2023 AMC 10 B

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? （2023 AMC10B Q1）

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? （2023 AMC10B Q2）

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? （2023 AMC10B Q3）

Jackson's paintbrush makes a narrow strip with a width of 6.5 milimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint? （2023 AMC10B Q4）

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? （2023 AMC10B Q5）

, and for How many terms in the sequence , , , , are even? (2023 AMC10B Q6)

Square ABCD is rotated 20° clockwise about its center to obtain square EFGH, as shown below. What is the degree measure of ？(2023 AMC10B Q7)​​​

What is the units digit of +? (2023 AMC10B Q8)

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? (2023 AMC10B Q9)

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? (2023 AMC10B Q10)

Suzanne went to the bank and withdrew . The teller gave her this amount using bills, bills, and bills, with at least one of each denomination. How many different collections of bills could Suzanne have received? (2023 AMC10B Q11)

When the roots of the polynomial

are removed from the real number line, what remains is the union of 11 disjoint open intervals. On how many of those intervals is P(x) positive? (2023 AMC10B Q12)

What is the area of the region in the coordinate plane defined by the inequality

(2023 AMC10B Q13)

How many ordered pairs of integers (m, n) satisfy the equation (2023 AMC10B Q14)

What is the least positive integer such that is a perfect square? (2023 AMC10B Q15)

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 and 8620 is a . Let U equal the total number of and let D equal the total number of . What is|U - D|? (2023 AMC10B Q16)

A rectangular box P has distinct edge lengths a, b, and c. The sum of the lengths of all 12 edge 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? (2023 AMC10B Q17)

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,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. (2023 AMC10B Q18)

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? (2023 AMC10B Q19)

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 curcve is . What is n? (2023 AMC10B Q20)

Each of 2023 balls is placed in on of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? (2023 AMC10B Q21)

How many distinct values of x satisfy where [x] denotes the largest integer less than or equal to x? (2023 AMC10B Q22)

An arithmetic sequence has n≥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 was 222. What was (2023 AMC10B Q23)

What is the perimeter of the boundary of the region consisting of all points which can be expressed as with , , and (2023 AMC10B Q24)

A regular pentagon with area is printed on paper and cut out. The five vertices of pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? (2023 AMC10B Q25)