MathDB

For triangle

\vartriangle ABC

, define its

A

-excircle to be the circle that is externally tangent to line segment

BC

and extensions of

\overleftrightarrow{AB}

and

\overleftrightarrow{AC}

, and define the

B

-excircle and

C

-excircle likewise. Then, define the

A

-veryexcircle to be the unique circle externally tangent to both the

A

-excircle as well as the extensions of

\overleftrightarrow{AB}

and

\overleftrightarrow{AC}

, but that shares no points with line

\overleftrightarrow{BC}

, and define the

B

-veryexcircle and

C

-veryexcircle likewise. Compute the smallest integer

N \ge 337

such that for all

N_1 \ge N

, the area of a triangle with lengths

3N^2_1

3N^2_1 + 1

, and

2022N_1

is at most

\frac{1}{22022}

times the area of the triangle formed by connecting the centers of its three veryexcircles. If your submitted estimate is a positive number

E

and the true value is

A

, then your score is given by

\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^3\right\rfloor \right)

25