2024A 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 9901·101-99·10101?
💡 解题思路
The likely fastest method will be direct computation. $9901\cdot101$ evaluates to $1000001$ and $99\cdot10101$ evaluates to $999999$ . The difference is $\boxed{\textbf{(A) }2}.$
2
第 2 题
统计
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form T=aL+bG, where a and b are constants, T is the time in minutes, L is the length of the trail in miles, and G is the altitude gain in feet. The model estimates that it will take 69 minutes to hike to the top if a trail is 1.5 miles long and ascends 800 feet, as well as if a trail is 1.2 miles long and ascends 1100 feet. How many minutes does the model estimates it will take to hike to the top if the trail is 4.2 miles long and ascends 4000 feet?
💡 解题思路
Plug in the values into the equation to give you the following two equations: \begin{align*} 69&=1.5a+800b, \\ 69&=1.2a+1100b. \end{align*} Solving for the values $a$ and $b$ gives you that $a=30$ and
3
第 3 题
规律与数列
The number 2024 is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
💡 解题思路
Since we want the least number of two-digit numbers, we maximize the two-digit numbers by choosing as many $99$ s as possible. Since $2024=99\cdot20+44\cdot1,$ we choose twenty $99$ s and one $44,$ fo
4
第 4 题
数论
What is the least value of n such that n! is a multiple of 2024 ?
💡 解题思路
Note that $2024=2^3\cdot11\cdot23$ in the prime factorization. Since $23!$ is a multiple of $2^3, 11,$ and $23,$ we conclude that $23!$ is a multiple of $2024.$ Therefore, we have $n=\boxed{\textbf{(D
5
第 5 题
统计
A data set containing 20 numbers, some of which are 6 , has mean 45 . When all the 6s are removed, the data set has mean 66 . How many 6s were in the original data set?
💡 解题思路
Because the set has $20$ numbers and mean $45$ , the sum of the terms in the set is $45\cdot 20=900$ .
6
第 6 题
规律与数列
The product of three integers is 60 . What is the least possible positive sum of the three integers?
💡 解题思路
We notice that the optimal solution involves two negative numbers and a positive number. Thus we may split $60$ into three factors and choose negativity. We notice that $10\cdot6\cdot1=10\cdot(-6)\cdo
7
第 7 题
规律与数列
In \Delta ABC , \angle ABC = 90^\circ and BA = BC = √(2) . Points P_1, P_2, \dots, P_{2024} lie on hypotenuse \overline{AC} so that AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C . What is the length of the vector sum \[\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}?\]
💡 解题思路
Let us find an expression for the $x$ - and $y$ -components of $\overrightarrow{BP_i}$ . Note that $AP_1+P_1P_2+\dots+P_{2023}P_{2024}+P_{2024}C=AC=2$ , so $AP_1=P_1P_2=\dots=P_{2023}P_{2024}=P_{2024}
8
第 8 题
几何·角度
How many angles \theta with 0\le\theta\le2π satisfy \log(\sin(3\theta))+\log(\cos(2\theta))=0 ?
💡 解题思路
Note that this is equivalent to $\sin(3\theta)\cos(2\theta)=1$ , which is clearly only possible when $\sin(3\theta)=\cos(2\theta)=\pm1$ . (If either one is between $1$ and $-1$ , the other one must be
9
第 9 题
几何·面积
Let M be the greatest integer such that both M+1213 and M+3773 are perfect squares. What is the units digit of M ?
💡 解题思路
Let $M+1213=P^2$ and $M+3773=Q^2$ for some positive integers $P$ and $Q.$ We subtract the first equation from the second, then apply the difference of squares: \[(Q+P)(Q-P)=2560.\] Note that $Q+P$ and
10
第 10 题
几何·面积
Let \alpha be the radian measure of the smallest angle in a 3{-}4{-}5 right triangle. Let \beta be the radian measure of the smallest angle in a 7{-}24{-}25 right triangle. In terms of \alpha , what is \beta ?
💡 解题思路
We are given that \[\tan\alpha=\frac{3}{4} \text{ and } \tan\beta=\frac{7}{24}.\] We have \begin{align*} \tan(\alpha+\beta) &= \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta} \\ &= \frac{\frac{3}{4
11
第 11 题
数论
There are exactly K positive integers b with 5 ≤ b ≤ 2024 such that the base- b integer 2024_b is divisible by 16 (where 16 is in base ten). What is the sum of the digits of K ?
💡 解题思路
$2024_b = 2b^3+2b+4\equiv 0\pmod{16}\implies b^3+b+2\equiv 0\pmod 8$ . If $b$ is even, then $b+2\equiv 0\pmod 8\implies b\equiv 6\pmod 8$ . If $b$ is odd, then $b^2\equiv 1\pmod 8$ * $\implies b^3+b+2
12
第 12 题
规律与数列
The first three terms of a geometric sequence are the integers a, 720, and b, where a<720<b. What is the sum of the digits of the least possible value of b?
💡 解题思路
For a geometric sequence, we have $ab=720^2=2^8 3^4 5^2$ , and we can test values for $b$ . We find that $b=768$ and $a=675$ works, and we can test multiples of $5$ in between the two values. Finding
13
第 13 题
坐标几何
The graph of y=e^{x+1}+e^{-x}-2 has an axis of symmetry. What is the reflection of the point (-1,\tfrac{1}{2}) over this axis?
💡 解题思路
The line of symmetry is probably of the form $x=a$ for some constant $a$ . A vertical line of symmetry at $x=a$ for a function $f$ exists if and only if $f(a-b)=f(a+b)$ ; we substitute $a-b$ and $a+b$
14
第 14 题
行程问题
The numbers, in order, of each row and the numbers, in order, of each column of a 5 × 5 array of integers form an arithmetic progression of length 5 . The numbers in positions (5, 5), (2,4), (4,3), and (3, 1) are 0, 48, 16, and 12 , respectively. What number is in position (1, 2)? https://youtu.be/bA95oaAbEbY
💡 解题思路
Start from the $0$ . Going up, let the common difference be $a$ , and going left, let the common difference be $b$ . Therefore, we have \[\begin{bmatrix} . & ? &.&.&4a \\ .&.&.&48&3a\\ 12&.&.&.&2a\\ .
15
第 15 题
综合
💡 解题思路
You can factor $(p^2 + 4)(q^2 + 4)(r^2 + 4)$ as $(p+2i)(p-2i)(q+2i)(q-2i)(r+2i)(r-2i)$ .
16
第 16 题
数论
A set of 12 tokens — 3 red, 2 white, 1 blue, and 6 black — is to be distributed at random to 3 game players, 4 tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as \frac{m}{n} , where m and n are relatively prime positive integers. What is m+n ?
💡 解题思路
We have $\binom{12}{4,4,4}$ ways to handle the red/white/blue tokens distribution on the denominator. Now we simply $\binom{6}{1}$ $\binom{5}{2}$ $3!$ for the numerator in order to handle the black to
17
第 17 题
整数运算
Integers a , b , and c satisfy ab + c = 100 , bc + a = 87 , and ca + b = 60 . What is ab + bc + ca ?
💡 解题思路
Subtracting the first two equations yields $(a-c)(b-1)=13$ . Notice that both factors are integers, so $b-1$ could equal one of $13,1,-1,-13$ and $b=14,2,0,-12$ . We consider each case separately:
18
第 18 题
时间问题
On top of a rectangular card with sides of length 1 and 2+√(3) , an identical card is placed so that two of their diagonals line up, as shown ( \overline{AC} , in this case). [图] Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled B in the figure?
💡 解题思路
Let the midpoint of $AC$ be $P$ .
19
第 19 题
几何·角度
Cyclic quadrilateral ABCD has lengths BC=CD=3 and DA=5 with \angle CDA=120^\circ . What is the length of the shorter diagonal of ABCD ?
💡 解题思路
[asy] import geometry; size(200); pair A = (-1.66, 0.33); pair B = (-9.61277, 1.19799); pair C = (-7.83974, 3.61798); pair D = (-4.88713, 4.14911); draw(circumcircle(A, B, C)); draw(A--C); draw(A--D);
20
第 20 题
几何·面积
Points P and Q are chosen uniformly and independently at random on sides \overline {AB} and \overline{AC}, respectively, of equilateral triangle \triangle ABC. Which of the following intervals contains the probability that the area of \triangle APQ is less than half the area of \triangle ABC?
💡 解题思路
Let $\overline{AP}=x$ and $\overline{AQ}=y$ . Applying the sine formula for a triangle's area, we see that \[[\Delta APQ]=\dfrac12\cdot x\cdot y\sin(\angle PAQ)=\dfrac{xy}2\sin(60^\circ)=\dfrac{\sqrt3
21
第 21 题
规律与数列
Suppose that a_1 = 2 and the sequence (a_n) satisfies the recurrence relation \[\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}\] for all n \ge 2. What is the greatest integer less than or equal to \[\sum^{100}_{n=1} a_n^2?\]
💡 解题思路
Multiply both sides of the recurrence to find that $n(a_n-1)=(n-1)(a_{n-1}+1)=(n-1)(a_{n-1}-1)+(n-1)(2)$ .
22
第 22 题
综合
💡 解题思路
Observations:
23
第 23 题
综合
What is the value of \[\tan^2 \frac {π}{16} · \tan^2 \frac {3π}{16} + \tan^2 \frac {π}{16} · \tan^2 \frac {5π}{16}+\tan^2 \frac {3π}{16} · \tan^2 \frac {7π}{16}+\tan^2 \frac {5π}{16} · \tan^2 \frac {7π}{16}?\]
💡 解题思路
First, notice that
24
第 24 题
几何·面积
A \textit{disphenoid} is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
💡 解题思路
Notice that any scalene $\textit{acute}$ triangle can be the faces of a $\textit{disphenoid}$ . (See proof in Solution 2.)
25
第 25 题
坐标几何
A graph is \textit{symmetric} about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers (a,b,c,d) , where |a|,|b|,|c|,|d|\le5 and c and d are not both 0 , is the graph of \[y=\frac{ax+b}{cx+d}\] symmetric about the line y=x ?
💡 解题思路
Symmetric about the line $y=x$ implies that the inverse function $y^{-1}=y$ . Then we split the question into several cases to find the final answer.