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Bulgaria Contests
Bulgarian Spring Mathematical Competition
2023 Bulgarian Spring Mathematical Competition
10.3
10.3
Part of
2023 Bulgarian Spring Mathematical Competition
Problems
(1)
Operations on triangulated polygon
Source: Bulgarian Spring Tournament 2023 10.3
3/25/2023
Given is a convex octagon
A
1
A
2
…
A
8
A_1A_2 \ldots A_8
A
1
A
2
…
A
8
. Given a triangulation
T
T
T
, one can take two triangles
△
A
i
A
j
A
k
\triangle A_iA_jA_k
△
A
i
A
j
A
k
and
△
A
i
A
k
A
l
\triangle A_iA_kA_l
△
A
i
A
k
A
l
and replace them with
△
A
i
A
j
A
l
\triangle A_iA_jA_l
△
A
i
A
j
A
l
and
△
A
j
A
l
A
k
\triangle A_jA_lA_k
△
A
j
A
l
A
k
. Find the minimal number of operations
k
k
k
we have to do so that for any pair of triangulations
T
1
,
T
2
T_1, T_2
T
1
,
T
2
, we can reach
T
2
T_2
T
2
from
T
1
T_1
T
1
using at most
k
k
k
operations.
combinatorics