2016 AMC 10B — Official Competition Problems (February 2016)
📅 2016 B 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
综合
What is the value of \frac{2a^{-1}+\frac{a^{-1}}{2}}{a} when a= \tfrac{1}{2} ?
💡 解题思路
Factorizing the numerator, $\frac{\frac{1}{a}\cdot(2+\frac{1}{2})}{a}$ then becomes $\frac{\frac{5}{2}}{a^{2}}$ which is equal to $\frac{5}{2}\cdot 2^2$ which is $\boxed{\textbf{(D) }10}$ .
2
第 2 题
综合
If n\heartsuit m=n^3m^2 , what is \frac{2\heartsuit 4}{4\heartsuit 2} ?
💡 解题思路
$\frac{2^3(2^2)^2}{(2^2)^32^2}=\frac{2^7}{2^8}=\frac12$ which is $\textbf{(B)}$ .
3
第 3 题
方程
Let x=-2016 . What is the value of \Bigg\vert\Big\vert |x|-x\Big\vert-|x|\Bigg\vert-x ?
Zoey reads 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each book taking her 1 more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day the week did she finish her 15 th book?
💡 解题思路
The process took $1+2+3+\ldots+13+14+15=120$ days, so the last day was $119$ days after the first day. Since $119$ is divisible by $7$ , both must have been the same day of the week, so the answer is
5
第 5 题
统计
The mean age of Amanda's 4 cousins is 8 , and their median age is 5 . What is the sum of the ages of Amanda's youngest and oldest cousins?
💡 解题思路
The sum of the ages of the cousins is $4$ times the mean, or $32$ . There are an even number of cousins, so there is no single median, so $5$ must be the mean of the two in the middle. Therefore the s
6
第 6 题
规律与数列
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number S . What is the smallest possible value for the sum of the digits of S ?
💡 解题思路
Let the two three-digit numbers she added be $a$ and $b$ with $a+b=S$ and $a<b$ . The hundreds digits of these numbers must be at least $1$ and $2$ , so $a\ge 100$ and $b\ge 200$ .
7
第 7 题
分数与比例
The ratio of the measures of two acute angles is 5:4 , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
💡 解题思路
We can set up a system of equations where $x$ and $y$ are the two acute angles. WLOG, assume that $x$ $<$ $y$ in order for the complement of $x$ to be greater than the complement of $y$ . Therefore, $
8
第 8 题
数字运算
What is the tens digit of 2015^{2016}-2017?
💡 解题思路
Notice that, for $n\ge 2$ , $2015^n\equiv 15^n$ is congruent to $25\pmod{100}$ when $n$ is even and $75\pmod{100}$ when $n$ is odd. (Check for yourself). Since $2016$ is even, $2015^{2016} \equiv 25\p
9
第 9 题
几何·面积
All three vertices of \bigtriangleup ABC lie on the parabola defined by y=x^2 , with A at the origin and \overline{BC} parallel to the x -axis. The area of the triangle is 64 . What is the length of BC ?
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of 5 inches. Which of the following is closest to the weight, in ounces, of the second piece?
💡 解题思路
It is known that \( Density = \frac{mass}{volume} \). You can remember this through \( D = \frac{m}{v} \) (DMV - Like the car lol).
11
第 11 题
几何·面积
Carl decided to fence in his rectangular garden. He bought 20 fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly 4 yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
💡 解题思路
If the dimensions are $4a\times 4b$ , then one side will have $a+1$ posts (including corners) and the other $b+1$ (including corners).
12
第 12 题
概率
Two different numbers are selected at random from \{1, 2, 3, 4, 5\} and multiplied together. What is the probability that the product is even?
💡 解题思路
The product will be even if at least one selected number is even, and odd if none are. Using complementary counting, the chance that both numbers are odd is $\frac{\tbinom32}{\tbinom52}=\frac3{10}$ ,
13
第 13 题
数论
At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these 1000 babies were in sets of quadruplets?
💡 解题思路
We can set up a system of equations where $a$ is the sets of twins, $b$ is the sets of triplets, and $c$ is the sets of quadruplets. \[\begin{split} 2a + 3b + 4c & = 1000 \\ b & = 4c \\ a & = 3b \end{
14
第 14 题
几何·面积
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line y=π x , the line y=-0.1 and the line x=5.1?
💡 解题思路
The region is a right triangle which contains the following lattice points: $(0,0); (1,0)\rightarrow(1,3); (2,0)\rightarrow(2,6); (3,0)\rightarrow(3,9); (4,0)\rightarrow(4,12); (5,0)\rightarrow(5,15)$
15
第 15 题
几何·面积
All the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a 3×3 array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to 18 . What is the number in the center?
💡 解题思路
Consecutive numbers share an edge. That means that it is possible to walk from $1$ to $9$ by single steps north, south, east, or west. Consequently, the squares in the diagram with different shades ha
16
第 16 题
规律与数列
The sum of an infinite geometric series is a positive number S , and the second term in the series is 1 . What is the smallest possible value of S?
💡 解题思路
The sum of an infinite geometric series is of the form: \[\begin{split} S & = \frac{a_1}{1-r} \end{split}\] where $a_1$ is the first term and $r$ is the ratio whose absolute value is less than 1.
17
第 17 题
规律与数列
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
💡 解题思路
Let us call the six faces of our cube $a,b,c,d,e,$ and $f$ (where $a$ is opposite $d$ , $c$ is opposite $e$ , and $b$ is opposite $f$ ). Thus, for the eight vertices, we have the following products: $
18
第 18 题
计数
In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?
💡 解题思路
Factor $345=3\cdot 5\cdot 23$ .
19
第 19 题
几何·面积
Rectangle ABCD has AB=5 and BC=4 . Point E lies on \overline{AB} so that EB=1 , point G lies on \overline{BC} so that CG=1 , and point F lies on \overline{CD} so that DF=2 . Segments \overline{AG} and \overline{AC} intersect \overline{EF} at Q and P , respectively. What is the value of \frac{PQ}{EF} ? [图]
💡 解题思路
First, we will define point $D$ as the origin. Then, we will find the equations of the following three lines: $AG$ , $AC$ , and $EF$ . The slopes of these lines are $-\frac{3}{5}$ , $-\frac{4}{5}$ , a
20
第 20 题
几何·面积
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius 2 centered at A(2,2) to the circle of radius 3 centered at A’(5,6) . What distance does the origin O(0,0) , move under this transformation?
💡 解题思路
The center of dilation must lie on the line $A A'$ , which can be expressed as $y = \dfrac{4x}{3} - \dfrac{2}{3}$ . Note that the center of dilation must have an $x$ -coordinate less than $2$ ; if the
21
第 21 题
几何·面积
What is the area of the region enclosed by the graph of the equation x^2+y^2=|x|+|y|?
💡 解题思路
Without loss of generality (WLOG) note that if a point in the first quadrant satisfies the equation, so do its corresponding points in the other three quadrants. Therefore, we can assume that $x, y \g
22
第 22 题
综合
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won 10 games and lost 10 games; there were no ties. How many sets of three teams \{A, B, C\} were there in which A beat B , B beat C , and C beat A?
💡 解题思路
There are $10 \cdot 2+1=21$ teams. Any of the $\tbinom{21}3=1330$ sets of three teams must either be a fork (in which one team beat both the others) or a cycle:
23
第 23 题
几何·面积
In regular hexagon ABCDEF , points W , X , Y , and Z are chosen on sides \overline{BC} , \overline{CD} , \overline{EF} , and \overline{FA} respectively, so lines AB , ZW , YX , and ED are parallel and equally spaced. What is the ratio of the area of hexagon WCXYFZ to the area of hexagon ABCDEF ?
💡 解题思路
We draw a diagram to make our work easier: [asy] pair A,B,C,D,E,F,W,X,Y,Z; A=(0,0); B=(1,0); C=(3/2,sqrt(3)/2); D=(1,sqrt(3)); E=(0,sqrt(3)); F=(-1/2,sqrt(3)/2); X=(4/3,2sqrt(3)/3); W=(4/3,sqrt(3)/3);
24
第 24 题
规律与数列
How many four-digit integers abcd , with a ≠ 0 , have the property that the three two-digit integers ab<bc<cd form an increasing arithmetic sequence? One such number is 4692 , where a=4 , b=6 , c=9 , and d=2 .
💡 解题思路
The numbers are $10a+b, 10b+c,$ and $10c+d$ . Note that only $d$ can be zero, the numbers $ab$ , $bc$ , and $cd$ cannot start with a zero, and $a\le b\le c$ .
25
第 25 题
规律与数列
Let f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor) , where \lfloor r \rfloor denotes the greatest integer less than or equal to r . How many distinct values does f(x) assume for x \ge 0 ?