2019B 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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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
分数与比例
Alicia had two containers. The first was \tfrac{5}{6} full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was \tfrac{3}{4} full of water. What is the ratio of the volume of the first container to the volume of the second container?
💡 解题思路
Let the first jar's volume be $A$ and the second's be $B$ . It is given that $\frac{5}{6}A=\frac{3}{4}B$ . We find that $\frac{A}{B}=\frac{\left(\frac{3}{4}\right)}{\left(\frac{5}{6}\right)}=\boxed{\t
2
第 2 题
数论
Consider the statement, "If n is not prime, then n-2 is prime." Which of the following values of n is a counterexample to this statement?
💡 解题思路
Since a counterexample must be a value of $n$ which is not prime, $n$ must be composite, so we eliminate $\text{A}$ and $\text{C}$ . Now we subtract $2$ from the remaining answer choices, and we see t
3
第 3 题
计数
Which of the following rigid transformations (isometries) maps the line segment \overline{AB} onto the line segment \overline{A'B'} so that the image of A(-2, 1) is A'(2, -1) and the image of B(-1, 4) is B'(1, -4) ? (A) reflection in the y -axis (B) counterclockwise rotation around the origin by 90^{\circ}(C) translation by 3 units to the right and 5 units down (D) reflection in the x -axis (E) clockwise rotation about the origin by 180^{\circ}
💡 解题思路
We can simply graph the points, or use coordinate geometry, to realize that both $A'$ and $B'$ are, respectively, obtained by rotating $A$ and $B$ by $180^{\circ}$ about the origin. Hence the rotation
4
第 4 题
规律与数列
There is a positive integer n such that (n+1)! + (n+2)! = n! · 440 . What is the sum of the digits of n ?
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or n pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of n ?
💡 解题思路
If he has enough money to buy $12$ pieces of red candy, $14$ pieces of green candy, and $15$ pieces of blue candy, then the smallest amount of money he could have is $\text{lcm}{(12,14,15)} = 420$ cen
6
第 6 题
几何·面积
In a given plane, points A and B are 10 units apart. How many points C are there in the plane such that the perimeter of \triangle ABC is 50 units and the area of \triangle ABC is 100 square units?
💡 解题思路
Notice that whatever point we pick for $C$ , $AB$ will be the base of the triangle. Without loss of generality, let points $A$ and $B$ be $(0,0)$ and $(10,0)$ , since for any other combination of poin
7
第 7 题
统计
What is the sum of all real numbers x for which the median of the numbers 4,6,8,17, and x is equal to the mean of those five numbers?
💡 解题思路
The mean is $\frac{4+6+8+17+x}{5}=\frac{35+x}{5}$ .
8
第 8 题
规律与数列
Let f(x) = x^{2}(1-x)^{2} . What is the value of the sum \[f (\frac{1}{2019} )-f (\frac{2}{2019} )+f (\frac{3}{2019} )-f (\frac{4}{2019} )+·s + f (\frac{2017}{2019} ) - f (\frac{2018}{2019} )?\]
💡 解题思路
First, note that $f(x) = f(1-x)$ . We can see this since \[f(x) = x^2(1-x)^2 = (1-x)^2x^2 = (1-x)^{2}\left(1-\left(1-x\right)\right)^{2} = f(1-x)\] Using this result, we regroup the terms accordingly:
9
第 9 题
几何·面积
For how many integral values of x can a triangle of positive area be formed having side lengths \log_{2} x, \log_{4} x, 3 ?
💡 解题思路
For these lengths to form a triangle of positive area, the Triangle Inequality tells us that we need \[\log_2{x} + \log_4{x} > 3\] \[\log_2{x} + 3 > \log_4{x}\] \[\log_4{x} + 3 > \log_2{x}.\] The seco
10
第 10 题
行程问题
The figure below is a map showing 12 cities and 17 roads connecting certain pairs of cities. Paula wishes to travel along exactly 13 of those roads, starting at city A and ending at city L , without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.) [图] How many different routes can Paula take?
💡 解题思路
Note that of the $12$ cities, $6$ of them ( $2$ on the top, $2$ on the bottom, and $1$ on each side) have $3$ edges coming into/out of them (i.e., in graph theory terms, they have degree $3$ ). Theref
11
第 11 题
规律与数列
How many unordered pairs of edges of a given cube determine a plane?
💡 解题思路
Without loss of generality, choose one of the $12$ edges of the cube to be among the two selected. We now calculate the probability that a randomly-selected second edge makes the pair satisfy the cond
12
第 12 题
几何·面积
Right triangle ACD with right angle at C is constructed outwards on the hypotenuse \overline{AC} of isosceles right triangle ABC with leg length 1 , as shown, so that the two triangles have equal perimeters. What is \sin(2\angle BAD) ? [图]
💡 解题思路
First, note by the Pythagorean Theorem in $\triangle ABC$ that $AC = \sqrt{2}$ . Now, the equal perimeter condition means that $BC + BA = 2 = CD + DA$ , since side $AC$ is common to both triangles and
13
第 13 题
综合
💡 解题思路
We see that the total probability will eventually sum to 1 from the infinite geometric series sum formula. Now, if the green ball goes into the \( k=1 \) bin with a \( 1/2 \) chance, then the red ball
14
第 14 题
整数运算
Let S be the set of all positive integer divisors of 100,000. How many numbers are the product of two distinct elements of S?
💡 解题思路
The prime factorization of $100,000$ is $2^5 \cdot 5^5$ . Thus, we choose two numbers $2^a5^b$ and $2^c5^d$ where $0 \le a,b,c,d \le 5$ and $(a,b) \neq (c,d)$ , whose product is $2^{a+c}5^{b+d}$ , whe
15
第 15 题
几何·面积
As shown in the figure, line segment \overline{AD} is trisected by points B and C so that AB=BC=CD=2. Three semicircles of radius 1,\overarc{AEB},\overarc{BFC}, and \overarc{CGD}, have their diameters on \overline{AD}, lie in the same halfplane determined by line AD , and are tangent to line EG at E,F, and G, respectively. A circle of radius 2 has its center on F. The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \[\frac{a}{b}·π-√(c)+d,\] where a,b,c, and d are positive integers and a and b are relatively prime. What is a+b+c+d ? [图]
💡 解题思路
This solution is essentially the same, but is much more clear and easier to understand, than Solution 2. I will TeX this at a later time, but if anyone wants to help in the mean time, please do -- tha
16
第 16 题
概率
There are lily pads in a row numbered 0 to 11 , in that order. There are predators on lily pads 3 and 6 , and a morsel of food on lily pad 10 . Fiona the frog starts on pad 0 , and from any given lily pad, has a \frac{1}{2} chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6 ?
💡 解题思路
Firstly, notice that if Fiona jumps over the predator on pad $3$ , she must land on pad $4$ . Similarly, she must land on $7$ if she makes it past $6$ . Thus, we can split the problem into $3$ smaller
17
第 17 题
几何·面积
How many nonzero complex numbers z have the property that 0, z, and z^3, when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
💡 解题思路
Convert $z$ and $z^3$ into modulus-argument (polar) form, giving $z=r\text{cis}(\theta)$ for some $r$ and $\theta$ . Thus, by De Moivre's Theorem, $z^3=r^3\text{cis}(3\theta)$ . Since the distance fro
18
第 18 题
几何·面积
Square pyramid ABCDE has base ABCD , which measures 3 cm on a side, and altitude AE perpendicular to the base, which measures 6 cm. Point P lies on BE , one third of the way from B to E ; point Q lies on DE , one third of the way from D to E ; and point R lies on CE , two thirds of the way from C to E . What is the area, in square centimeters, of \triangle{PQR} ?
💡 解题思路
Using the given data, we can label the points $A(0, 0, 0), B(3, 0, 0), C(3, 3, 0), D(0, 3, 0),$ and $E(0, 0, 6)$ . We can also find the points $P = B + \frac{1}{3} \overrightarrow{BE} = (3,0,0) + \fra
19
第 19 题
概率
Raashan, Sylvia, and Ted play the following game. Each starts with \1 . A bell rings every 15 seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives \1 to that player. What is the probability that after the bell has rung 2019 times, each player will have \1 ? (For example, Raashan and Ted may each decide to give \1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have \0 , Sylvia will have \2 , and Ted will have \1 , and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their \1 to, and the holdings will be the same at the end of the second round.)
💡 解题思路
On the first turn, each player starts off with $\$1$ . Each turn after that, there are only two possibilities: either everyone stays at $\$1$ , which we will write as $(1-1-1)$ , or the distribution o
20
第 20 题
几何·面积
Points A=(6,13) and B=(12,11) lie on circle \omega in the plane. Suppose that the tangent lines to \omega at A and B intersect at a point on the x -axis. What is the area of \omega ?
💡 解题思路
First, observe that the two tangent lines are of identical length. Therefore, supposing that the point of intersection is $(x, 0)$ , the Pythagorean Theorem gives $\sqrt{(x-6)^2 + 13^2} = \sqrt{(x-12)
21
第 21 题
逻辑推理
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is ax^2+bx+c,a≠ 0, and the roots are r and s, then the requirement is that \{a,b,c\}=\{r,s\} .)
💡 解题思路
Firstly, if $r=s$ , then $a=b=c$ , so the equation becomes $ax^2 + ax + a = 0 \Rightarrow x^2 + x + 1=0$ , which has no real roots.
22
第 22 题
规律与数列
Define a sequence recursively by x_0=5 and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers n. Let m be the least positive integer such that \[x_m≤ 4+\frac{1}{2^{20}}.\] In which of the following intervals does m lie?
💡 解题思路
We first prove that $x_n > 4$ for all $n \ge 0$ , by induction. Observe that \[x_{n+1} - 4 = \frac{x_n^2 + 5x_n + 4 - 4(x_n+6)}{x_n+6} = \frac{(x_n - 4)(x_n+5)}{x_n+6}.\] so (since $x_n$ is clearly po
23
第 23 题
规律与数列
How many sequences of 0 s and 1 s of length 19 are there that begin with a 0 , end with a 0 , contain no two consecutive 0 s, and contain no three consecutive 1 s? This problem is just a simplified version of 2001 AIME I, Problem 14 , and a slightly harder version of CLMC 2025 C4 .
💡 解题思路
Let $f(n)$ be the number of valid sequences of length $n$ (satisfying the conditions given in the problem).
24
第 24 题
几何·面积
Let \omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3. Let S denote all points in the complex plane of the form a+b\omega+c\omega^2, where 0≤ a ≤ 1,0≤ b≤ 1, and 0≤ c≤ 1. What is the area of S ?
💡 解题思路
Notice that $\omega=e^{\frac{2i\pi}{3}}$ , which is one of the cube roots of unity. We wish to find the span of $(a+b\omega+c\omega^2)$ for reals $0\le a,b,c\le 1$ . Observe also that if $a,b,c>0$ , t
25
第 25 题
几何·面积
Let ABCD be a convex quadrilateral with BC=2 and CD=6. Suppose that the centroids of \triangle ABC,\triangle BCD, and \triangle ACD form the vertices of an equilateral triangle. What is the maximum possible value of the area of ABCD ?
💡 解题思路
Place an origin at $A$ , and assign position vectors of $B = \vec{p}$ and $D = \vec{q}$ . Since $AB$ is not parallel to $AD$ , vectors $\vec{p}$ and $\vec{q}$ are linearly independent, so we can write