2023 AMC 10B Problems
| 2023 AMC 10B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
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Instructions
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
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 
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?
Problem 2
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 
on every pair of shoes. Carlos also knew that he had to pay a 
sales tax on the discounted price. He had 
dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
Problem 3
A 
right triangle is inscribed in circle 
, and a 
right triangle is inscribed in circle 
. What is the ratio of the area of circle to the area of circle ?
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?
Problem 5
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds 
to each number in the list and finds that the sum of her new numbers is 
. Lara multiplies each number in the list by and finds that the sum of her new numbers is also . How many numbers are written on the blackboard?
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?
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?
Problem 8
What is the units digit of 
?
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?
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?
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Define an 
to be a positive integer of 2 or more digits where the digits are strictly
increasing moving left to right. Similarly, define a 
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 
and let 𝑑 equal the total number of 
. What is |𝑈 − 𝐷|?
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 
, and the volume of 𝒫 is 
. What is the length of the longest interior diagonal connecting two vertices of 𝒫 ?
Problem 18
Suppose 𝑎, 𝑏, and 𝑐 are positive integers such that
.
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.
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?
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 
. What is 𝑛?
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?
Problem 22
Problem 23
An arithmetic sequence of positive integers has 
terms, initial term 
, and common difference 
. Carl wrote down all the terms in this sequence correctly except for one term, which was off by 
. The sum of the terms he wrote down was 
. What is 
?
Problem 24
What is the perimeter of the boundary of the region consisting of all points which can be expressed as 
with 
, 
and 
?
Problem 25
A regular pentagon with area 
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?
See also
| 2023 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by 2022 AMC 10A Problems |
Followed by 2024 AMC 10A Problems | |
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| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
