Problem

Source: 2023 IGMO Round 2 p6 International Gamma Mathematics Olympiad

Tags: geometry



Pepe has a large garden for frogs. He plays a game with Lintz the pig in the garden. The rules are as follows: 1. A positive real number $s$ is randomly generated by a computer system, called FROG (short form for “Fast, Random, Outstanding Generator of numbers”). 2. Lintz then draws a circle $K_1$ in Pepe’s garden (the garden is assumed to be an Euclidean plane). 3. Pepe then draws another circle $K_2$ in his garden. 4. After that, Lintz colours a finite number of arcs of $K_1$ in yellow, such that the total length of yellow arcs is more than two-third of the circumference of $K_1$. He also colours a finite number of arcs of $K_2$ in purple, such that the total length of purple arcs is more than two-third of the circumference of $K_2$. 5. Pepe puts three frogs, $A_1$, $B_1$, $C_1$ on the yellow arcs of $K_1$. He also puts three frogs, $A_2$, $B_2$, $C_2$ on the purple arcs of $K_2$. (Assume that the frogs are points, i.e. they have no areas.) Define $m_{A_1}$ , $m_{B_1}$ , $m_{C_1}$ , $h_{A_1}$ , $h_{B_1}$ , $h_{C_1}$ as the lengths of the $A$-median, $B$-median, $C$-median, $A$-altitude, $B$-altitude and $C$-altitude of $\vartriangle A_1B_1C_1$ respectively. $m_{A_2}$ , $m_{B_2}$ , $m_{C_2}$ , $h_{A_2}$ , $h_{B_2}$ , $h_{C_2}$ are defined similarly for $\vartriangle A_2B_2C_2$. Pepe wins if $$m^2_{A_2} (m^2_{B_1} + m^2_{C_1}- m^2_{A_1}) + m^2_{B_2} (m^2_{C_1} + m^2_{A_1}- m^2_{B_1}) + m^2_{C_2} (m^2_{A_1} + m^2_{B_1}- m^2_{C_1})$$$$+ \frac{1}{h^2_{A_2}}\left( \frac{1}{h^2_{B_1}}+\frac{1}{h^2_{C_1}}-\frac{1}{h^2_{A_1}}\right) + \frac{1}{h^2_{B_2}}\left( \frac{1}{h^2_{C_1}}+\frac{1}{h^2_{A_1}}-\frac{1}{h^2_{B_1}}\right) +\frac{1}{h^2_{C_2}}\left( \frac{1}{h^2_{A_1}}+\frac{1}{h^2_{B_1}}-\frac{1}{h^2_{C_1}}\right)= s$$Otherwise, Lintz wins. Find the range of $s$ where Pepe has a winning strategy and the range of $s$ where Lintz has a winning strategy. Prove your claim.

HIDE: PS I added last equation as an attachment also because it had too many indices, in case I mistyped any of them.


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