Diagnostic - AMC8

What is the value of the following expression? $1+2-3+4+5-6+7+8-9+10+11-12$

Casey went on a road trip that covered $100$ miles, stopping only for a lunch break along the way. The trip took $3$ hours in total and her average speed while driving was $40$ miles per hour. In minutes, how long was the lunch break?

Peter lives near a rectangular field that is filled with blackberry bushes. The field is 10 meters long and 8 meters wide, and Peter can reach any blackberries that are within 1 meter of an edge of the field. The portion of the field he can reach is shaded in the figure below. What fraction of the area of the field can Peter reach?

What is the value of this expression? \[\frac{\sqrt{16\sqrt{81}}}{\sqrt{81\sqrt{16}}}\]

Squares of side length $1, 1, 2, 3,$ and $5$ are arranged to form the rectangle shown below. A curve is drawn by inscribing a quarter circle in each square and joining the quarter circles in order, from shortest to longest. What is the length of the curve?

The figure below shows a tiling of $1 \times 1$ unit squares. Each row of unit squares is shifted horizontally by half a unit relative to the row above it. A shaded square is drawn on top of the tiling. Each vertex of the shaded square is a vertex of one of the unit squares. In square units, what is the area of the shaded square?

Miguel is walking with his dog, Luna. When they reach the entrance to a park, Miguel throws a ball straight ahead and continues to walk at a steady pace. Luna sprints toward the ball, which stops by a tree. As soon as the dog reaches the ball, she brings it back to Miguel. Luna runs 5 times faster than Miguel walks. What fraction of the distance between the entrance and the tree does Miguel cover by the time Luna brings him the ball?

The land of Catania uses gold coins and silver coins. Gold coins are $1$ mm think and silver coins are $3$ mm thick. In how many ways can Taylor make a stack of coins that is $8$ mm tall using any arrangement of gold and silver coins, assuming order matters?

The notation $n!$ (read "n factorial") is defined as the product of the first $n$ positive integers. (For example, $3!=1 \cdot 2 \cdot 3 = 6$). Define the superfactorial of a positive integer, denoted by $n^!$, to be the product of the factorials of the first $n$ integers. (For example, $3^!=1! \cdot 2! \cdot 3! = 12$). How many factors of $7$ appear in the prime factorization of $51^!$, the superfactorial of $51$?

In an equiangular hexagon, all interior angles measure 120°. An example of such a hexagon with side lengths 2, 3, 1, 3, 2, and 2 is shown below, inscribed in equilateral triangle ABC. Consider all equiangular hexagons with positive integer side lengths that can be inscribed in triangle ABC, with all six vertices on the sides of the triangle. What is the total number of such hexagons? Hexagons that differ only by a rotation or a reflection are considered the same.