Difference between revisions of "1973 Canadian MO Problems/Problem 4"
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==Problem== | ==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: <math>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}</math>. In how many ways can these triangles be labeled with the names <math>\triangle_{1}, \triangle_{2}, \triangle_{3}, \triangle_{4}, \triangle_{5}, \triangle_{6}, \triangle_{7}</math> so that <math>P_{i}</math> is a vertex of triangle <math>\triangle_{i}</math> for <math>i = 1, 2, 3, 4, 5, 6, 7</math>? Justify your answer. | The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: <math>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}</math>. In how many ways can these triangles be labeled with the names <math>\triangle_{1}, \triangle_{2}, \triangle_{3}, \triangle_{4}, \triangle_{5}, \triangle_{6}, \triangle_{7}</math> so that <math>P_{i}</math> is a vertex of triangle <math>\triangle_{i}</math> for <math>i = 1, 2, 3, 4, 5, 6, 7</math>? Justify your answer. | ||
Revision as of 17:17, 8 October 2014
Problem
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: . In how many ways can these triangles be labeled with the names so that is a vertex of triangle for ? Justify your answer.
Solution
See also
1973 Canadian MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 | Followed by Problem 5 |