Diagnostic - AMC8

Brynn's savings decreased by $20\%$ in July, then increased by $50\%$ in August. Brynn's savings are now what percent of the original amount?

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?

Mika would like to estimate how far she can ride a new model of electric bike on a fully charged battery. She completed two trips totaling 40 miles. The first trip used $\frac{1}{2}$ of the total battery power, while the second trip used $\frac{3}{10}$ of the total battery power. How many miles can this electric bike go on a fully charged battery?

A poll asked a number of people if they liked solving mathematics problems. Exactly $74\%$ answered "yes." What is the fewest possible number of people who could have been asked the question?

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?

Consider all positive four-digit integers consisting of only even digits. What fraction of these integers are divisible by $4$?

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.