2023 AMC 12A
2023 AMC 10A
GT Math Comp
100
Cities 
and 
are 
miles apart. Alicia lives in and Beth lives in . Alicia bikes towards at 18 miles per hour. Leaving at the same time, Beth bikes toward at 12 miles per hour.
27 miles
How many miles from City will they be when they meet?
100
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. The 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
What is C? How much is 1/6?
100
The maximum number of bishops that can be placed on a 4 × 4 chess board without any pair attacking each other. Note: bishops attack each other if they lie on the same diagonal.
How many is 6?
200
The weight of of a large pizza together with 
cups of orange slices is the same as the weight of 
of a large pizza together with 
cup of orange slices. A cup of orange slices weighs 
of a pound.
The weight, in pounds, of a large pizza:
How much is 9/5 (or 1 4/5) pounds or what is A?
200
You are playing a game. A 
rectangle covers two adjacent squares (oriented either horizontally or vertically) of a 
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. The minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle
How many is 4? What is C?
200
The polynomial f(x) = x 3 − ax − b has a root at x = −1 and a root at x = 2. What is a + b?
How much is 5?
300
Many positive perfect squares less than 
are divisible by 
How many is 8? What is A?
300
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. The original (before discount) price of the most expensive shoes he could afford to buy
What is B? How much is $50?
300
Let m be a real number. Define set A = {1, 2, m}, set B = {a 2 : a ∈ A}, and set C = A ∪ B. Assume the sum of all elements in C is 6. What is the product of all elements in C?
Options: −8. (Options: No such m exists, −16, −64, −4.
What is -8?
400
A square of area 
is inscribed in a square of area 
, creating four congruent triangles, as shown below. 
The ratio of the shorter leg to the longer leg in the shaded right triangle
What is C? What is 2 - sqrt(3)?
400
In the 
-plane, a circle of radius 
with center on the positive 
-axis is tangent to the 
-axis at the origin, and a circle with radius 
with center on the positive -axis is tangent to the -axis at the origin. The slope of the line passing through the two points at which these circles intersect.
How steep is 2/5? What is E?
400
The number of three-digit positive integers x there are such that 5x =y^3 for some positive integer y
How many is 2?
If 5x =^3 , then 5 divides y ^3 . Since 5 is prime, it must divide y, so y = 5k for some k ∈ N. Then x = 25k^3 . Now, x is divisible by 25, so x = 25m for some m ∈ N. Next, from 100 ≤ x ≤ 999 we get 4 ≤ m ≤ 39. There are only two cubes in this interval: 8 and 27. So, there are two possible values of x. They are 25 · 8 = 200 and 25 · 27 = 675.