2023 AMC 10A — Official Competition Problems (November 2023)
📅 2023 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
行程问题
Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B . Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many miles from City A will they be when they meet?
💡 解题思路
This is a $d=st$ problem, so let $x$ be the time it takes to meet. We can write the following equation: \[12x+18x=45\] Solving gives us $x=1.5$ . The $18x$ is Alicia so $18\times1.5=\boxed{\textbf{(E)
2
第 2 题
统计
The weight of \frac{1}{3} of a large pizza together with 3 \frac{1}{2} cups of orange slices is the same as the weight of \frac{3}{4} of a large pizza together with \frac{1}{2} cup of orange slices. A cup of orange slices weighs \frac{1}{4} of a pound. What is the weight, in pounds, of a large pizza?
💡 解题思路
Use a system of equations. Let $x$ be the weight of a pizza and $y$ be the weight of a cup of orange slices. We have \[\frac{1}{3}x+\frac{7}{2}y=\frac{3}{4}x+\frac{1}{2}y.\] Rearranging, we get \begin
3
第 3 题
几何·面积
How many positive perfect squares less than 2023 are divisible by 5 ?
💡 解题思路
Since $\left \lfloor{\sqrt{2023}}\right \rfloor = 44$ , there are $\left \lfloor{\frac{44}{5}}\right \rfloor = \boxed{\textbf{(A) 8}}$ perfect squares less than 2023 that are divisible by 5.
4
第 4 题
几何·面积
A quadrilateral has all integer sides lengths, a perimeter of 26 , and one side of length 4 . What is the greatest possible length of one side of this quadrilateral?
💡 解题思路
Let's use the triangle inequality. We know that for a triangle, the sum of the 2 shorter sides must always be longer than the longest side. This is because if the longest side were to be as long as th
5
第 5 题
数字运算
How many digits are in the base-ten representation of 8^5 · 5^{10} · 15^5 ?
💡 解题思路
Prime factorizing this gives us $2^{15}\cdot3^{5}\cdot5^{15}=10^{15}\cdot3^5=243\cdot10^{15}$ .
6
第 6 题
综合
💡 解题思路
Each of the vertices is counted $3$ times because each vertex is shared by three different edges. Each of the edges is counted $2$ times because each edge is shared by two different faces. Since the s
7
第 7 题
概率
Janet rolls a standard 6 -sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3 ?
💡 解题思路
There are $3$ cases where the running total will equal $3$ : one roll; two rolls; or three rolls:
8
第 8 题
几何·角度
Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at 110 degrees Fahrenheit, which is 0 degrees on the Breadus scale. Bread is baked at 350 degrees Fahrenheit, which is 100 degrees on the Breadus scale. Bread is done when its internal temperature is 200 degrees Fahrenheit. What is this in degrees on the Breadus scale?
💡 解题思路
To solve this question, you can use $f(x) = mx + b$ where the $x$ is Fahrenheit and the $y$ is Breadus. We have $(110,0)$ and $(350,100)$ . We want to find the value of $y$ in $(200,y)$ that falls on
9
第 9 题
行程问题
A digital display shows the current date as an 8 -digit integer consisting of a 4 -digit year, followed by a 2 -digit month, followed by a 2 -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in 2023 does each digit appear an even number of times in the 8 -digital display for that date?
💡 解题思路
Do careful casework by each month. Make sure to start with 2023. In the month and the date, we need a $0$ , a $3$ , and two digits repeated (which has to be $1$ and $2$ after consideration). After the
10
第 10 题
综合
💡 解题思路
Let $a$ represent the amount of tests taken previously and $x$ the mean of the scores taken previously.
11
第 11 题
几何·面积
A square of area 2 is inscribed in a square of area 3 , creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle? [图]
💡 解题思路
The side lengths of the inner square and outer square are $\sqrt{2}$ and $\sqrt{3}$ respectively. Let the shorter side of our triangle be $x$ , thus the longer leg is $\sqrt{3}-x$ . Hence, by the Pyth
12
第 12 题
数字运算
How many three-digit positive integers N satisfy the following properties?
💡 解题思路
Multiples of $5$ will always end in $0$ or $5$ , and since the numbers have to be a three-digit numbers, it cannot start with 0 (otherwise it would be a two-digit number), narrowing our choices to 3-d
13
第 13 题
几何·面积
Abdul and Chiang are standing 48 feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures 60^\circ . What is the square of the distance (in feet) between Abdul and Bharat?
💡 解题思路
Let $\theta=\angle ACB$ and $x=\overline{AB}$ .
14
第 14 题
数论
A number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by 11 ?
💡 解题思路
In order for the divisor chosen to be a multiple of $11$ , the original number chosen must also be a multiple of $11$ . Among the first $100$ positive integers, there are 9 multiples of 11; 11, 22, 33
15
第 15 题
几何·面积
An even number of circles are nested, starting with a radius of 1 and increasing by 1 each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius 2 but outside the circle of radius 1. An example showing 8 circles is displayed below. What is the least number of circles needed to make the total shaded area at least 2023π ? [图]
💡 解题思路
Notice that the area of the shaded region is $(2^2\pi-1^2\pi)+(4^2\pi-3^2\pi)+(6^2\pi-5^2\pi)+ \cdots + (n^2\pi-(n-1)^2 \pi)$ for any even number $n$ .
16
第 16 题
综合
In a table tennis tournament, every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was 40\% more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
💡 解题思路
We know that the total amount of games must be the sum of games won by left and right handed players. Then, we can write $g = l + r$ , and since $l = 1.4r$ , $g = 2.4r$ . Given that $r$ and $g$ are bo
17
第 17 题
几何·面积
Let ABCD be a rectangle with AB = 30 and BC = 28 . Point P and Q lie on \overline{BC} and \overline{CD} respectively so that all sides of \triangle{ABP}, \triangle{PCQ}, and \triangle{QDA} have integer lengths. What is the perimeter of \triangle{APQ} ?
💡 解题思路
[asy] /* ~ItsMeNoobieboy */ size(200); pair A, B, C, D, P, Q; A = (0,28/30); B = (1,28/30); C = (1,0); D = (0,0); P = (1,12/30); Q = (21/30,0); draw(A--B--C--D--cycle); draw(A--P--Q--cycle); dot("$A$"
18
第 18 题
几何·角度
A rhombic dodecahedron is a solid with 12 congruent rhombus faces. At every vertex, 3 or 4 edges meet, depending on the vertex. How many vertices have exactly 3 edges meet?
💡 解题思路
Note Euler's formula where $\text{Vertices}+\text{Faces}-\text{Edges}=2$ . There are $12$ faces. There are $24$ edges, because there are 12 faces each with four edges and each edge is shared by two fa
19
第 19 题
综合
The line segment formed by A(1, 2) and B(3, 3) is rotated to the line segment formed by A'(3, 1) and B'(4, 3) about the point P(r, s) . What is |r-s| ?
💡 解题思路
Due to rotations preserving an equal distance, we can bash the answer with the distance formula. $D(A, P) = D(A', P)$ , and $D(B, P) = D(B',P)$ . Thus we will square our equations to yield: $(1-r)^2+(
20
第 20 题
几何·面积
Each square in a 3×3 grid of squares is colored red, white, blue, or green so that every 2×2 square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible? [图]
💡 解题思路
Let a "tile" denote a \(1 \times 1\) square, and a "square" refer to a \(2 \times 2\) square.
21
第 21 题
数论
Let P(x) be the unique polynomial of minimal degree with the following properties: The roots of P(x) are integers, with one exception. The root that is not an integer can be written as \frac{m}{n} , where m and n are relatively prime integers. What is m+n ?
💡 解题思路
From the problem statement, we find $P(2-2)=0$ , $P(9)=0$ and $4P(4)=0$ . Therefore, we know that $0$ , $9$ , and $4$ are roots. So, we can factor $P(x)$ as $x(x - 9)(x - 4)(x - a)$ , where $a$ is the
22
第 22 题
几何·面积
Circle C_1 and C_2 each have radius 1 , and the distance between their centers is \frac{1}{2} . Circle C_3 is the largest circle internally tangent to both C_1 and C_2 . Circle C_4 is internally tangent to both C_1 and C_2 and externally tangent to C_3 . What is the radius of C_4 ? [图]
If the positive integer c has positive integer divisors a and b with c = ab , then a and b are said to be \textit{complementary} divisors of c . Suppose that N is a positive integer that has one complementary pair of divisors that differ by 20 and another pair of complementary divisors that differ by 23 . What is the sum of the digits of N ?
💡 解题思路
Consider positive integers $a, b$ with a difference of $20$ . Suppose $b = a-20$ . Then, we have $(a)(a-20) = n$ . If there is another pair of two integers that multiply to $n$ but have a difference o
24
第 24 题
综合
💡 解题思路
[asy] unitsize(1cm); pair A, B, C, D, E, F, W,X,Y,Z; real bigSide = 3; real smallSide = 1; real angle = 60; // Each external angle for the hexagon real offset = 3/7; // Offset for the smaller hexagons
25
第 25 题
几何·面积
If A and B are vertices of a polyhedron, define the distance d(A,B) to be the minimum number of edges of the polyhedron one must traverse in order to connect A and B . For example, if \overline{AB} is an edge of the polyhedron, then d(A, B) = 1 , but if \overline{AC} and \overline{CB} are edges and \overline{AB} is not an edge, then d(A, B) = 2 . Let Q , R , and S be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that d(Q, R) > d(R, S) ?
💡 解题思路
To find the total amount of vertices we first find the amount of edges, and that is $\frac{20 \times 3}{2}$ . Next, to find the amount of vertices we can use Euler's characteristic, $V - E + F = 2$ ,