Difference between revisions of "1973 Canadian MO Problems/Problem 4"
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==Problem== | ==Problem== | ||
| − | 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. | + | <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); | ||
| + | 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); | ||
| + | } | ||
| + | for(int k=0;k<9;++k) | ||
| + | { | ||
| + | pair A=dir(120+(40)*(k)); | ||
| + | MP("P_{"+string(k)+"}",A,11,A); | ||
| + | } | ||
| + | </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. | ||
==Solution== | ==Solution== | ||
Latest revision as of 17:24, 8 October 2014
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); 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); } for(int k=0;k<9;++k) { pair A=dir(120+(40)*(k)); MP("P_{"+string(k)+"}",A,11,A); } [/asy]](http://latex.artofproblemsolving.com/8/3/4/8344776ea853dd08ddb05d7aea1fc48e2f92214a.png)
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 |
