2016 AMC 10A — Official Competition Problems (February 2016)
📅 2016 A 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 \dfrac{11!-10!}{9!} ?
💡 解题思路
We can use subtraction of fractions to get \[\frac{11!-10!}{9!} = \frac{11!}{9!} - \frac{10!}{9!} = 110 -10 = \boxed{\textbf{(B)}\;100}.\]
2
第 2 题
综合
For what value of x does 10^{x}· 100^{2x}=1000^{5} ?
💡 解题思路
We can rewrite $10^{x}\cdot 100^{2x}=1000^{5}$ as $10^{5x}=10^{15}$ : \[\begin{split} 10^x\cdot100^{2x} & =10^x\cdot(10^2)^{2x} \\ 10^x\cdot10^{4x} & =(10^3)^5 \\ 10^{5x} & =10^{15} \end{split}\] Sinc
3
第 3 题
应用题
For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid \12.50$ more than David. How much did they spend in the bagel store together?
💡 解题思路
If Ben paid $\$ 12.50$ more than David, then he paid $\frac{12.5}{.25}= \$ 50.00$ . Thus, David paid $\$ 37.50$ , and they spent $50.00+37.50 =\$ 87.50 \implies \boxed{\textbf{(C) }\$ 87.50}$ .
4
第 4 题
数论
The remainder can be defined for all real numbers x and y with y ≠ 0 by \[rem (x ,y)=x-y \lfloor \frac{x}{y} \rfloor\] where \lfloor \tfrac{x}{y} \rfloor denotes the greatest integer less than or equal to \tfrac{x}{y} . What is the value of rem (\tfrac{3}{8}, -\tfrac{2}{5} ) ?
💡 解题思路
The value, by definition, is \begin{align*} \text{rem}\left(\frac{3}{8},-\frac{2}{5}\right) &= \frac{3}{8}-\left(-\frac{2}{5}\right)\left\lfloor\frac{\frac{3}{8}}{-\frac{2}{5}}\right\rfloor \\ &= \fra
5
第 5 题
分数与比例
A rectangular box has integer side lengths in the ratio 1: 3: 4 . Which of the following could be the volume of the box?
💡 解题思路
Let the smallest side length be $x$ . Then the volume is $x \cdot 3x \cdot 4x =12x^3$ . If $x=2$ , then $12x^3 = 96 \implies \boxed{\textbf{(D) } 96.}$
6
第 6 题
规律与数列
Ximena lists the whole numbers 1 through 30 once. Emilio copies Ximena's numbers, replacing each occurrence of the digit 2 by the digit 1 . Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
💡 解题思路
For every tens digit 2, we subtract 10, and for every units digit 2, we subtract 1. Because 2 appears 10 times as a tens digit and 2 appears 3 times as a units digit, the answer is $10\cdot 10+1\cdot
7
第 7 题
统计
The mean, median, and mode of the 7 data values 60, 100, x, 40, 50, 200, 90 are all equal to x . What is the value of x ?
💡 解题思路
Since $x$ is the mean, \begin{align*} x&=\frac{60+100+x+40+50+200+90}{7}\\ &=\frac{540+x}{7}. \end{align*}
8
第 8 题
概率
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays 40 coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
💡 解题思路
If you started backwards you would get: \[0\Rightarrow (+40)=40 , \Rightarrow \left(\frac{1}{2}\right)=20 , \Rightarrow (+40)=60 , \Rightarrow \left(\frac{1}{2}\right)=30 , \Rightarrow (+40)=70 , \Rig
9
第 9 题
概率
A triangular array of 2016 coins has 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to N coins in the N th row. What is the sum of the digits of N ?
💡 解题思路
We are trying to find the value of $N$ such that \[1+2+3\cdots+(N-1)+N=\frac{N(N+1)}{2}=2016.\] Noticing that $\frac{63\cdot 64}{2}=2016,$ we have $N=63,$ so our answer is $\boxed{\textbf{(D) } 9}.$
10
第 10 题
几何·面积
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is 1 foot wide on all four sides. What is the length in feet of the inner rectangle? [图]
Three distinct integers are selected at random between 1 and 2016 , inclusive. Which of the following is a correct statement about the probability p that the product of the three integers is odd?
💡 解题思路
For the product to be odd, all three factors have to be odd. The probability of this is $\frac{1008}{2016} \cdot \frac{1007}{2015} \cdot \frac{1006}{2014}$ .
13
第 13 题
综合
Five friends sat in a movie theater in a row containing 5 seats, numbered 1 to 5 from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
💡 解题思路
Assume that Edie and Dee were originally in seats 3 and 4. If this were so, there is no possible position for which Bea can move 2 seats to the right. The same applies for seats 2 and 3. This means th
14
第 14 题
计数
How many ways are there to write 2016 as the sum of twos and threes, ignoring order? (For example, 1008· 2 + 0· 3 and 402· 2 + 404· 3 are two such ways.)
💡 解题思路
The amount of twos in our sum ranges from $0$ to $1008$ , with differences of $3$ because $2 \cdot 3 = \operatorname{lcm}(2, 3)$ .
15
第 15 题
几何·面积
Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie? [图]
💡 解题思路
The big cookie has radius $3$ , since the center of the center cookie is the same as that of the large cookie. The difference in areas of the big cookie and the seven small ones is $3^2\pi-7\pi=9\pi-7
16
第 16 题
几何·面积
A triangle with vertices A(0, 2) , B(-3, 2) , and C(-3, 0) is reflected about the x -axis, then the image \triangle A'B'C' is rotated counterclockwise about the origin by 90^{\circ} to produce \triangle A''B''C'' . Which of the following transformations will return \triangle A''B''C'' to \triangle ABC ? (A) counterclockwise rotation about the origin by 90^{\circ} . (B) clockwise rotation about the origin by 90^{\circ} . (C) reflection about the x -axis (D) reflection about the line y = x(E) reflection about the y -axis.
💡 解题思路
Consider a point $(x, y)$ . Reflecting it about the $x$ -axis will map it to $(x, -y)$ , and rotating it counterclockwise about the origin by $90^{\circ}$ will map it to $(y, x)$ . The operation that
17
第 17 题
数论
Let N be a positive multiple of 5 . One red ball and N green balls are arranged in a line in random order. Let P(N) be the probability that at least \tfrac{3}{5} of the green balls are on the same side of the red ball. Observe that P(5)=1 and that P(N) approaches \tfrac{4}{5} as N grows large. What is the sum of the digits of the least value of N such that P(N) < \tfrac{321}{400} ?
💡 解题思路
Let $n = \frac{N}{5}$ . Then, consider $5$ blocks of $n$ green balls in a line, along with the red ball. Shuffling the line is equivalent to choosing one of the $N + 1$ positions between the green bal
18
第 18 题
统计
Each vertex of a cube is to be labeled with an integer 1 through 8 , with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
💡 解题思路
Note that the sum of the numbers on each face must be 18, because $\frac{1+2+\cdots+8}{2}=18$ .
19
第 19 题
几何·面积
In rectangle ABCD,AB=6 and BC=3 . Point E between B and C , and point F between E and C are such that BE=EF=FC . Segments \overline{AE} and \overline{AF} intersect \overline{BD} at P and Q , respectively. The ratio BP:PQ:QD can be written as r:s:t where the greatest common factor of r,s, and t is 1. What is r+s+t ?
For some particular value of N , when (a+b+c+d+1)^N is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b,c, and d , each to some positive power. What is N ?
💡 解题思路
All the desired terms are in the form $a^xb^yc^zd^w1^t$ , where $x + y + z + w + t = N$ (the $1^t$ part is necessary to make stars and bars work better.) Since $x$ , $y$ , $z$ , and $w$ must be at lea
21
第 21 题
几何·面积
Circles with centers P, Q and R , having radii 1, 2 and 3 , respectively, lie on the same side of line l and are tangent to l at P', Q' and R' , respectively, with Q' between P' and R' . The circle with center Q is externally tangent to each of the other two circles. What is the area of triangle PQR ?
For some positive integer n , the number 110n^3 has 110 positive integer divisors, including 1 and the number 110n^3 . How many positive integer divisors does the number 81n^4 have?
💡 解题思路
Since the prime factorization of $110$ is $2 \cdot 5 \cdot 11$ , we have that the number is equal to $2 \cdot 5 \cdot 11 \cdot n^3$ . This has $2 \cdot 2 \cdot 2=8$ factors when $n=1$ . This needs a m
23
第 23 题
数论
A binary operation \diamondsuit has the properties that a \diamondsuit (b \diamondsuit c) = (a \diamondsuit b)· c and that a \diamondsuit a=1 for all nonzero real numbers a, b, and c . (Here · represents multiplication). The solution to the equation 2016 \diamondsuit (6 \diamondsuit x)=100 can be written as \tfrac{p}{q} , where p and q are relatively prime positive integers. What is p+q?
💡 解题思路
We see that $a \, \diamondsuit \, a = 1$ , and think of division. Testing, we see that the first condition $a \, \diamondsuit \, (b \, \diamondsuit \, c) = (a \, \diamondsuit \, b) \cdot c$ is satisfi
24
第 24 题
几何·面积
A quadrilateral is inscribed in a circle of radius 200√(2) . Three of the sides of this quadrilateral have length 200 . What is the length of the fourth side?
💡 解题思路
[asy] size(250); defaultpen(linewidth(0.4)); //Variable Declarations real RADIUS; pair A, B, C, D, E, F, O; RADIUS=3; //Variable Definitions A=RADIUS*dir(148.414); B=RADIUS*dir(109.471); C=RADIUS*dir(
25
第 25 题
数论
How many ordered triples (x,y,z) of positive integers satisfy lcm(x,y) = 72, lcm(x,z) = 600, and lcm(y,z)=900 ?
💡 解题思路
We prime factorize $72,600,$ and $900$ . The prime factorizations are $2^3\times 3^2$ , $2^3\times 3\times 5^2$ and $2^2\times 3^2\times 5^2$ , respectively. Let $x=2^a\times 3^b\times 5^c$ , $y=2^d\t