IMO Shortlist 1967 problem 4
Kvaliteta:
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Avg: 0,0 (i) Solve the equation:
(ii) Supposing the solutions are in the form of arcs
with one end at the point
, the beginning of the arcs of the trigonometric circle, and
a regular polygon inscribed in the circle with one vertex in
, find:
1) The subsets of arcs having the other end in
in one of the vertices of the regular dodecagon.
2) Prove that no solution can have the end
in one of the vertices of polygon
whose number of sides is prime or having factors other than 2 or 3.
![\sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.](/media/m/d/7/4/d7405f5a82b73c386b7ef15c62b9fdb4.png)
(ii) Supposing the solutions are in the form of arcs
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
1) The subsets of arcs having the other end in
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
2) Prove that no solution can have the end
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1967