2016A AMC 12 — Official Competition Problems (February 2024)
📅 2024 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
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
4
第 4 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*}
5
第 5 题
数论
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, 2016=13+2003 ). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?
💡 解题思路
In this case, a counterexample is a number that would prove Goldbach's conjecture false. The conjecture asserts what can be done with even integers greater than 2. Therefore the solution is \[{\textbf
6
第 6 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 $\boxed{\textbf{(D) } 9}.$
7
第 7 题
坐标几何
Which of these describes the graph of x^2(x+y+1)=y^2(x+y+1) ?
💡 解题思路
The equation $x^2(x+y+1)=y^2(x+y+1)$ tells us $x^2=y^2$ or $x+y+1=0$ . $x^2=y^2$ generates two lines $y=x$ and $y=-x$ . $x+y+1=0$ is another straight line. The only intersection of $y=x$ and $y=-x$ is
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is \tfrac{a-√(2)}{b} , where a and b are positive integers. What is a+b ? [图]
💡 解题思路
Let $s$ be the side length of the small squares.
10
第 10 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
11
第 11 题
规律与数列
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?
💡 解题思路
Let $a$ be the number of students that can only sing, $b$ can only dance, and $c$ can only act.
12
第 12 题
几何·面积
In \triangle ABC , AB = 6 , BC = 7 , and CA = 8 . Point D lies on \overline{BC} , and \overline{AD} bisects \angle BAC . Point E lies on \overline{AC} , and \overline{BE} bisects \angle ABC . The bisectors intersect at F . What is the ratio AF : FD ? [图]
💡 解题思路
By the angle bisector theorem, $\frac{AB}{AE} = \frac{CB}{CE}$
13
第 13 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
14
第 14 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 $\frac{1+2+\cdots+8}{2}=18$ .
15
第 15 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 ?
The graphs of y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x, and y=\log_x \dfrac{1}{3} are plotted on the same set of axes. How many points in the plane with positive x -coordinates lie on two or more of the graphs?
💡 解题思路
Setting the first two equations equal to each other, $\log_3 x = \log_x 3$ .
17
第 17 题
几何·面积
Let ABCD be a square. Let E, F, G and H be the centers, respectively, of equilateral triangles with bases \overline{AB}, \overline{BC}, \overline{CD}, and \overline{DA}, each exterior to the square. What is the ratio of the area of square EFGH to the area of square ABCD ?
💡 解题思路
The center of an equilateral triangle is its centroid, where the three medians meet.
18
第 18 题
整数运算
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
19
第 19 题
数论
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process \frac{a}{b}, where a and b are relatively prime positive integers. What is a + b? (For example, he succeeds if his sequence of tosses is HTHHHHHH. )
💡 解题思路
For $6$ to $8$ heads, we are guaranteed to hit $4$ , so the sum here is $\binom{8}{2}+\binom{8}{1}+\binom{8}{0}=28+8+1=37$ .
20
第 20 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
21
第 21 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(
22
第 22 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
23
第 23 题
几何·面积
Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?
💡 解题思路
Because we can let the the sides of the triangle be any variable we want, to make it easier for us when solving, let’s let the side lengths be $x,y,$ and $a$ . WLOG assume $a$ is the largest. Then, $x
24
第 24 题
整数运算
There is a smallest positive real number a such that there exists a positive real number b such that all the roots of the polynomial x^3-ax^2+bx-a are real. In fact, for this value of a the value of b is unique. What is this value of b ?
💡 解题思路
The acceleration must be zero at the $x$ -intercept; this intercept must be an inflection point for the minimum $a$ value. Derive $f(x)$ so that the acceleration $f''(x)=0$ . Using the power rule, \be
25
第 25 题
几何·面积
Let k be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with k+1 digits. Every time Bernardo writes a number, Silvia erases the last k digits of it. Bernardo then writes the next perfect square, Silvia erases the last k digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let f(k) be the smallest positive integer not written on the board. For example, if k = 1 , then the numbers that Bernardo writes are 16, 25, 36, 49, 64 , and the numbers showing on the board after Silvia erases are 1, 2, 3, 4, and 6 , and thus f(1) = 5 . What is the sum of the digits of f(2) + f(4)+ f(6) + \dots + f(2016) ?
💡 解题思路
Consider $f(2)$ . The numbers left on the blackboard will show the hundreds place at the end. In order for the hundreds place to differ by 2, the difference between two perfect squares needs to be at