Diagnostic - AMC8

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

Haruki has a piece of wire that is $24$ centimeters long. He wants to bend it to form each of the following shapes, one at a time: A regular hexagon with side length $5$ cm. A square with area $36 \hspace{3pt} \text{cm}^2$. A right triangle whose legs are $6$ and $8$ cm long. Which of the shapes can Haruki make?

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

Five runners completed the grueling Xmarathon: Luke, Melina, Nico, Olympia, and Pedro. Nico finished $11$ minutes behind Pedro. Olympia finished $2$ minutes ahead of Melina, but $3$ minutes behind Pedro. Olympia finished $6$ minutes ahead of Luke. Which runner finished fourth?

In the figure below, each circle will be filled with a digit from 1 to 6. Each digit must appear exactly once. The sum of the digits in neighboring circles is shown in the box between them. What digit must be placed in the top circle?

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?

Four students are seated in a row. They chat with the people sitting next to them, then rearrange themselves so that they are no longer seated next to any of the same people. How many rearrangements are possible?

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?

Lakshmi has $5$ round coins of diameter $4$ centimeters. She arranges the coins in $2$ rows on a table top, as shown below, and wraps an elastic band tightly around them. In centimeters, what will be the length of the band?

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 $51$?