2024B 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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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
行程问题
The national debt of the United States is on track to reach 5×10^{13} dollars by 2033 . How many digits does this number of dollars have when written as a numeral in base 5 ? (The approximation of \log_{10} 5 as 0.7 is sufficient for this problem)
💡 解题思路
Generally, the number of digits of number $n$ in base $b$ is \[\lfloor \log_b n \rfloor + 1.\] In this question, it is $\lfloor \log_{5} (5\times 10^{13})\rfloor+1$ . Note that \begin{align*} \log_{5}
7
第 7 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}{
8
第 8 题
综合
What value of x satisfies \[\frac{\log_2x · \log_3x}{\log_2x+\log_3x}=2?\]
💡 解题思路
We have \begin{align*} \log_2x\cdot\log_3x&=2(\log_2x+\log_3x) \\ 1&=\frac{2(\log_2x+\log_3x)}{\log_2x\cdot\log_3x} \\ 1&=2(\frac{1}{\log_3x}+\frac{1}{\log_2x}) \\ 1&=2(\log_x3+\log_x2) \\ \log_x6&=\f
9
第 9 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
10
第 10 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
$\textbf{First Case}$
11
第 11 题
统计
Let x_n = \sin^2(n^{\circ}) . What is the mean of x_1,x_2,x_3,\dots,x_{90} ?
💡 解题思路
Add up $x_1$ with $x_{89}$ , $x_2$ with $x_{88}$ , and $x_i$ with $x_{90-i}$ . Notice \[x_i+x_{90-i}=\sin^2(i^{\circ})+\sin^2((90-i)^{\circ})=\sin^2(i^{\circ})+\cos^2(i^{\circ})=1\] by the Pythagorean
12
第 12 题
几何·面积
Suppose z is a complex number with positive imaginary part, with real part greater than 1 , and with |z| = 2 . In the complex plane, the four points 0 , z , z^{2} , and z^{3} are the vertices of a quadrilateral with area 15 . What is the imaginary part of z ?
💡 解题思路
By making a rough estimate of where $z$ , $z^2$ , and $z^3$ are on the complex plane, we can draw a pretty accurate diagram (like above.)
13
第 13 题
方程
There are real numbers x,y,h and k that satisfy the system of equations \[x^2 + y^2 - 6x - 8y = h\] \[x^2 + y^2 - 10x + 4y = k\] What is the minimum possible value of h+k ?
💡 解题思路
Adding up the first and second equation, we get: \begin{align*} h + k &= 2x^2 + 2y^2 - 16x - 4y \\ &= 2x^2 - 16x + 2y^2 - 4y \\ &= 2(x^2 - 8x) + 2(y^2 - 2y) \\ &= 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 -
14
第 14 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 $125.$
15
第 15 题
几何·面积
A triangle in the coordinate plane has vertices A(\log_21,\log_22) , B(\log_23,\log_24) , and C(\log_27,\log_28) . What is the area of \triangle ABC ?
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
17
第 17 题
概率
Integers a and b are randomly chosen without replacement from the set of integers with absolute value not exceeding 10 . What is the probability that the polynomial x^3 + ax^2 + bx + 6 has 3 distinct integer roots?
💡 解题思路
Since $-10 \le a,b \le 10$ , there are 21 integers to choose from, and $P(21,2) = 21 \times 20 = 420$ equally likely ordered pairs $(a,b)$ .
18
第 18 题
综合
💡 解题思路
The first $20$ terms are $F_n = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765$
19
第 19 题
几何·面积
Equilateral \triangle ABC with side length 14 is rotated about its center by angle \theta , where 0 < \theta < 60^{\circ} , to form \triangle DEF . See the figure. The area of hexagon ADBECF is 91√(3) . What is \tan\theta ? [图]
💡 解题思路
Let O be circumcenter of the equilateral triangle
20
第 20 题
几何·面积
Suppose A , B , and C are points in the plane with AB=40 and AC=42 , and let x be the length of the line segment from A to the midpoint of \overline{BC} . Define a function f by letting f(x) be the area of \triangle ABC . Then the domain of f is an open interval (p,q) , and the maximum value r of f(x) occurs at x=s . What is p+q+r+s ?
💡 解题思路
Let the midpoint of $BC$ be $M$ , and let the length $BM = CM = a$ . We know there are limits to the value of $x$ , and these limits can probably be found through the triangle inequality. But the tria
21
第 21 题
几何·面积
The measures of the smallest angles of three different right triangles sum to 90^\circ . All three triangles have side lengths that are primitive Pythagorean triples. Two of them are 3-4-5 and 5-12-13 . What is the perimeter of the third triangle?
💡 解题思路
Let $\alpha$ and $\beta$ be the smallest angles of the $3-4-5$ and $5-12-13$ triangles respectively. We have \[\tan(\alpha)=\frac{3}{4} \text{ and } \tan(\beta)=\frac{5}{12}\] Then \[\tan(\alpha+\beta
22
第 22 题
几何·面积
Let \triangle{ABC} be a triangle with integer side lengths and the property that \angle{B} = 2\angle{A} . What is the least possible perimeter of such a triangle?
💡 解题思路
Let $AB=c$ , $BC=a$ , $AC=b$ . According to the law of sines, \[\frac{b}{a}=\frac{\sin \angle B}{\sin \angle A}=2\cos \angle A\]
23
第 23 题
几何·面积
A right pyramid has regular octagon ABCDEFGH with side length 1 as its base and apex V. Segments \overline{AV} and \overline{DV} are perpendicular. What is the square of the height of the pyramid?
💡 解题思路
To find the height of the pyramid, we need the length from the center of the octagon (denote as $I$ ) to its vertices and the length of AV.
24
第 24 题
几何·面积
What is the number of ordered triples (a,b,c) of positive integers, with a\le b\le c\le 9 , such that there exists a (non-degenerate) triangle \triangle ABC with an integer inradius for which a , b , and c are the lengths of the altitudes from A to \overline{BC} , B to \overline{AC} , and C to \overline{AB} , respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
💡 解题思路
First we derive the relationship between the inradius of a triangle $r$ , and its three altitudes $a, b, c$ . Using an area argument, we can get the following well known result \[\left(\frac{AB+BC+AC}
25
第 25 题
数论
Pablo will decorate each of 6 identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the 12 decisions he must make. After the paint dries, he will place the 6 balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as \frac mn, where m and n are relatively prime positive integers. What is m? (Recall that two events A and B are independent if P(A and B) = P(A) · P(B). )
💡 解题思路
Let $a$ be the number of balls that are both striped and red, $x$ be the number of balls that are striped but blue, and $y$ be the number of balls that are red but dotted. Then there must be $6-a-x-y$