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Difference between revisions of "2023 AMC 10B Problems"

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==Problem 23==
 
==Problem 23==
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An arithmetic sequence of positive integers has <math>n\ge3</math> terms, initial term <math>a</math>, and common difference <math>d>1</math>. Carl wrote down all the terms in this sequence correctly except for one term, which was off by <math>1</math>. The sum of the terms he wrote down was <math>222</math>. What is <math>a+d+n</math>?
  
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==Solution==
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There are <math>n</math> terms, the <math>x</math>th term is <math>a+(x-1)d</math>, summation is <math>an+dn(n-1)/2=n(a+\frac{d(n-1)}{2})</math>.
  
 +
 +
The summation of the set is <math>222 \pm 1 = 221,223</math>. First, <math>221</math>: its only possible factors are <math>1,13,17,221</math>, and as said by the problem, <math>n\ge3</math>, so <math>n</math> must be <math>13,17,</math>or<math>223</math>. Let's start with <math>n=13</math>. Then, <math>a+6d=17</math>, and this means <math>a=5</math>,<math>d=2</math>. Summing gives <math>13+5+2=20</math>. We don't need to test any more cases, since the problem writes that all <math>a+d+n</math> are the same, so we don't need to test other cases.
 
[[2023 AMC 10B Problems/Problem 23|Solution]]
 
[[2023 AMC 10B Problems/Problem 23|Solution]]
  

Revision as of 15:50, 15 November 2023

2023 AMC 10B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

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?

$\textbf{(A) }\frac{1}{12}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{1}{8}\qquad\textbf{(E) }\frac{2}{9}$

Solution

Problem 2

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?

$\textbf{(A) }$46\qquad\textbf{(B) }$47\qquad\textbf{(C) }$48\qquad\textbf{(D) }$49\qquad\textbf{(E) }$50$

Solution

Problem 3

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 $B$?

$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{25}{169}\qquad\textbf{(C) }\frac{4}{25}\qquad\textbf{(D) }\frac{1}{5}\qquad\textbf{(E) }\frac{9}{25}$

Solution

Problem 4

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?

$\textbf{(A) }162.5\qquad\textbf{(B) }1,625\qquad\textbf{(C) }16,250\qquad\textbf{(D) }162,500\qquad\textbf{(E) }1,625,000$

Solution

Problem 5

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?

$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Solution

Problem 6

Let L₁ = 1, L₂ = 3, and Lₙ₊₂ = Lₙ₊₁ +Lₙ for n≥1. How many terms in the sequence L₁, L₂, L₃,…, L₂₀₂₃ are even?

Solution

Problem 7

Square ABCD is rotated 20 degrees clockwise about its center to obtain square EFGH, as shown below. What is the degree measure of <EAB?

$\textbf{(A) }24\qquad\textbf{(B) }35\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }20$

Solution

Problem 8

What is the units digit of $2022^{2023} + 2023^{2022}$?

$\text{(A)}\ 7 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 3$

Solution

Problem 9

The numbers 16 and 25 are a pair of consecutive postive squares whose difference is 9. How many pairs of consecutive positive perfect squares have a difference of less than or equal to 2023?

$\text{(A)}\ 674 \qquad \text{(B)}\ 1011 \qquad \text{(C)}\ 1010 \qquad \text{(D)}\ 2019 \qquad \text{(E)}\ 2017$

Solution

Problem 10

You are playing a game. A 2 x 1 rectangle covers two adjacent squares oriented either horizontally or vertically) of a 3 x 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?

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Define an $upno$ to be a positive integer of 2 or more digits where the digits are strictly increasing moving left to right. Similarly, define a $downno$ 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|?

Solution

Problem 17

A rectangular box 𝒫 has distinct edge lengths 𝑎, 𝑏, and 𝑐. The sum of the lengths of all 12 edges of 𝒫 is 13, the sum of the areas of all 6 faces of 𝒫 is $\dfrac{11}{2}$, and the volume of 𝒫 is $\dfrac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of 𝒫 ?

Solution

Problem 18

Suppose 𝑎, 𝑏, and 𝑐 are positive integers such that $\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}$.

Which of the following statements are necessarily true?

I. If gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both, then gcd(𝑐, 21) = 1.

II. If gcd(𝑐, 21) = 1, then gcd(𝑎, 14) = 1 or gcd(𝑏, 15) = 1 or both.

III. gcd(𝑐, 21) = 1 if and only if gcd(𝑎, 14) = gcd(𝑏, 15) = 1.

Solution

Problem 19

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 he choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?

Solution

Problem 20

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 $\pi\sqrt{n}$. What is 𝑛?

Solution

Problem 21

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?

Solution

Problem 22

Solution

Problem 23

An arithmetic sequence of positive integers has $n\ge3$ 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 down was $222$. What is $a+d+n$?

Solution

There are $n$ terms, the $x$th term is $a+(x-1)d$, summation is $an+dn(n-1)/2=n(a+\frac{d(n-1)}{2})$.


The summation of the set is $222 \pm 1 = 221,223$. First, $221$: its only possible factors are $1,13,17,221$, and as said by the problem, $n\ge3$, so $n$ must be $13,17,$or$223$. Let's start with $n=13$. Then, $a+6d=17$, and this means $a=5$,$d=2$. Summing gives $13+5+2=20$. We don't need to test any more cases, since the problem writes that all $a+d+n$ are the same, so we don't need to test other cases. Solution

Problem 24

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 $0\le w\le1$?

Solution

Problem 25

A regular pentagon with area $1+\sqrt5$ is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

Solution

See also

2023 AMC 10B (Problems • Answer Key • Resources)
Preceded by
2022 AMC 10A Problems 		Followed by
2024 AMC 10A Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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