Math Problems

Most of these problems will require proofs as answers, so you should know how to write a proof. Many of the solutions are direct proofs, so a good understanding of geometry and algebra (at least it's core aspects) is essential. Additionally, the following branches of math should be very useful in specific cases:

• Calculus
• Linear Algebra
• Number Theory / Discrete Math
• Functional Equations

While I encourage that you limit your resources, you shouldn't restrict them entirely. I highly encourage the use of graphing utilities (such as Desmos) for all problems.

The purpose of these questions is to challenge your classical education and test your ability to determine what tools you need/have. For example, if you think you will need vector equations but have never worked with them, look it up! Do your research for each problem, just don't look up the problem directly.

Another example, let's say your solution requires you to solve a complicated differential equation. I encourage you to first analyze the equation without tools and ask questions such as "Is this solvable?" and "Does this represent my problem?". After you have analyzed all you can, don't restrict yourself to doing complicated algebra from memory. Use a calculator or MATLAB to solve your equation; as long as you know what it means and what the calculator is doing.

I want you to feel like a mathematician, but it's 2025 after all.