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?

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?

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?

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?

Elijah has a large collection of identical wooden cubes which are white on 4 faces and gray on 2 faces that share an edge. He glues some cubes together face-to-face. The figure below shows 2 cubes being glued together, leaving 3 gray faces visible. What is the fewest number of cubes that he could glue together to ensure that no gray faces are visible, no matter how he rotates the figure?

In how many ways can $60$ be written as the sum of two or more consecutive odd positive integers that are arranged in increasing order?

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?

Charlotte the spider is walking along a web shaped like a $5$-pointed star, shown in the figure below. The web has $5$ outer points and $5$ inner points. Each time Charlotte reaches a point, she randomly chooses a neighboring point and moves to that point. Charlotte starts at one of the outer points and makes $3$ moves (re-visiting points is allowed). What is the probability she is now at one of the outer points of the star?

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.