2024 AMC 10A — 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
数论
Let n be the least prime number that can be written as the sum of 5 distinct prime numbers. What is the sum of the digits of n ?
💡 解题思路
Let the requested sum be $S.$ Recall that $2$ is the only even (and the smallest) prime, so $S$ is odd. It follows that the five distinct primes are all odd. The first few odd primes are $3,5,7,11,13,
4
第 4 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
5
第 5 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
6
第 6 题
规律与数列
What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA? (For example, 3 swaps are required to change ABC to CBA; one such sequence of swaps is ABC\to BAC\to BCA\to CBA. )
💡 解题思路
Procedurally, it takes:
7
第 7 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
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第 8 题
数论
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at 1:00 \ PM and were able to pack 4 , 3 , and 3 packages, respectively, every 3 minutes. At some later time, Daria joined the group, and Daria was able to pack 5 packages every 4 minutes. Together, they finished packing 450 packages at exactly 2:45\ PM . At what time did Daria join the group?
💡 解题思路
Note that Amy, Bomani, and Charlie pack a total of $4+3+3=10$ packages every $3$ minutes.
9
第 9 题
计数
In how many ways can 6 juniors and 6 seniors form 3 disjoint teams of 4 people so that each team has 2 juniors and 2 seniors?
💡 解题思路
The number of ways in which we can choose the juniors for the team are ${6\choose2}{4\choose2}{2\choose2}=15\cdot6\cdot1=90$ . Similarly, the number of ways to choose the seniors are the same, so the
10
第 10 题
数论
Consider the following operation. Given a positive integer n , if n is a multiple of 3 , then you replace n by \frac{n}{3} . If n is not a multiple of 3 , then you replace n by n+10 . Then continue this process. For example, beginning with n=4 , this procedure gives 4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to ·s . Suppose you start with n=100 . What value results if you perform this operation exactly 100 times?
💡 解题思路
Let $s$ be the number of times the operation is performed. Notice the sequence goes $100 \to 110 \to 120 \to 40 \to 50 \to 60 \to 20 \to 30 \to 10 \to 20 \to \cdots$ . Thus, for $s \equiv 1 \pmod{3}$
11
第 11 题
整数运算
How many ordered pairs of integers (m, n) satisfy √(n^2 - 49) = m ?
💡 解题思路
Note that $m$ is a nonnegative integer.
12
第 12 题
统计
Zelda played the Adventures of Math game on August 1 and scored 1700 points. She continued to play daily over the next 5 days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was 1700 + 80 = 1780 points.) What was Zelda's average score in points over the 6 days?
💡 解题思路
Going through the table, we see her scores over the six days were: $1700$ , $1700+80=1780$ , $1780-90=1690$ , $1690-10=1680$ , $1680+60=1740$ , and $1740-40=1700$ .
13
第 13 题
坐标几何
Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane: Of the 6 pairs of distinct transformations from this list, how many commute?
💡 解题思路
Label the given transformations $T_1, T_2, T_3,$ and $T_4,$ respectively. The rules of transformations are:
14
第 14 题
几何·面积
One side of an equilateral triangle of height 24 lies on line \ell . A circle of radius 12 is tangent to line \ l and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line \ell can be written as a √(b) - c π , where a , b , and c are positive integers and b is not divisible by the square of any prime. What is a + b + c ? [图] ~MRENTHUSIASM
💡 解题思路
Call the bottom vertices of the triangle $B$ and $C$ (the one closer to the circle is $C$ ) and the top vertex $A$ . The tangency point between the circle and the side of the triangle is $D$ , and the
15
第 15 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
16
第 16 题
几何·面积
All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length AB ?
💡 解题思路
Using the rectangle with area $1$ , let its short side be $x$ and the long side be $y$ . Observe that for every rectangle, since ratios of the side length of the rectangles are directly proportional t
17
第 17 题
概率
Two teams are in a best-two-out-of-three playoff: the teams will play at most 3 games, and the winner of the playoff is the first team to win 2 games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a \frac{2}{3} chance of winning at home, and its probability of winning when playing away from home is p . Outcomes of the games are independent. The probability that Team A wins the playoff is \frac{1}{2} . Then p can be written in the form \frac{1}{2}(m - √(n)) , where m and n are positive integers. What is m + n ?
💡 解题思路
We only have three cases where A wins: AA, ABA, and BAA (A denotes a team A win and B denotes a team B win). Knowing this, we can sum up the probability of each case. Thus the total probability is $\f
18
第 18 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
19
第 19 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
20
第 20 题
综合
Let S be a subset of \{1, 2, 3, \dots, 2024\} such that the following two conditions hold: What is the maximum possible number of elements in S ?
💡 解题思路
All lists are organized in ascending order:
21
第 21 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\\ .
22
第 22 题
几何·面积
Let \mathcal K be the kite formed by joining two right triangles with legs 1 and √(3) along a common hypotenuse. Eight copies of \mathcal K are used to form the polygon shown below. What is the area of triangle \triangle ABC ?
💡 解题思路
First, we should find the length of $AB$ . In order to do this, as we see in the diagram, it can be split into 4 equal sections. Since diagram $K$ shows us that it is made up of two ${30,60,90}$ trian
23
第 23 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:
24
第 24 题
概率
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled A^+, A^-, B^+, B^-, C^+, and C^- is rolled. Suppose the bee occupies the point (a,b,c). If the die shows A^+ , then the bee moves to the point (a+1,b,c) and if the die shows A^-, then the bee moves to the point (a-1,b,c). Analogous moves are made with the other four outcomes. Suppose the bee starts at the point (0,0,0) and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube? Diagrams have been moved to the bottom of the solutions.
💡 解题思路
We start by imagining the three dimensional plane.