MrMineev

Geometry

Number Theory

Algebra

• \(f(f(x + y)) = f(x) + f(y) - y\)
• \(f(f(x + y)) = f(x) + f(y)\) where \(f : \mathbb{Q} \rightarrow \mathbb{Q}\)
• \(f(x^5 - 4x^3 + f(y)) + f(y) = f(x^3 + xf(y))\)

• \(3b^2 \leq \sqrt[4]{\left( \frac{1}{c^2} + b^2 + c^2 \right) \left( \frac{1}{c^6} + \frac{1}{b^6} + \frac{1}{c^2} \right) \left( \frac{b^6}{c^2} + b^4 + c^4 \right)^2}\) for any \(b,c \in \mathbb{R}\)

Combinatorics

• Alice and Bob received a gift for Christmas, which was a board game where there were 2024 cities. Along with the board, a figurine knight was given. They decided to play a game, the knight starts in city 1, and then every turn, you can move the knight to another city, but every time you do so, the knight leaves a mark on the road between the two cities. The knight is not allowed to travel on previously marked roads. Here is the twist: the knight does not want to go to City 2024, meaning if a player moves the knight to City 2024, they lose. Alice moves first. Who can guarantee a victory?