2016B 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
📋 答题说明
共 25 道题，每题从 A、B、C、D、E 五个选项中选一个答案，点击选项即可选择
答题过程中可随时更改选项，选完后点击底部「提交答案」统一批改
提交后显示对错、正确答案和简短解题思路
点击题目右侧 ⭐ 可收藏难题，方便后续复习
题目涉及图形的部分，原题以文字描述代替（图形题建议配合原版试卷使用）
1
第 1 题
综合
What is the value of \frac{2a^{-1}+\frac{a^{-1}}{2}}{a} when a= \frac{1}{2} ?
💡 解题思路
By: Dragonfly
2
第 2 题
统计
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of 1 and 2016 is closest to which integer?
💡 解题思路
Since the harmonic mean is $2$ times their product divided by their sum, we get the equation
3
第 3 题
方程
Let x=-2016 . What is the value of \bigg|||x|-x|-|x|\bigg|-x ?
💡 解题思路
By: dragonfly
4
第 4 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
By: dragonfly
5
第 5 题
分数与比例
The War of 1812 started with a declaration of war on Thursday, June 18 , 1812 . The peace treaty to end the war was signed 919 days later, on December 24 , 1814 . On what day of the week was the treaty signed?
💡 解题思路
By: dragonfly
6
第 6 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 ?
💡 解题思路
By: Albert471
7
第 7 题
综合
Josh writes the numbers 1,2,3,\dots,99,100 . He marks out 1 , skips the next number (2) , marks out 3 , and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number (2) , skips the next number (4) , marks out 6 , skips 8 , marks out 10 , and so on to the end. Josh continues in this manner until only one number remains. What is that number?
💡 解题思路
Following the pattern, you are crossing out...
8
第 8 题
几何·面积
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
By: dragonfly
9
第 9 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
To start, use algebra to determine the number of posts on each side. You have (the long sides count for $2$ because there are twice as many) $6x = 20 + 4$ (each corner is double counted so you must ad
10
第 10 题
几何·面积
A quadrilateral has vertices P(a,b) , Q(b,a) , R(-a, -b) , and S(-b, -a) , where a and b are integers with a>b>0 . The area of PQRS is 16 . What is a+b ?
💡 解题思路
Note that the slope of $PQ$ is $\frac{a-b}{b-a}=-1$ and the slope of $PS$ is $\frac{b+a}{a+b}=1$ . Hence, $PQ\perp PS$ and we can similarly prove that the other angles are right angles. This means tha
11
第 11 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
Solution by e_power_pi_times_i Revised by Kinglogic and RJ5303707
12
第 12 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
Solution by Mlux: Draw a $3\times3$ matrix. Notice that no adjacent numbers could be in the corners since two consecutive numbers must share an edge. Now find 4 nonconsecutive numbers that add up to $
13
第 13 题
行程问题
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is 30^\circ from Alice's position and 60^\circ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
💡 解题思路
Let's set the altitude = z, distance from Alice to airplane's ground position (point right below airplane)=y and distance from Bob to airplane's ground position=x
14
第 14 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 second term in a geometric series is $a_2 = a \cdot r$ , where $r$ is the common ratio for the series and $a$ is the first term of the series. So we know that $a\cdot r = 1$ and we wish to find th
15
第 15 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
First assign each face the letters $a,b,c,d,e,f$ . The sum of the product of the faces is $abc+acd+ade+aeb+fbc+fcd+fde+feb$ . We can factor this into $(a+f)(b+d)(c+e)$ which is the product of the sum
16
第 16 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
We proceed with this problem by considering two cases, when: 1) There are an odd number of consecutive numbers, 2) There are an even number of consecutive numbers.
17
第 17 题
几何·面积
In \triangle ABC shown in the figure, AB=7 , BC=8 , CA=9 , and \overline{AH} is an altitude. Points D and E lie on sides \overline{AC} and \overline{AB} , respectively, so that \overline{BD} and \overline{CE} are angle bisectors, intersecting \overline{AH} at Q and P , respectively. What is PQ ? [图]
💡 解题思路
Get the area of the triangle by Heron's Formula : \[\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{(12)(3)(4)(5)} = 12\sqrt{5}\] Use the area to find the height $AH$ with known base $BC$ : \[Area = 12\sqrt{5} = \fra
18
第 18 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
Consider the case when $x \geq 0$ , $y \geq 0$ . \[x^2+y^2=x+y\] \[(x - \frac{1}{2})^2+(y - \frac{1}{2})^2=\frac{1}{2}\] Notice the circle intersects the axes at points $(0, 1)$ and $(1, 0)$ . Find th
19
第 19 题
概率
Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?
💡 解题思路
By: dragonfly
20
第 20 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
We use complementary counting. First, because each team played $20$ other teams, there are $21$ teams total. All sets that do not have $A$ beat $B$ , $B$ beat $C$ , and $C$ beat $A$ have one team that
21
第 21 题
几何·面积
Let ABCD be a unit square. Let Q_1 be the midpoint of \overline{CD} . For i=1,2,\dots, let P_i be the intersection of \overline{AQ_i} and \overline{BD} , and let Q_{i+1} be the foot of the perpendicular from P_i to \overline{CD} . What is \[\sum_{i=1}^{\infty} Area of \triangle DQ_i P_i ?\]
💡 解题思路
We start with $DQ_i = 1/2$ for $i = 1.$ $\triangle BP_iA \sim \triangle DP_iQ_i$ and $\triangle DP_iQ_{i+1} \sim \triangle DBC$ so we have \[\frac{DQ_i}{AB} = \frac{DP_i}{P_iB} = \frac{1}{2} \implies
22
第 22 题
分数与比例
For a certain positive integer n less than 1000 , the decimal equivalent of \frac{1}{n} is 0.\overline{abcdef} , a repeating decimal of period of 6 , and the decimal equivalent of \frac{1}{n+6} is 0.\overline{wxyz} , a repeating decimal of period 4 . In which interval does n lie?
💡 解题思路
Solution by e_power_pi_times_i
23
第 23 题
立体几何
What is the volume of the region in three-dimensional space defined by the inequalities |x|+|y|+|z|\le1 and |x|+|y|+|z-1|\le1 ?
💡 解题思路
The first inequality refers to the interior of a regular octahedron with top and bottom vertices $(0,0,1),\ (0,0,-1)$ . Its volume is $8\cdot\tfrac16=\tfrac43$ . The second inequality describes an ide
24
第 24 题
数论
There are exactly 77,000 ordered quadruplets (a, b, c, d) such that \gcd(a, b, c, d) = 77 and \operatorname{lcm}(a, b, c, d) = n . What is the smallest possible value for n ?
💡 解题思路
Let $A=\frac{a}{77},\ B=\frac{b}{77}$ , etc., so that $\gcd(A,B,C,D)=1$ . Then for each prime power $p^k$ in the prime factorization of $N=\frac{n}{77}$ , at least one of the prime factorizations of $
25
第 25 题
规律与数列
The sequence (a_n) is defined recursively by a_0=1 , a_1=\sqrt[19]{2} , and a_n=a_{n-1}a_{n-2}^2 for n≥ 2 . What is the smallest positive integer k such that the product a_1a_2·s a_k is an integer?
💡 解题思路
Let $b_i=19\text{log}_2a_i$ . Then $b_0=0, b_1=1,$ and $b_n=b_{n-1}+2b_{n-2}$ for all $n\geq 2$ . The characteristic polynomial of this linear recurrence is $x^2-x-2=0$ , which has roots $2$ and $-1$