Competition · AMC preparation · step 4 of 4

AMC 12 2023B: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 12 2023B #1 grade 5+ rate-ratio
Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses comp…
AMC 12 2023B #2 grade 6+ arithmetic
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair…
AMC 12 2023B #3 grade 7+ geometry-2d
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 rati…
AMC 12 2023B #4 grade 5+ geometry-2d
Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 m…
AMC 12 2023B #5 grade 4+ geometry-2d
You are playing a game. A 2 × 1 rectangle covers two adjacent squares (oriented either horizontally or vertically) of a…
AMC 12 2023B #6 grade 8+ algebra
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 rem…
AMC 12 2023B #7 grade 11+ algebra
For how many integers n does the expression√(log (n²) - (log n)²)/(log n - 3)represent a real number, where log denotes…
AMC 12 2023B #8 grade 11+ counting
How many nonempty subsets B of {0, 1, 2, 3, …, 12} have the property that the number of elements in B is equal to the le…
AMC 12 2023B #9 grade 8+ geometry-2d
What is the area of the region in the coordinate plane defined by | | x | - 1 | + | | y | - 1 | ≤ 1?
AMC 12 2023B #10 grade 10+ geometry-2d
In the xy-plane, a circle of radius 4 with center on the positive x-axis is tangent to the y-axis at the origin, and a c…
AMC 12 2023B #11 grade 9+ geometry-2d
What is the maximum area of an isosceles trapezoid that has legs of length 1 and one base twice as long as the other?
AMC 12 2023B #12 grade 11+ algebra
For complex number u = a+bi and v = c+di (where i=√(-1)), define the binary operation u otimes v = ac + bdi Suppose z is…
AMC 12 2023B #13 grade 8+ geometry-3d
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 areas…
AMC 12 2023B #14 grade 11+ algebra
For how many ordered pairs (a,b) of integers does the polynomial x³+ax²+bx+6 have 3 distinct integer roots?
AMC 12 2023B #15 grade 8+ number-theory
Suppose a, b, and c are positive integers such thata/14+b/15=c/210.Which of the following statements are necessarily tru…
AMC 12 2023B #16 grade 6+ number-theory
In the state of Coinland, coins have values 6,10, and 15 cents. Suppose x is the value in cents of the most expensive it…
AMC 12 2023B #17 grade 11+ geometry-2d
Triangle ABC has side lengths in arithmetic progression, and the smallest side has length 6. If the triangle has an angl…
AMC 12 2023B #18 grade 7+ algebra, logic
Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizz…
AMC 12 2023B #19 grade 8+ probability
Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each…
AMC 12 2023B #20 grade 11+ probability
Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands les…
AMC 12 2023B #21 grade 10+ geometry-3d
A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustu…
AMC 12 2023B #22 grade 11+ algebra
A real-valued function f has the property that for all real numbers a and b, f(a + b) + f(a - b) = 2f(a) f(b). Which one…
AMC 12 2023B #23 grade 11+ number-theory, counting
When n standard six-sided dice are rolled, the product of the numbers rolled can be any of 936 possible values. What is…
AMC 12 2023B #24 grade 8+ number-theory
Suppose that a, b, c and d are positive integers satisfying all of the following relations. abcd=2⁶· 3⁹· 5⁷ lcm(a,b)=2³·…
AMC 12 2023B #25 grade 8+ geometry-2d
A regular pentagon with area √5+1 is printed on paper and cut out. The five vertices of the pentagon are folded into the…

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