Here's the mock: clickie. Have fun! Spoilers below.

To be honest, this was really fun to make and see how some people did decently well and some people thought it was really hard.

Problem 1

This problem asks you to show that there is a number \(x\) such that \(x^2 \equiv -7 \pmod{2^n}\) for all \(n\) (those who know will call this a quadratic residue). This is a similar to Hensel-proof induction argument, easy to come up with even if you have not seen it before (add \(2^{n-1}\) or stay the same).

Problem 2

This was originally from the MEMO (I believe), the problem did not give the value \(6n^2\), and was originally a 2-part problem. I tried to make it easier by including this value, asking one to only show the bound. Still it is quite a nontrivial problem.

Problem 3

Trigbashable geometry, as is (or used to be?) usual for the RMO.

Problem 4

Inequality. Yeah. I don't know about this one.

Problem 5

Very interesting and nice problem from the Canada MO. Many intuitively think of this as an \(n\)-dimensional space.

Problem 6

Objectively the easiest problem on the mock.