2023A 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
行程问题
Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B . Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many miles from City A will they be when they meet?
💡 解题思路
This is a $d=st$ problem, so let $x$ be the time it takes to meet. We can write the following equation: \[12x+18x=45\] Solving gives us $x=1.5$ . The $18x$ is Alicia so $18\times1.5=\boxed{\textbf{(E)
2
第 2 题
统计
The weight of \frac{1}{3} of a large pizza together with 3 \frac{1}{2} cups of orange slices is the same as the weight of \frac{3}{4} of a large pizza together with \frac{1}{2} cup of orange slices. A cup of orange slices weighs \frac{1}{4} of a pound. What is the weight, in pounds, of a large pizza?
💡 解题思路
Use a system of equations. Let $x$ be the weight of a pizza and $y$ be the weight of a cup of orange slices. We have \[\frac{1}{3}x+\frac{7}{2}y=\frac{3}{4}x+\frac{1}{2}y.\] Rearranging, we get \begin
3
第 3 题
几何·面积
How many positive perfect squares less than 2023 are divisible by 5 ?
💡 解题思路
Since $\left \lfloor{\sqrt{2023}}\right \rfloor = 44$ , there are $\left \lfloor{\frac{44}{5}}\right \rfloor = \boxed{\textbf{(A) 8}}$ perfect squares less than 2023 that are divisible by 5.
4
第 4 题
数字运算
How many digits are in the base-ten representation of 8^5 · 5^{10} · 15^5 ?
💡 解题思路
Prime factorizing this gives us $2^{15}\cdot3^{5}\cdot5^{15}=10^{15}\cdot3^5=243\cdot10^{15}$ .
5
第 5 题
概率
Janet rolls a standard 6 -sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3 ?
💡 解题思路
There are $3$ cases where the running total will equal $3$ : one roll; two rolls; or three rolls:
6
第 6 题
坐标几何
Points A and B lie on the graph of y=\log_{2}x . The midpoint of \overline{AB} is (6, 2) . What is the positive difference between the x -coordinates of A and B ?
💡 解题思路
Let $A(6+m,2+n)$ and $B(6-m,2-n)$ , since $(6,2)$ is their midpoint. Thus, we must find $2m$ . We find two equations due to $A,B$ both lying on the function $y=\log_{2}x$ . The two equations are then
7
第 7 题
行程问题
A digital display shows the current date as an 8 -digit integer consisting of a 4 -digit year, followed by a 2 -digit month, followed by a 2 -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in 2023 does each digit appear an even number of times in the 8 -digital display for that date?
💡 解题思路
Do careful casework by each month. Make sure to start with 2023. In the month and the date, we need a $0$ , a $3$ , and two digits repeated (which has to be $1$ and $2$ after consideration). After the
8
第 8 题
综合
💡 解题思路
Let $a$ represent the amount of tests taken previously and $x$ the mean of the scores taken previously.
9
第 9 题
几何·面积
A square of area 2 is inscribed in a square of area 3 , creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle? [图]
💡 解题思路
The side lengths of the inner square and outer square are $\sqrt{2}$ and $\sqrt{3}$ respectively. Let the shorter side of our triangle be $x$ , thus the longer leg is $\sqrt{3}-x$ . Hence, by the Pyth
10
第 10 题
综合
💡 解题思路
Because $y^3=x^2$ , set $x=a^3$ , $y=a^2$ ( $a\neq 0$ ). Put them in $(y-x)^2=4y^2$ we get $(a^2(a-1))^2=4a^4$ which implies $a^2-2a+1=4$ . Solve the equation to get $a=3$ or $-1$ . Since $x$ and $y$
11
第 11 题
坐标几何
What is the degree measure of the acute angle formed by lines with slopes 2 and \frac{1}{3} ?
💡 解题思路
Remind that $\text{slope}=\dfrac{\Delta y}{\Delta x}=\tan \theta$ where $\theta$ is the angle between the slope and $x$ -axis. $k_1=2=\tan \alpha$ , $k_2=\dfrac{1}{3}=\tan \beta$ . The angle formed by
12
第 12 题
综合
What is the value of \[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\]
💡 解题思路
To solve this problem, we will be using difference of cube, sum of squares and sum of arithmetic sequence formulas.
13
第 13 题
综合
In a table tennis tournament, every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was 40\% more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
💡 解题思路
We know that the total amount of games must be the sum of games won by left and right handed players. Then, we can write $g = l + r$ , and since $l = 1.4r$ , $g = 2.4r$ . Given that $r$ and $g$ are bo
14
第 14 题
方程
How many complex numbers satisfy the equation z^5=\overline{z} , where \overline{z} is the conjugate of the complex number z ?
💡 解题思路
When $z^5=\overline{z}$ , there are two conditions: either $z=0$ or $z\neq 0$ . When $z\neq 0$ , since $|z^5|=|\overline{z}|$ , $|z|=1$ . $z^5\cdot z=z^6=\overline{z}\cdot z=|z|^2=1$ . Consider the $r
15
第 15 题
规律与数列
Usain is walking for exercise by zigzagging across a 100 -meter by 30 -meter rectangular field, beginning at point A and ending on the segment \overline{BC} . He wants to increase the distance walked by zigzagging as shown in the figure below (APQRS) . What angle \theta = \angle PAB=\angle QPC=\angle RQB=·s will produce a length that is 120 meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.) [图]
💡 解题思路
By "unfolding" $APQRS$ into a straight line, we get a right angled triangle $ABS'$ .
16
第 16 题
数论
Consider the set of complex numbers z satisfying |1+z+z^{2}|=4 . The maximum value of the imaginary part of z can be written in the form \tfrac{√(m)}{n} , where m and n are relatively prime positive integers. What is m+n ?
💡 解题思路
First, substitute in $z=a+bi$ .
17
第 17 题
概率
Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance m with probability \frac{1}{2^m} . What is the probability that Flora will eventually land at 10?
💡 解题思路
Initially, the probability of landing at $10$ and landing past $10$ (summing the infinte series) are exactly the same. Landing before 10 repeats this initial condition, with a different irrelevant sca
18
第 18 题
几何·面积
Circle C_1 and C_2 each have radius 1 , and the distance between their centers is \frac{1}{2} . Circle C_3 is the largest circle internally tangent to both C_1 and C_2 . Circle C_4 is internally tangent to both C_1 and C_2 and externally tangent to C_3 . What is the radius of C_4 ? [图]
What is the product of all solutions to the equation \[\log_{7x}2023· \log_{289x}2023=\log_{2023x}2023\]
💡 解题思路
For $\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023$ , transform it into $\dfrac{\ln 289+\ln 7}{\ln 7 + \ln x}\cdot \dfrac{\ln 289+\ln 7}{\ln 289 + \ln x}=\dfrac{\ln 289+\ln 7}{\ln 289+\ln 7+\ln
20
第 20 题
规律与数列
Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below. [图] Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digits of the sum of the 2023 numbers in the 2023rd row?
💡 解题思路
First, let $R(n)$ be the sum of the $n$ th row. Now, with some observation and math instinct, we can guess that $R(n) = 2^n - n$ .
21
第 21 题
几何·面积
If A and B are vertices of a polyhedron, define the distance d(A,B) to be the minimum number of edges of the polyhedron one must traverse in order to connect A and B . For example, if \overline{AB} is an edge of the polyhedron, then d(A, B) = 1 , but if \overline{AC} and \overline{CB} are edges and \overline{AB} is not an edge, then d(A, B) = 2 . Let Q , R , and S be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that d(Q, R) > d(R, S) ?
💡 解题思路
To find the total amount of vertices we first find the amount of edges, and that is $\frac{20 \times 3}{2}$ . Next, to find the amount of vertices we can use Euler's characteristic, $V - E + F = 2$ ,
22
第 22 题
规律与数列
Let f be the unique function defined on the positive integers such that \[\sum_{d\mid n}d· f(\frac{n}{d})=1\] for all positive integers n . What is f(2023) ?
💡 解题思路
First, we note that $f(1) = 1$ , since the only divisor of $1$ is itself.
23
第 23 题
方程
How many ordered pairs of positive real numbers (a,b) satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\]
💡 解题思路
Using the AM-GM inequality on the two terms in each factor on the left-hand side, we get \[(1+2a)(2+2b)(2a+b) \ge 8\sqrt{2a \cdot 4b \cdot 2ab}= 32ab,\] This means the equality condition must be satis
24
第 24 题
数论
Let K be the number of sequences A_1 , A_2 , \dots , A_n such that n is a positive integer less than or equal to 10 , each A_i is a subset of \{1, 2, 3, \dots, 10\} , and A_{i-1} is a subset of A_i for each i between 2 and n , inclusive. For example, \{\} , \{5, 7\} , \{2, 5, 7\} , \{2, 5, 7\} , \{2, 5, 6, 7, 9\} is one such sequence, with n = 5 .What is the remainder when K is divided by 10 ?
💡 解题思路
Consider any sequence with $n$ terms. Every number has such choices: never appear, appear the first time in the first spot, appear the first time in the second spot ..., and appear the first time in t
25
第 25 题
方程
There is a unique sequence of integers a_1, a_2, ·s a_{2023} such that \[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + ·s + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x ·s + a_{2022} \tan^{2022} x}\] whenever \tan 2023x is defined. What is a_{2023}?
💡 解题思路
\begin{align*} \cos 2023 x + i \sin 2023 x &= (\cos x + i \sin x)^{2023}\\ &= \cos^{2023} x + \binom{2023}{1} \cos^{2022} x (i\sin x) + \binom{2023}{2} \cos^{2021} x (i \sin x)^{2} +\binom{2023}{3} \c