2020B 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 in simplest form of the following expression? \[√(1) + √(1+3) + √(1+3+5) + √(1+3+5+7)\]
💡 解题思路
We have \[\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7} = \sqrt{1} + \sqrt{4} + \sqrt{9} + \sqrt{16}\ = 1 + 2 + 3 + 4 = \boxed{\textbf{(C) } 10}.\] Note: This comes from the fact that the sum
2
第 2 题
综合
What is the value of the following expression? \[\frac{100^2-7^2}{70^2-11^2} · \frac{(70-11)(70+11)}{(100-7)(100+7)}\]
💡 解题思路
Using difference of squares to factor the left term, we get \[\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)} = \frac{(100-7)(100+7)}{(70-11)(70+11)} \cdot \frac{(70-11)(70+11)
3
第 3 题
分数与比例
The ratio of w to x is 4:3 , the ratio of y to z is 3:2 , and the ratio of z to x is 1:6 . What is the ratio of w to y?
💡 解题思路
We have \[\frac wy = \frac wx \cdot \frac xz \cdot \frac zy = \frac43\cdot\frac61\cdot\frac23=\frac{16}{3},\] from which $w:y=\boxed{\textbf{(E)}\ 16:3}.$
4
第 4 题
几何·面积
The acute angles of a right triangle are a^{\circ} and b^{\circ} , where a>b and both a and b are prime numbers. What is the least possible value of b ?
💡 解题思路
Since the three angles of a triangle add up to $180^{\circ}$ and one of the angles is $90^{\circ}$ because it's a right triangle, $a^{\circ} + b^{\circ} = 90^{\circ}$ .
5
第 5 题
综合
Teams A and B are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team A has won \tfrac{2}{3} of its games and team B has won \tfrac{5}{8} of its games. Also, team B has won 7 more games and lost 7 more games than team A. How many games has team A played?
💡 解题思路
Suppose team $A$ has played $g$ games in total so that it has won $\frac23g$ games. It follows that team $B$ has played $g+14$ games in total so that it has won $\frac23g+7$ games.
6
第 6 题
计数
For all integers n ≥ 9, the value of \[\frac{(n+2)!-(n+1)!}{n!}\] is always which of the following?
💡 解题思路
We first expand the expression: \[\frac{(n+2)!-(n+1)!}{n!} = \frac{(n+2)(n+1)n!-(n+1)n!}{n!}.\] We can now divide out a common factor of $n!$ from each term of the numerator: \[(n+2)(n+1)-(n+1).\] Fac
7
第 7 题
坐标几何
Two nonhorizontal, non vertical lines in the xy -coordinate plane intersect to form a 45^{\circ} angle. One line has slope equal to 6 times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?
💡 解题思路
Let the intersection point be the origin. Let $(a,b)$ be a point on the line of lesser slope. The mutliplication of $a+bi$ by cis 45. $(a+bi)(\frac{1}{\sqrt 2 }+i*\frac{1}{\sqrt 2 })=\frac{1}{\sqrt 2
8
第 8 题
方程
How many ordered pairs of integers (x, y) satisfy the equation \[x^{2020}+y^2=2y?\]
💡 解题思路
Rearranging the terms and and completing the square for $y$ yields the result $x^{2020}+(y-1)^2=1$ . Then, notice that $x$ can only be $0$ , $1$ and $-1$ because any value of $x^{2020}$ that is greate
9
第 9 题
几何·面积
A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? [图]
💡 解题思路
Notice that when the cone is created, the 2 shown radii when merged will become the slant height of the cone and the intact circumference of the circle will become the circumference of the base of the
10
第 10 题
几何·面积
In unit square ABCD, the inscribed circle \omega intersects \overline{CD} at M, and \overline{AM} intersects \omega at a point P different from M. What is AP? [图] ~MRENTHUSIASM
💡 解题思路
Call the midpoint of $\overline{AB}$ point $N.$ Draw in $\overline{NM}$ and $\overline{NP}.$ Note that $\angle{NPM}=90^{\circ}$ due to Thales's Theorem. [asy] /* Made by QIDb602; edited by MRENTHUSIAS
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第 11 题
几何·面积
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles?
💡 解题思路
[asy] real x=sqrt(3); real y=2sqrt(3); real z=3.5; real a=x/2; real b=0.5; real c=3a; pair A, B, C, D, E, F; A = (1,0); B = (3,0); C = (4,x); D = (3,y); E = (1,y); F = (0,x); fill(A--B--C--D--E--F--A-
12
第 12 题
几何·面积
Let \overline{AB} be a diameter in a circle of radius 5\sqrt2. Let \overline{CD} be a chord in the circle that intersects \overline{AB} at a point E such that BE=2\sqrt5 and \angle AEC = 45^{\circ}. What is CE^2+DE^2? [图] ~Shihan ~MRENTHUSIASM
💡 解题思路
Let $O$ be the center of the circle, and $X$ be the midpoint of $\overline{CD}$ . Let $CX=a$ and $EX=b$ . This implies that $DE = a - b$ . Since $CE = CX + EX = a + b$ , we now want to find $(a+b)^2+(
13
第 13 题
综合
Which of the following is the value of √(\log_2{6)+\log_3{6}}?
💡 解题思路
Recall that:
14
第 14 题
计数
Bela and Jenn play the following game on the closed interval [0, n] of the real number line, where n is a fixed integer greater than 4 . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval [0, n] . Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
💡 解题思路
We use game theory.
15
第 15 题
几何·面积
There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other?
💡 解题思路
Consider the $10$ people to be standing in a circle, where two people opposite of each other form a diameter of the circle.
16
第 16 题
分数与比例
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
💡 解题思路
Suppose that we have a deck, currently containing just one black card. We then insert $n$ red cards one-by-one into the deck at random positions. It is easy to see using induction, that the black card
17
第 17 题
综合
How many polynomials of the form x^5 + ax^4 + bx^3 + cx^2 + dx + 2020 , where a , b , c , and d are real numbers, have the property that whenever r is a root, so is \frac{-1+i√(3)}{2} · r ? (Note that i=√(-1) )
💡 解题思路
Let $P(x) = x^5+ax^4+bx^3+cx^2+dx+2020$ . We first notice that $\frac{-1+i\sqrt{3}}{2} = e^{2\pi i / 3}$ . That is because of Euler's Formula : $e^{ix} = \cos(x) + i \cdot \sin(x)$ . $\frac{-1+i\sqrt{
18
第 18 题
几何·面积
In square ABCD , points E and H lie on \overline{AB} and \overline{DA} , respectively, so that AE=AH. Points F and G lie on \overline{BC} and \overline{CD} , respectively, and points I and J lie on \overline{EH} so that \overline{FI} \perp \overline{EH} and \overline{GJ} \perp \overline{EH} . See the figure below. Triangle AEH , quadrilateral BFIE , quadrilateral DHJG , and pentagon FCGJI each has area 1. What is FI^2 ? [图]
💡 解题思路
Since the total area is $4$ , the side length of square $ABCD$ is $2$ . We see that since triangle $HAE$ is a right isosceles triangle with area 1, we can determine sides $HA$ and $AE$ both to be $\sq
19
第 19 题
几何·面积
Square ABCD in the coordinate plane has vertices at the points A(1,1), B(-1,1), C(-1,-1), and D(1,-1). Consider the following four transformations: \bullet L, a rotation of 90^{\circ} counterclockwise around the origin; \bullet R, a rotation of 90^{\circ} clockwise around the origin; \bullet H, a reflection across the x -axis; and \bullet V, a reflection across the y -axis. Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying R and then V would send the vertex A at (1,1) to (-1,-1) and would send the vertex B at (-1,1) to itself. How many sequences of 20 transformations chosen from \{L, R, H, V\} will send all of the labeled vertices back to their original positions? (For example, R, R, V, H is one sequence of 4 transformations that will send the vertices back to their original positions.)
💡 解题思路
For each transformation:
20
第 20 题
概率
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?
💡 解题思路
Define two ways of painting to be in the same $class$ if one can be rotated to form the other.
21
第 21 题
整数运算
How many positive integers n satisfy \[\dfrac{n+1000}{70} = \lfloor √(n) \rfloor?\] (Recall that \lfloor x\rfloor is the greatest integer not exceeding x .)
💡 解题思路
We can first consider the equation without a floor function:
22
第 22 题
综合
What is the maximum value of \frac{(2^t-3t)t}{4^t} for real values of t?
💡 解题思路
We proceed by using AM-GM. We get $\frac{(2^t-3t) + 3t}{2}$ $\ge \sqrt{(2^t-3t)(3t)}$ . Thus, squaring gives us that $4^{t-1} \ge (2^t-3t)(3t)$ . Remembering what we want to find, we divide both sides
23
第 23 题
几何·面积
How many integers n ≥ 2 are there such that whenever z_1, z_2, ..., z_n are complex numbers such that \[|z_1| = |z_2| = ... = |z_n| = 1 and z_1 + z_2 + ... + z_n = 0,\] then the numbers z_1, z_2, ..., z_n are equally spaced on the unit circle in the complex plane?
💡 解题思路
For $n=2$ , we see that if $z_{1}+z_{2}=0$ , then $z_{1}=-z_{2}$ , so they are evenly spaced along the unit circle.
24
第 24 题
数论
Let D(n) denote the number of ways of writing the positive integer n as a product \[n = f_1· f_2·s f_k,\] where k\ge1 , the f_i are integers strictly greater than 1 , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as 6 , 2· 3 , and 3·2 , so D(6) = 3 . What is D(96) ?
💡 解题思路
Note that $96 = 2^5 \cdot 3$ . Since there are at most six not necessarily distinct factors $>1$ multiplying to $96$ , we have six cases: $k=1, 2, ..., 6.$ Now we look at each of the six cases.
25
第 25 题
概率
For each real number a with 0 ≤ a ≤ 1 , let numbers x and y be chosen independently at random from the intervals [0, a] and [0, 1] , respectively, and let P(a) be the probability that \[\sin^2{(π x)} + \sin^2{(π y)} > 1\] What is the maximum value of P(a)?
💡 解题思路
Let's start first by manipulating the given inequality.