1973 Canadian MO Problems/Problem 4

Problem

$[asy] size(200); pair A=dir(120), B=dir(80); for(int k=0;k<9;++k) { pair C=dir(120-(40)*(k+2)); D(A--B); MP("P_{"+string(k)+"}",A,11,A); A=B;B=C; } for(int k=0;k<3;++k) { pair A1=dir(120-(40)*(3*k)); pair B1=dir(120-(40)*(3*k+2)); pair C1=dir(120-(40)*(3*k+3)); D(A1--B1); D(A1--C1); } [/asy]$

The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}, P_{0}P_{3}P_{6}, P_{0}P_{6}P_{7}, P_{0}P_{7}P_{8}, P_{1}P_{2}P_{3}, P_{3}P_{4}P_{6}, P_{4}P_{5}P_{6}$ . In how many ways can these triangles be labeled with the names $\triangle_{1}, \triangle_{2}, \triangle_{3}, \triangle_{4}, \triangle_{5}, \triangle_{6}, \triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$ ? Justify your answer.

1973 Canadian MO (Problems)
Preceded by Problem 3	1 • 2 • 3 • 4 • 5	Followed by Problem 5

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1973 Canadian MO Problems/Problem 4

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