2021 National MathCounts Sprint #21 (video)

Four lines are drawn through the figure shown (see problem statement in video). What is the maximum number of non-overlapping regions created inside the figure?

2021 National MathCounts Team Round #9 (video solution)

Gretchen labels each of the six faces of a cube with a distinct positive integer so that for each vertex of the cube, the product of the three numbers on the faces touching the vertex is a perfect square. What is the least possible value of the sum of the numbers on this cube?

2021 MathCounts National Target Round #7 (video solution)

This one uses a rarely used formula for the area of a triangle.

What is the largest possible perimeter of a triangle whose sides have integer side lengths and that can fit inside a circle of radius 20 cm?

Fall 2021 AMC 10 B #24 video solution

Here I introduce 2 time-saving tips and simplified notation for tracking the sides of a cube after rotation.

A cube is constructed from white unit cubes and blue unit cubes. How many different ways are there to construct the cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

Fall 2021 AMC 10 B #20 video solution

In a particular game, each of players rolls a standard -sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo’s first roll was a , given that he won the game?

Fall 2021 AMC 10 A #20 video solution

I like how this problem nicely combines properties of quadratic equations with solving inequalities.

How many ordered pairs of positive integers exist where both and do not have distinct, real solutions?

2013 MathCounts Chapter Target #3

This problem is a good example of finding a bijection or a 1:1 correspondence between a set that is difficult to count and another that is easier to count, in this case using binary numbers.

A circular spinner has seven sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner by colored?

2020 AIME II #8 Video Solution

Define a sequence recursively by and for integers . Find the least value of such that the sum of the zeros of exceeds .

2020 AIME II #9 Video Solution

While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.

2020 AIME II #12 video solution

I could not find a great solution to the 2020 AIME II #12, so I created a video solution myself. It is a nice geometric problem with lots of inequalities.