6th Vietnamese Mathematical Olympiad 1967 Problems

6th Vietnamese Mathematical Olympiad 1967 Problems

1.  Draw the graph of the function y = | x³ - x² - 2x | /3 - | x + 1 |.
2.  A river flows at speed u. A boat has speed v relative to the water. If its velocity is at an angle α relative the direction of the river, what is its speed relative to the river bank? What α minimises the time taken to cross the river?

3.  (1) ABCD is a rhombus. A tangent to the inscribed circle meets AB, DA, BC, CD at M, N, P, Q respectively. Find a relationship between BM and DN.
(2) ABCD is a rhombus and P a point inside. The circles through P with centers A, B, C, D meet the four sides AB, BC, CD, DA in eight points. Find a property of the resulting octagon. Use it to construct an equiangular octagon.
(3) Rotate the figure about the line AC to form a solid. State a similar result.