2024 AMC 10B — Official Competition Problems (February 2024)
📅 2024 B 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
📋 答题说明
共 25 道题，每题从 A、B、C、D、E 五个选项中选一个答案，点击选项即可选择
答题过程中可随时更改选项，选完后点击底部「提交答案」统一批改
提交后显示对错、正确答案和简短解题思路
点击题目右侧 ⭐ 可收藏难题，方便后续复习
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
统计
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line? In a long line of people arranged left to right, the 1015th person from the left is also the 1010th person from the right. How many people are in line?
💡 解题思路
If the person is the 1015th from the left, that means there is 1014 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there a
For how many integer values of x is |2x| ≤ 7 π For how many integer values of x is |2x| ≤ 6 π
💡 解题思路
Use the fact that $\pi \approx 3.14$ , and thus you can get $6\pi \approx 18.84$ . We could easily see that the answer is $\{-9,-8,...,8,9\}\implies\boxed{\text{(C) }19}$
4
第 4 题
综合
Balls numbered 1, 2, 3, ... are deposited in 5 bins, labeled A, B, C, D, and E, using the following procedure. Ball 1 is deposited in bin A, and balls 2 and 3 are deposited in bin B. The next 3 balls are deposited in bin C, the next 4 in bin D, and so on, cycling back to bin A after balls are deposited in bin E. (For example, balls numbered 22, 23, ..., 28 are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited?
In the following expression, Melanie changed some of the plus signs to minus signs: \[1+3+5+7+...+97+99\] When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
💡 解题思路
Recall that the sum of the first $n$ odd numbers is $n^2$ . Thus \[1 + 3 + 5 + 7+ \dots + 97 + 99 = 50^2 = 2500.\]
6
第 6 题
几何·面积
A rectangle has integer length sides and an area of 2024. What is the least possible perimeter of the rectangle?
💡 解题思路
We can start by assigning the values x and y for both sides. Here is the equation representing the area:
7
第 7 题
数论
What is the remainder when 7^{2024}+7^{2025}+7^{2026} is divided by 19 ?
💡 解题思路
We can factor the expression as
8
第 8 题
数字运算
Let N be the product of all the positive integer divisors of 42 . What is the units digit of N ?
💡 解题思路
The factors of $42$ are $1, 2, 3, 6, 7, 14, 21, 42$ . Multiply the unit digits to get $\boxed{\textbf{(D) } 6}$
9
第 9 题
统计
Real numbers a, b, and c have arithmetic mean 0 . The arithmetic mean of a^2, b^2, and c^2 is 10 . What is the arithmetic mean of ab, ac, and bc ?
💡 解题思路
If $\frac{a+b+c}{3} = 0$ , that means $a+b+c=0$ , and $(a+b+c)^2=0$ . Expanding that gives \[(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc\] If $\frac{a^2+b^2+c^2}{3} = 10$ , then $a^2+b^2+c^2=30$ . Thus, we have
10
第 10 题
几何·面积
Quadrilateral ABCD is a parallelogram, and E is the midpoint of the side \overline{AD} . Let F be the intersection of lines EB and AC . What is the ratio of the area of quadrilateral CDEF to the area of \triangle CFB ?
💡 解题思路
Let $AB = CD$ have length $b$ and let the altitude of the parallelogram perpendicular to $\overline{AD}$ have length $h$ .
11
第 11 题
几何·面积
In the figure below WXYZ is a rectangle with WX=4 and WZ=8 . Point M lies \overline{XY} , point A lies on \overline{YZ} , and \angle WMA is a right angle. The areas of \triangle WXM and \triangle WAZ are equal. What is the area of \triangle WMA ? [图] Note: On certain tests that took place in China, the problem asked for the area of \triangle MAY .
💡 解题思路
We know that $WX = 4$ , $WZ = 8$ , so $YZ = 4$ and $YX = 8$ . Since $\angle WMA = 90^\circ$ , triangles $WXM$ and $MYA$ are similar. Therefore, $\frac{WX}{MY} = \frac{XM}{YA}$ , which gives $\frac{4}{
12
第 12 题
计数
A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students A and B , student A speaks some language that student B does not speak, and student B speaks some language that student A does not speak. What is the least possible total number of languages spoken by all the students?
💡 解题思路
We think of this problem like boxes. First start with 9. We see that we can arrange the groups of people into the 9 boxes. We take 9 people of different languages and arrange them in each of the boxes
13
第 13 题
方程
Positive integers x and y satisfy the equation √(x) + √(y) = √(1183) . What is the minimum possible value of x+y ?
💡 解题思路
Note that $\sqrt{1183}=13\sqrt7$ . Since $x$ and $y$ are positive integers, and $\sqrt{x}+\sqrt{y}=\sqrt{1183}$ we can represent each value of $\sqrt{x}$ and $\sqrt{y}$ as the product of a positive in
14
第 14 题
坐标几何
A dartboard is the region B in the coordinate plane consisting of points (x, y) such that |x| + |y| \le 8 . A target T is the region where (x^2 + y^2 - 25)^2 \le 49 . A dart is thrown and lands at a random point in B. The probability that the dart lands in T can be expressed as \frac{m}{n} · π , where m and n are relatively prime positive integers. What is m + n ? [图] ~Elephant200
💡 解题思路
Inequalities of the form $|x|+|y| \le 8$ are well-known and correspond to a square in space with centre at origin and vertices at $(8, 0)$ , $(-8, 0)$ , $(0, 8)$ , $(0, -8)$ . The diagonal length of t
15
第 15 题
统计
A list of 9 real numbers consists of 1 , 2.2 , 3.2 , 5.2 , 6.2 , and 7 , as well as x , y , and z with x\ley\lez . The range of the list is 7 , and the mean and the median are both positive integers. How many ordered triples ( x , y , z ) are possible?
💡 解题思路
$\textbf{First Case}$
16
第 16 题
分数与比例
Jerry likes to play with numbers. One day, he wrote all the integers from 1 to 2024 on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase 1 , 2 , 3 , and 5 , and then write either 11 , their sum, or 30 , their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?
💡 解题思路
Consider the numbers as $1,0,1,0,...,1,0$ . Note that the number of odd integers is monotonously decreasing.
17
第 17 题
综合
In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?
💡 解题思路
Intuitive solution: We perform casework based on how many snails tie. Let's say we're dealing with the following snails: $A,B,C,D,E$ .
18
第 18 题
数论
How many different remainders can result when the 100 th power of an integer is divided by 125 ?
💡 解题思路
First note that the Euler's totient function of $125$ is $100$ . We can set up two cases, which depend on whether a number is relatively prime to $125.$
19
第 19 题
坐标几何
In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"? ~Cattycute
💡 解题思路
Let's look at zero slope first. All lines of such form will be expressed in the form $y=k$ , where $k$ is some real number. If $k$ is an integer, the line passes through infinitely many lattice points
20
第 20 题
计数
Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?
💡 解题思路
Let $A_R, A_L, B_R, B_L, C_R, C_L$ denote the shoes.
21
第 21 题
规律与数列
Two straight pipes (circular cylinders), with radii 1 and \frac{1}{4} , lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? [图]
💡 解题思路
Notice that the sum of radii of two circles tangent to each other will equal to the distance from center to center. Set the center of the big circle be at $(0,1).$ Since both circles are tangent to a
22
第 22 题
数论
A group of 16 people will be partitioned into 4 indistinguishable 4 -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as 3^{r}M , where r and M are positive integers and M is not divisible by 3 . What is r ?
💡 解题思路
There are ${16 \choose 4}$ ways to choose the first committee, ${12 \choose 4}$ ways to choose the second, ${8 \choose 4}$ for the third, and ${4 \choose 4}=1$ for the fourth. Since the committees are
23
第 23 题
综合
💡 解题思路
The first $20$ terms are $F_n = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765$
24
第 24 题
整数运算
Let \[P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}\] How many of the values P(2022) , P(2023) , P(2024) , and P(2025) are integers?
💡 解题思路
First, we know that $P(2022)$ and $P(2024)$ must be integers since they are both divisible by $2$ .
25
第 25 题
数论
Each of 27 bricks (right rectangular prisms) has dimensions a × b × c , where a , b , and c are pairwise relatively prime positive integers. These bricks are arranged to form a 3 × 3 × 3 block, as shown on the left below. A 28 th brick with the same dimensions is introduced, and these bricks are reconfigured into a 2 × 2 × 7 block, shown on the right. The new block is 1 unit taller, 1 unit wider, and 1 unit deeper than the old one. What is a + b + c ?
💡 解题思路
The $3$ x $3$ x $3$ block has side lengths of $3a, 3b, 3c$ . The $2$ x $2$ x $7$ block has side lengths of $2b, 2c, 7a$ .