2023 AMC 10B — Official Competition Problems (November 2023)
📅 2023 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
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1
第 1 题
分数与比例
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only \frac{1}{3} full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
💡 解题思路
The first three glasses each have a full glass. Let's assume that each glass has "1 unit" of juice. It won't matter exactly how much juice everyone has because we're dealing with ratios, and that woul
2
第 2 题
计数
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20\% on every pair of shoes. Carlos also knew that he had to pay a 7.5\% sales tax on the discounted price. He had \43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
💡 解题思路
Let the price originally be \( x \). Then, after a \(20\) percent discount, the price is now \( x - \frac{1}{5}x = \frac{4}{5}x \).
3
第 3 题
几何·面积
A 3-4-5 right triangle is inscribed in circle A , and a 5-12-13 right triangle is inscribed in circle B . What is the ratio of the area of circle A to the area of circle B ?
💡 解题思路
Because the triangles are right triangles, we know the hypotenuses are diameters of circles $A$ and $B$ . Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply $\pi$ to
4
第 4 题
几何·面积
Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint?
💡 解题思路
$6.5$ millimeters is equal to $0.65$ centimeters. $25$ meters is $2500$ centimeters. The answer is $0.65 \times 2500$ , so the answer is $\boxed{\textbf{(C) 1,625}}$ .
5
第 5 题
规律与数列
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds 3 to each number in the list and finds that the sum of her new numbers is 45 . Lara multiplies each number in the list by 3 and finds that the sum of her new numbers is also 45 . How many numbers are written on the blackboard?
💡 解题思路
Let there be $n$ numbers in the list of numbers, and let their sum be $S$ . Then we have the following
6
第 6 题
规律与数列
Let L_{1}=1, L_{2}=3 , and L_{n+2}=L_{n+1}+L_{n} for n≥ 1 . How many terms in the sequence L_{1}, L_{2}, L_{3},...,L_{2023} are even?
💡 解题思路
We calculate more terms: \[1,3,4,7,11,18,\ldots.\] We find a pattern: if $n+2$ is a multiple of $3$ , then the term is even, or else it is odd. There are $\left\lfloor \frac{2023}{3} \right\rfloor =\b
7
第 7 题
几何·面积
Square ABCD is rotated 20^{\circ} clockwise about its center to obtain square EFGH , as shown below. What is the degree measure of \angle EAB ? [图] (A)\ 24^{\circ} (B)\ 35^{\circ} (C)\ 30^{\circ} (D)\ 32^{\circ} (E)\ 20^{\circ}
💡 解题思路
First, let's call the center of both squares $I$ . Then, $\angle{AIE} = 20$ , and since $\overline{EI} = \overline{AI}$ , $\angle{AEI} = \angle{EAI} = 80$ . Then, we know that $AI$ bisects angle $\ang
8
第 8 题
数字运算
What is the units digit of 2022^{2023} + 2023^{2022} ? (A)\ 7 (B)\ 1 (C)\ 9 (D)\ 5 (E)\ 3
The numbers 16 and 25 are a pair of consecutive positive squares whose difference is 9 . How many pairs of consecutive positive perfect squares have a difference of less than or equal to 2023 ? (A)\ 674 (B)\ 1011 (C)\ 1010 (D)\ 2019 (E)\ 2017
💡 解题思路
Let $x$ be the square root of the smaller of the two perfect squares. Then, $(x+1)^2 - x^2 =x^2+2x+1-x^2 = 2x+1 \le 2023$ . Thus, $x \le 1011$ . So there are $\boxed{\text{(B)}1011}$ numbers that sati
10
第 10 题
几何·面积
You are playing a game. A 2 × 1 rectangle covers two adjacent squares (oriented either horizontally or vertically) of a 3 × 3 grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
💡 解题思路
Notice that the $3\times3$ square grid has a total of $12$ possible $2\times1$ rectangles.
11
第 11 题
综合
Suzanne went to the bank and withdrew \800 . The teller gave her this amount using \20 bills, \50 bills, and \100 bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
💡 解题思路
We let the number of $\$20$ , $\$50$ , and $\$100$ bills be $a,b,$ and $c,$ respectively.
12
第 12 题
整数运算
When the roots of the polynomial \[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 · · · · (x-10)^{10}\] are removed from the number line, what remains is the union of 11 disjoint open intervals. On how many of these intervals is P(x) positive?
💡 解题思路
The expressions to the power of even powers are always positive, so we don't need to care about those. We only need to care about $(x-1)^1(x-3)^3(x-5)^5(x-7)^7(x-9)^9$ . We need 0, 2, or 4 of the expr
13
第 13 题
几何·面积
What is the area of the region in the coordinate plane defined by | | x | - 1 | + | | y | - 1 | \le 1 ? (A)\ 2 (B)\ 8 (C)\ 4 (D)\ 15 (E)\ 12 ~diagram by grogg007
💡 解题思路
First consider, $|x-1|+|y-1| \le 1.$ We can see that it is a square with a side length of $\sqrt{2}$ (diagonal $2$ ). The area of the square is $\sqrt{2}^2 = 2.$
14
第 14 题
方程
How many ordered pairs of integers (m, n) satisfy the equation m^2+mn+n^2 = m^2n^2 ?
💡 解题思路
Let's use 10th grade math to solve this. After all, it is called the AMC 10 for a reason!
15
第 15 题
几何·面积
What is the least positive integer m such that m·2!·3!·4!·5!...16! is a perfect square?
💡 解题思路
We want $m\cdot2!\cdot3!\cdot4!\cdot\dots\cdot16!$ to be a perfect square. Notice that we can rewrite and pair up certain elements:
16
第 16 题
几何·角度
Define an \textit{upno} to be a positive integer of 2 or more digits where the digits are strictly increasing moving left to right. Similarly, define a \textit{downno} to be a positive integer of 2 or more digits where the digits are strictly decreasing moving left to right. For instance, the number 258 is an upno and 8620 is a downno. Let U equal the total number of upnos and let D equal the total number of downnos . What is |U-D| ?
💡 解题思路
First, we know that $D$ is greater than $U$ , since there are less $upnos$ than $downnos$ . To see why, we examine what determines an upno or $downno$ .
17
第 17 题
几何·面积
A rectangular box \mathcal{P} has distinct edge lengths a , b , and c . The sum of the lengths of all 12 edges of \mathcal{P} is 13 , the areas of all 6 faces of \mathcal{P} is \frac{11}{2} , and the volume of \mathcal{P} is \frac{1}{2} . What is the length of the longest interior diagonal connecting two vertices of \mathcal{P} ?
💡 解题思路
[asy] import geometry; pair A = (-3, 4); pair B = (-3, 5); pair C = (-1, 4); pair D = (-1, 5); pair AA = (0, 0); pair BB = (0, 1); pair CC = (2, 0); pair DD = (2, 1); draw(D--AA,dashed); draw(A--B); d
18
第 18 题
数论
Suppose a , b , and c are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true? I. If \gcd(a,14)=1 or \gcd(b,15)=1 or both, then \gcd(c,210)=1 . II. If \gcd(c,210)=1 , then \gcd(a,14)=1 or \gcd(b,15)=1 or both. III. \gcd(c,210)=1 if and only if \gcd(a,14)=\gcd(b,15)=1 .
💡 解题思路
We examine each of the conditions.
19
第 19 题
几何·面积
Sonya the frog chooses a point uniformly at random lying within the square [0, 6]×[0, 6] in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from [0, 1] and a direction uniformly at random from {north, south, east, west}. All of her choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?
💡 解题思路
WLOG, we assume Sonya jumps $0.5$ units every time, since that is her expected value.
20
第 20 题
几何·面积
Four congruent semicircles are drawn on the surface of a sphere with radius 2 , as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is π√(n) . What is n ?
💡 解题思路
Focus on 2 of the points. Let the center of the Sphere be \( A \). Label two points that form the diameter of one of the four semicircles \( M \) and \( C \) respectively.
21
第 21 题
概率
Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? Some of the solutions below are not quite correct, though they ultimately lead to the same answer choice. See the Talk page for more details .
💡 解题思路
Because each bin will have an odd number, they will have at least one ball. So we can put one ball in each bin prematurely. We then can add groups of 2 balls into each bin, meaning we now just have to
22
第 22 题
整数运算
How many distinct values of x satisfy \lfloor{x}\rfloor^2-3x+2=0 , where \lfloor{x}\rfloor denotes the largest integer less than or equal to x ?
💡 解题思路
To further grasp at this equation, we rearrange the equation into \[\lfloor{x}\rfloor^2=3x-2.\] Thus, $3x-2$ is a perfect square and nonnegative. It is now much more apparent that $x \ge 2/3,$ and tha
23
第 23 题
规律与数列
An arithmetic sequence of positive integers has n \ge 3 terms, initial term a , and common difference d > 1 . Carl wrote down all the terms in this sequence correctly except for one term, which was off by 1 . The sum of the terms he wrote was 222 . What is a + d + n ?
💡 解题思路
Since one of the terms was either $1$ more or $1$ less than it should have been, the sum should have been $222-1=221$ or $222+1=223.$
24
第 24 题
几何·面积
What is the perimeter of the boundary of the region consisting of all points which can be expressed as (2u-3w, v+4w) with 0\le u\le1 , 0\le v\le1, and 0\le w\le1 ?
💡 解题思路
[asy] import geometry; pair A = (-3, 4); pair B = (-3, 5); pair C = (-1, 4); pair D = (-1, 5); pair AA = (0, 0); pair BB = (0, 1); pair CC = (2, 0); pair DD = (2, 1); //draw(A--B--D--C--cycle); draw(A
25
第 25 题
几何·面积
A regular pentagon with area √(5)+1 is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
💡 解题思路
Since $A$ is folded onto $O$ , $AM = MO$ where $M$ is the intersection of $AO$ and the creaseline between $A$ and $O$ . Note that the inner pentagon is regular, and therefore similar to the original p