/

AMC12 2013 B

You are not logged in. After submit, your report may not be available on other devices. Login

AMC12 · 2013 (B)

Q1
On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was $3^{\circ}$. In degrees, what was the low temperature in Lincoln that day?
在某个特定的1月一天，林肯（内布拉斯加州）最高气温比最低气温高16度，而最高和最低气温的平均值为$3^{\circ}$。那天林肯的最低气温是多少度？
Q2
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr. Green’s steps is 2 feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
格林先生通过步行测量他的矩形花园的两条边，发现是15步乘20步。格林先生每步长2英尺。他期望花园每平方英尺产半磅土豆。格林先生期望从他的花园收获多少磅土豆？
Q3
When counting from 3 to 201, 53 is the 51st number counted. When counting backwards from 201 to 3, 53 is the $n$th number counted. What is $n$?
从3数到201时，53是第51个数。从201反向数到3时，53是第$n$个数。$n$是多少？
Q4
Ray’s car averages 40 miles per gallon of gasoline, and Tom’s car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars’ combined rate of miles per gallon of gasoline?
雷的车平均每加仑汽油行驶40英里，汤姆的车平均每加仑汽油行驶10英里。雷和汤姆各开相同的里程。两车的综合每加仑汽油里程数是多少？
Q5
The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders?
33名五年级学生的平均年龄是11岁。他们55位家长的平均年龄是33岁。所有这些家长和五年级学生的平均年龄是多少？
Q6
Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x -6y -34$. What is $x + y$?
实数 $x$ 和 $y$ 满足方程 $x^2 + y^2 = 10x -6y -34$。$x + y$ 等于多少？
Q7
Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saying “1”, so Blair follows by saying “1, 2”. Jo then says “1, 2, 3”, and so on. What is the 53rd number said?
Jo 和 Blair 轮流从 1 数到对方最后说的数加一。Jo 先说“1”，Blair 接着说“1, 2”。Jo 然后说“1, 2, 3”，依此类推。第 53 个说的数是多少？
Q8
Line $\ell_1$ has equation $3x -2y = 1$ and goes through $A = (-1, -2)$. Line $\ell_2$ has equation $y = 1$ and meets line $\ell_1$ at point $B$. Line $\ell_3$ has positive slope, goes through point $A$, and meets $\ell_2$ at point $C$. The area of $\triangle ABC$ is 3. What is the slope of $\ell_3$?
直线 $\ell_1$ 的方程为 $3x -2y = 1$，经过点 $A = (-1, -2)$。直线 $\ell_2$ 的方程为 $y = 1$，与直线 $\ell_1$ 相交于点 $B$。直线 $\ell_3$ 有正斜率，经过点 $A$，与 $\ell_2$ 相交于点 $C$。三角形 $\triangle ABC$ 的面积为 3。求 $\ell_3$ 的斜率。
Q9
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$?
求能整除 $12!$ 的最大的完全平方的平方根的质因数的指数之和。
Q10
Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
Alex 有 75 个红代币和 75 个蓝代币。有一个摊位，Alex 可以交 2 个红代币换取 1 个银代币和 1 个蓝代币；另一个摊位，可以交 3 个蓝代币换取 1 个银代币和 1 个红代币。Alex 继续交换直到无法再交换。最终 Alex 有多少银代币？
Q11
Two bees start at the same spot and fly at the same rate in the following directions. Bee A travels 1 foot north, then 1 foot east, then 1 foot upwards, and then continues to repeat this pattern. Bee B travels 1 foot south, then 1 foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly 10 feet away from each other?
两只蜜蜂从同一位置开始，以相同的速度向以下方向飞行。蜜蜂A向北飞行1英尺，然后向东1英尺，然后向上1英尺，然后继续重复此模式。蜜蜂B向南飞行1英尺，然后向西1英尺，然后继续重复此模式。当它们彼此相距恰好10英尺时，它们正在向哪些方向飞行？
Q12
Cities A, B, C, D, and E are connected by roads AB, AD, AE, BC, BD, CD, and DE. How many different routes are there from A to B that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
城市A、B、C、D和E通过道路AB、AD、AE、BC、BD、CD和DE连接。从A到B使用每条道路恰好一次的不同路径有多少条？（这样的路径必然会多次访问某些城市。）
stem
Q13
The internal angles of quadrilateral ABCD form an arithmetic progression. Triangles ABD and DCB are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of ABCD?
四边形ABCD的内角形成一个等差数列。三角形ABD和DCB相似，且$\angle DBA = \angle DCB$，$\angle ADB = \angle CBD$。此外，这两个三角形中的角度也形成等差数列。ABCD的两个最大角之和的最大可能值为多少度？
Q14
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$?
两个非递减的非负整数序列具有不同的首项。每个序列从第三项开始，每项是前两项之和，且每个序列的第七项均为$N$。$N$的最小可能值为多少？
Q15
The number 2013 is expressed in the form \[ 2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!}, \] where $a_1\ge a_2\ge\cdots\ge a_m$ and $b_1\ge b_2\ge\cdots\ge b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is $|a_1-b_1|$?
数 $2013$ 可以表示为 \[ 2013=\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!}, \] 其中 $a_1\ge a_2\ge\cdots\ge a_m$ 与 $b_1\ge b_2\ge\cdots\ge b_n$ 为正整数，并且在所有这样的表示中，$a_1+b_1$ 尽可能小。求 $|a_1-b_1|$。
Q16
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$?
设$ABCDE$为一个周长为$1$的等角凸五边形。将五边形各边所在的直线向外延长，这些直线两两相交所确定的交点构成一个五角星形多边形。设该星形的周长为$s$。问$s$的最大可能值与最小可能值之差是多少？
Q17
Let $a$, $b$, and $c$ be real numbers such that $$ \begin{cases} a+b+c=2, \text{ and}\\ a^2+b^2+c^2=12. \end{cases} $$ What is the difference between the maximum and minimum possible values of $c$?
设 $a$、$b$、$c$ 为实数，满足 $$ \begin{cases} a+b+c=2，\text{且}\\ a^2+b^2+c^2=12。 \end{cases} $$ 求 $c$ 的最大可能值与最小可能值之差是多少？
Q18
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with 2013 coins and when the game starts with 2014 coins?
Barbara 和 Jenna 玩下面这个轮流进行的游戏。桌上有若干枚硬币。轮到 Barbara 时，她必须拿走 2 枚或 4 枚硬币；但如果桌上只剩 1 枚硬币，则她这一回合无法行动并失去回合。轮到 Jenna 时，她必须拿走 1 枚或 3 枚硬币。由掷硬币决定谁先手。拿走最后一枚硬币的人赢得游戏。假设双方都采用最优策略：当游戏开始时有 2013 枚硬币，以及当开始时有 2014 枚硬币，分别是谁会获胜？
Q19
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $AD\perp BC$, $DE\perp AC$, and $AF\perp BF$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
在三角形 $ABC$ 中，$AB=13$，$BC=14$，$CA=15$。不同的点 $D$、$E$、$F$ 分别在线段 $\overline{BC}$、$\overline{CA}$、$\overline{DE}$ 上，并满足 $AD\perp BC$，$DE\perp AC$，且 $AF\perp BF$。线段 $\overline{DF}$ 的长度可以表示为 $\frac{m}{n}$，其中 $m$ 与 $n$ 为互质的正整数。求 $m+n$。
Q20
For $135^\circ<x<180^\circ$, points $P=(\cos x,\cos^2 x)$, $Q=(\cot x,\cot^2 x)$, $R=(\sin x,\sin^2 x)$, and $S=(\tan x,\tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$?
对于 $135^\circ<x<180^\circ$，点 $P=(\cos x,\cos^2 x)$、$Q=(\cot x,\cot^2 x)$、$R=(\sin x,\sin^2 x)$、$S=(\tan x,\tan^2 x)$ 构成一个梯形的四个顶点。求 $\sin(2x)$。
Q21
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in\{-2,-1,0,1,2\}$ and $b\in\{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
考虑如下定义的 30 条抛物线的集合：所有抛物线的焦点都是点 $(0,0)$，其准线为形如 $y=ax+b$ 的直线，其中 $a,b$ 为整数，且 $a\in\{-2,-1,0,1,2\}$、$b\in\{-3,-2,-1,1,2,3\}$。这些抛物线中任意三条没有公共点。问：平面上有多少个点恰好落在其中两条抛物线上？
Q22
Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation $$8(\log_{n} x)(\log_{m} x)-7\log_{n} x-6\log_{m} x-2013=0$$ is the smallest possible integer. What is $m+n$?
设 $m>1$ 且 $n>1$ 为整数。已知方程 $$8(\log_{n} x)(\log_{m} x)-7\log_{n} x-6\log_{m} x-2013=0$$ 的所有解 $x$ 的乘积是可能取得的最小整数。求 $m+n$。
Q23
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N=749$, Bernardo writes the numbers $10{,}444$ and $3{,}245$, and LeRoy obtains the sum $S=13{,}689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?
伯纳多选择一个三位正整数 $N$，并把它的五进制表示和六进制表示都写在黑板上。后来勒罗伊看到了伯纳多写下的两个数。他把这两个数都当作十进制整数相加，得到整数 $S$。例如，当 $N=749$ 时，伯纳多写下 $10{,}444$ 和 $3{,}245$，勒罗伊算得 $S=13{,}689$。问有多少个 $N$ 的取值，使得 $S$ 的最右边两位数字（按顺序）与 $2N$ 的最右边两位数字相同？
Q24
Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle ACB$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\triangle BXN$ is equilateral and $AC=2$. What is $BN^2$?
设$ABC$为一三角形，$M$为$\overline{AC}$的中点，$\overline{CN}$为$\angle ACB$的角平分线，且$N$在$\overline{AB}$上。设$X$为中线$\overline{BM}$与角平分线$\overline{CN}$的交点。另有$\triangle BXN$为等边三角形，且$AC=2$。求$BN^2$。
Q25
Let $G$ be the set of polynomials of the form $P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,$ where $c_1,c_2,\ldots,c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
设 $G$ 为如下形式多项式的集合： $P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,$ 其中 $c_1,c_2,\ldots,c_{n-1}$ 为整数，并且 $P(z)$ 有 $n$ 个互不相同的根，形如 $a+ib$，其中 $a,b$ 为整数。问集合 $G$ 中有多少个多项式？
Time Left 75:00