2022A 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
💡 解题思路
We have \begin{align*} 3+\frac{1}{3+\frac{1}{3+\frac13}} &= 3+\frac{1}{3+\frac{1}{\left(\frac{10}{3}\right)}} \\ &= 3+\frac{1}{3+\frac{3}{10}} \\ &= 3+\frac{1}{\left(\frac{33}{10}\right)} \\ &= 3+\fra
2
第 2 题
规律与数列
The sum of three numbers is 96. The first number is 6 times the third number, and the third number is 40 less than the second number. What is the absolute value of the difference between the first and second numbers?
💡 解题思路
Let $x$ be the third number. It follows that the first number is $6x,$ and the second number is $x+40.$
3
第 3 题
综合
💡 解题思路
The area of this square is equal to $6 + 8 + 30 + 14 + 6 = 64$ , and thus its side lengths are $8$ . The sum of the dimensions of the rectangles are $2 + 7 + 5 + 6 + 2 + 3 + 1 + 6 + 2 + 4 = 38$ . Thus
4
第 4 题
数论
The least common multiple of a positive integer n and 18 is 180 , and the greatest common divisor of n and 45 is 15 . What is the sum of the digits of n ?
💡 解题思路
Note that \begin{align*} 18 &= 2\cdot3^2, \\ 180 &= 2^2\cdot3^2\cdot5, \\ 45 &= 3^2\cdot5 \\ 15 &= 3\cdot5. \end{align*} Let $n = 2^a\cdot3^b\cdot5^c.$ It follows that:
5
第 5 题
坐标几何
The taxicab distance between points (x_1, y_1) and (x_2, y_2) in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\] For how many points P with integer coordinates is the taxicab distance between P and the origin less than or equal to 20 ? All possible locations of P are lattice points such that |x|+|y|≤ 20, whose graph is shown below: [图] ~MRENTHUSIASM
💡 解题思路
Let us consider the number of points for a certain $x$ -coordinate. For any $x$ , the viable points are in the range $[-20 + |x|, 20 - |x|]$ . This means that our total sum is equal to \begin{align*}
6
第 6 题
统计
A data set consists of 6 (not distinct) positive integers: 1 , 7 , 5 , 2 , 5 , and X . The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all positive values of X ?
💡 解题思路
First, note that $1+7+5+2+5=20$ . There are $3$ possible cases:
7
第 7 题
几何·面积
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible? [图]
💡 解题思路
The top left rectangle can be $5$ possible colors. Then the bottom left region can only be $4$ possible colors, and the bottom middle can only be $3$ colors since it is next to the top left and bottom
8
第 8 题
综合
The infinite product \[\sqrt[3]{10} · \sqrt[3]{\sqrt[3]{10}} · \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} ·s\] evaluates to a real number. What is that number?
💡 解题思路
We can write $\sqrt[3]{10}$ as $10 ^ \frac{1}{3}$ . Similarly, $\sqrt[3]{\sqrt[3]{10}} = (10 ^ \frac{1}{3}) ^ \frac{1}{3} = 10 ^ \frac{1}{3^2}$ .
9
第 9 题
计数
On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth?
💡 解题思路
Suppose that there are $T$ truth-tellers, $L$ liars, and $A$ alternaters who responded lie-truth-lie.
10
第 10 题
计数
How many ways are there to split the integers 1 through 14 into 7 pairs such that in each pair, the greater number is at least 2 times the lesser number?
💡 解题思路
Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
11
第 11 题
行程问题
What is the product of all real numbers x such that the distance on the number line between \log_6x and \log_69 is twice the distance on the number line between \log_610 and 1 ?
Let M be the midpoint of \overline{AB} in regular tetrahedron ABCD . What is \cos(\angle CMD) ? [图] ~MRENTHUSIASM
💡 解题思路
Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
13
第 13 题
几何·面积
Let \mathcal{R} be the region in the complex plane consisting of all complex numbers z that can be written as the sum of complex numbers z_1 and z_2 , where z_1 lies on the segment with endpoints 3 and 4i , and z_2 has magnitude at most 1 . What integer is closest to the area of \mathcal{R} ?
What is the value of \[(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)\] where \log denotes the base-ten logarithm?
💡 解题思路
Let $\text{log } 2 = x$ . The expression then becomes \[(1+x)^3+(1-x)^3+(3x)(-2x)=\boxed{2}.\]
15
第 15 题
立体几何
The roots of the polynomial 10x^3 - 39x^2 + 29x - 6 are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
💡 解题思路
Let $a$ , $b$ , $c$ be the three roots of the polynomial. The lengthened prism's volume is \[V = (a+2)(b+2)(c+2) = abc+2ac+2ab+2bc+4a+4b+4c+8 = abc + 2(ab+ac+bc) + 4(a+b+c) + 8.\] By Vieta's formulas,
16
第 16 题
几何·面积
A \emph{triangular number} is a positive integer that can be expressed in the form t_n = 1+2+3+·s+n , for some positive integer n . The three smallest triangular numbers that are also perfect squares are t_1 = 1 = 1^2 , t_8 = 36 = 6^2 , and t_{49} = 1225 = 35^2 . What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
💡 解题思路
We have $t_n = \frac{n (n+1)}{2}$ . If $t_n$ is a perfect square, then it can be written as $\frac{n (n+1)}{2} = k^2$ , where $k$ is a positive integer.
17
第 17 题
方程
Suppose a is a real number such that the equation \[a·(\sin{x}+\sin{(2x)}) = \sin{(3x)}\] has more than one solution in the interval (0, π) . The set of all such a that can be written in the form \[(p,q) \cup (q,r),\] where p, q, and r are real numbers with p < q< r . What is p+q+r ?
💡 解题思路
We are given that $a\cdot(\sin{x}+\sin{(2x)})=\sin{(3x)}$
18
第 18 题
坐标几何
Let T_k be the transformation of the coordinate plane that first rotates the plane k degrees counterclockwise around the origin and then reflects the plane across the y -axis. What is the least positive integer n such that performing the sequence of transformations T_1, T_2, T_3, ·s, T_n returns the point (1,0) back to itself?
💡 解题思路
Let $P=(r,\theta)$ be a point in polar coordinates, where $\theta$ is in degrees.
19
第 19 题
统计
Suppose that 13 cards numbered 1, 2, 3, \ldots, 13 are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the 13! possible orderings of the cards will the 13 cards be picked up in exactly two passes?
💡 解题思路
For $1\leq k\leq 12,$ suppose that cards $1, 2, \ldots, k$ are picked up on the first pass. It follows that cards $k+1,k+2,\ldots,13$ are picked up on the second pass.
20
第 20 题
几何·角度
Isosceles trapezoid ABCD has parallel sides \overline{AD} and \overline{BC}, with BC < AD and AB = CD. There is a point P in the plane such that PA=1, PB=2, PC=3, and PD=4. What is \tfrac{BC}{AD}?
💡 解题思路
Consider the reflection $P^{\prime}$ of $P$ over the perpendicular bisector of $\overline{BC}$ , creating two new isosceles trapezoids $DAPP^{\prime}$ and $CBPP^{\prime}$ . Under this reflection, $P^{
21
第 21 题
数论
Let \[P(x) = x^{2022} + x^{1011} + 1.\] Which of the following polynomials is a factor of P(x) ?
💡 解题思路
$P(x) = x^{2022} + x^{1011} + 1$ is equal to $\frac{x^{3033}-1}{x^{1011}-1}$ by difference of powers.
22
第 22 题
几何·面积
Let c be a real number, and let z_1 and z_2 be the two complex numbers satisfying the equation z^2 - cz + 10 = 0 . Points z_1 , z_2 , \frac{1}{z_1} , and \frac{1}{z_2} are the vertices of (convex) quadrilateral \mathcal{Q} in the complex plane. When the area of \mathcal{Q} obtains its maximum possible value, c is closest to which of the following?
💡 解题思路
Because $c$ is real, $z_2 = \bar z_1$ . We have \begin{align*} 10 & = z_1 z_2 \\ & = z_1 \bar z_1 \\ & = |z_1|^2 , \end{align*} where the first equality follows from Vieta's formula.
23
第 23 题
数论
Let h_n and k_n be the unique relatively prime positive integers such that \[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+·s+\frac{1}{n}=\frac{h_n}{k_n}.\] Let L_n denote the least common multiple of the numbers 1, 2, 3, \ldots, n . For how many integers with 1\le{n}\le{22} is k_n<L_n ?
💡 解题思路
We are given that \[\sum_{i=1}^{n}\frac1i = \frac{1}{L_n}\sum_{i=1}^{n}\frac{L_n}{i} = \frac{h_n}{k_n}.\] Since $k_n 1.$
24
第 24 题
数字运算
How many strings of length 5 formed from the digits 0 , 1 , 2 , 3 , 4 are there such that for each j \in \{1,2,3,4\} , at least j of the digits are less than j ? (For example, 02214 satisfies this condition because it contains at least 1 digit less than 1 , at least 2 digits less than 2 , at least 3 digits less than 3 , and at least 4 digits less than 4 . The string 23404 does not satisfy the condition because it does not contain at least 2 digits less than 2 .)
💡 解题思路
For some $n$ , let there be $n+1$ parking spaces counterclockwise in a circle. Consider a string of $n$ integers $c_1c_2 \ldots c_n$ each between $0$ and $n$ , and let $n$ cars come into this circle s
25
第 25 题
几何·面积
A circle with integer radius r is centered at (r, r) . Distinct line segments of length c_i connect points (0, a_i) to (b_i, 0) for 1 \le i \le 14 and are tangent to the circle, where a_i , b_i , and c_i are all positive integers and c_1 \le c_2 \le ·s \le c_{14} . What is the ratio \frac{c_{14}}{c_1} for the least possible value of r ?
💡 解题思路
Suppose that with a pair $(a_i,b_i)$ the circle is an excircle. Then notice that the hypotenuse must be $(r-x)+(r-y)$ , so it must be the case that \[a_i^2+b_i^2=(2r-a_i-b_i)^2.\] Similarly, if with a