## Wednesday, October 10, 2012

### Cool math Olympics - II

Cool math Olympics - 1960

1. Determine all three-digit numbers N having the property that N is divisible by 11. and N / 11 is equal to the sum of the squares of the digits of N.
2. For what values of the variable x does the following inequality hold: (4x^{2})/(1 -\sqrt{1 + 2x})^{2} < 2x + 9.
3. In a given right triangle ABC the hypotenuse BC of length a is divided into n equal parts (n an odd integer). Let \alpha be the acute angle subtending, from A that segment which contains the midpoint of the hypotenuse. Let h be the length of the altitude to the hypotenuse of the triangle.
Prove: tan(\alpha) = (4nh) / ((n^{2} - 1) * a).
4. Construct triangle ABC given h_{a}, h_{b} (the altitudes from A and B) and m_{a}, the median from vertex A.
5. Consider the cube ABCDA^{1}B^{1}C^{1}D^{1} (with face ABCD directly above face A^{1}B^{1}C^{1}D^{1}).
(a) Find the locus of the midpoints of segments XY where X is any point of AC and Y is any point of B^{1}D^{1}.
(b) Find the locus of points Z which lie on the segments XY of part (a) with ZY = 2XZ.
6. Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let V1 be the volume of the cone and V2 the volume of the cylinder.
(a) Prove that V1 != V2.
(b) Find the smallest number k for which V1 = kV2, for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.
7. An isosceles trapezoid with bases a and c and altitude h is given.
(a) On the axis of symmetry of this trapezoid, find all points P such that both legs of the trapezoid subtend right angles at P:
(b) Calculate the distance of P from either base.
(c) Determine under what conditions such points P actually exist. (Discuss various cases that might arise.)