of lattice points in $S$ that are in $S \cap T$ is $27$ times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?
$R$$S$ , 和 $T$ 为在坐标平面中的晶格点(即坐标均为整数的点)及其内部具有顶点的正方形。每个正方形的下边缘位于 $x$ -轴上。的左边缘 $R$ 和右边缘位于 $y$ -轴上, $R$ 并且包含 $\frac{9}{4}$ 的晶格点与 $S$ . $S$ 的顶部两个顶点 $T$ 位于 中 $R \cup S$ ,并且 $T$ 包含 $\frac{1}{4}$ $R \cup S.$ See the diagram 中包含的晶格点(未按比例绘制)。位于 中的 $S \cap T$ 晶格点 $S$ 的分数是 中位于 中的 $R \cap T$ 晶格点 $R$ 分数的乘 $27$ 以 。的边长 $R$ 加上 的边长 $S$ 加上 的 $T$ 边长 的最小可能值是多少?

$\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$