2022 AMC 10A — Official Competition Problems (November 2022)
📅 2022 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
Mike cycled 15 laps in 57 minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first 27 minutes?
💡 解题思路
Mike's speed is $\frac{15}{57}=\frac{5}{19}$ laps per minute.
3
第 3 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 $x+40.$
4
第 4 题
计数
In some countries, automobile fuel efficiency is measured in liters per 100 kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals m miles, and 1 gallon equals l liters. Which of the following gives the fuel efficiency in liters per 100 kilometers for a car that gets x miles per gallon?
💡 解题思路
The formula for fuel efficiency is \[\frac{\text{Distance}}{\text{Gas Consumption}}.\] Note that $1$ mile equals $\frac 1m$ kilometers. We have \[\frac{x\text{ miles}}{1\text{ gallon}} = \frac{\frac{x
5
第 5 题
几何·面积
Square ABCD has side length 1 . Points P , Q , R , and S each lie on a side of ABCD such that APQCRS is an equilateral convex hexagon with side length s . What is s ? [图] ~MRENTHUSIASM
💡 解题思路
Note that $BP=BQ=DR=DS=1-s.$ It follows that $\triangle BPQ$ and $\triangle DRS$ are congruent isosceles right triangles.
6
第 6 题
综合
Which expression is equal to \[|a-2-√((a-1)^2)|\] for a<0?
💡 解题思路
We have \begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|a-2-|a-1|\right| \\ &=\left|a-2-(1-a)\right| \\ &=\left|2a-3\right| \\ &=\boxed{\textbf{(A) } 3-2a}. \end{align*} ~MRENTHUSIASM
7
第 7 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:
8
第 8 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:
9
第 9 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
10
第 10 题
综合
💡 解题思路
[asy] /* Edited by MRENTHUSIASM */ size(250); real x, y; x = 6; y = 3; draw((0,0)--(x,0)); draw((0,0)--(0,y)); draw((0,y)--(x,y)); draw((x,0)--(x,y)); draw((0.5,0)--(0.5,0.5)--(0,0.5)); draw((x-0.5,y)
11
第 11 题
规律与数列
Ted mistakenly wrote 2^m·√(\frac{1){4096}} as 2·\sqrt[m]{\frac{1}{4096}}. What is the sum of all real numbers m for which these two expressions have the same value?
💡 解题思路
We are given that \[2^m\cdot\sqrt{\frac{1}{4096}} = 2\cdot\sqrt[m]{\frac{1}{4096}}.\] Converting everything into powers of $2$ and equating exponents, we have \begin{align*} 2^m\cdot(2^{-12})^{\frac12
12
第 12 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.
13
第 13 题
几何·面积
Let \triangle ABC be a scalene triangle. Point P lies on \overline{BC} so that \overline{AP} bisects \angle BAC. The line through B perpendicular to \overline{AP} intersects the line through A parallel to \overline{BC} at point D. Suppose BP=2 and PC=3. What is AD? [图] ~MRENTHUSIASM
💡 解题思路
Suppose that $\overline{BD}$ intersects $\overline{AP}$ and $\overline{AC}$ at $X$ and $Y,$ respectively. By Angle-Side-Angle, we conclude that $\triangle ABX\cong\triangle AYX.$
14
第 14 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 $14.$
15
第 15 题
几何·面积
Quadrilateral ABCD with side lengths AB=7, BC=24, CD=20, DA=15 is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form \frac{aπ-b}{c}, where a,b, and c are positive integers such that a and c have no common prime factor. What is a+b+c? [图] ~MRENTHUSIASM
💡 解题思路
Opposite angles of every cyclic quadrilateral are supplementary, so \[\angle B + \angle D = 180^{\circ}.\] We claim that $AC=25.$ We can prove it by contradiction:
16
第 16 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,
17
第 17 题
分数与比例
How many three-digit positive integers \underline{a} \ \underline{b} \ \underline{c} are there whose nonzero digits a,b, and c satisfy \[0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?\] (The bar indicates repetition, thus 0.\overline{\underline{a}~\underline{b}~\underline{c}} is the infinite repeating decimal 0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~·s )
💡 解题思路
We rewrite the given equation, then rearrange: \begin{align*} \frac{100a+10b+c}{999} &= \frac13\left(\frac a9 + \frac b9 + \frac c9\right) \\ 100a+10b+c &= 37a + 37b + 37c \\ 63a &= 27b+36c \\ 7a &= 3
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
数论
Define L_n as the least common multiple of all the integers from 1 to n inclusive. There is a unique integer h such that \[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+·s+\frac{1}{17}=\frac{h}{L_{17}}\] What is the remainder when h is divided by 17 ?
💡 解题思路
Notice that $L_{17}$ contains the highest power of every prime below $17$ since higher primes cannot divide $L_{17}$ . Thus, $L_{17}=16\cdot 9 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17$ .
20
第 20 题
规律与数列
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57 , 60 , and 91 . What is the fourth term of this sequence?
💡 解题思路
Let the arithmetic sequence be $a,a+d,a+2d,a+3d$ and the geometric sequence be $b,br,br^2,br^3.$
21
第 21 题
综合
💡 解题思路
We extend line segments $\ell,m,$ and $n$ to their point of concurrency, as shown below: [asy] /* Made by AoPS; edited by MRENTHUSIASM */ import three; size(225); currentprojection= orthographic(camer
22
第 22 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.
23
第 23 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^{
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
Let R , S , and T be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x -axis. The left edge of R and the right edge of S are on the y -axis, and R contains \frac{9}{4} as many lattice points as does S . The top two vertices of T are in R \cup S , and T contains \frac{1}{4} of the lattice points contained in R \cup S. See the figure (not drawn to scale). [图] The fraction of lattice points in S that are in S \cap T is 27 times the fraction of lattice points in R that are in R \cap T . What is the minimum possible value of the edge length of R plus the edge length of S plus the edge length of T ?
💡 解题思路
Let $r$ be the number of lattice points on the side length of square $R$ , $s$ be the number of lattice points on the side length of square $S$ , and $t$ be the number of lattice points on the side le