1. In the following figure, ΔABC is a triangle. ΔABF, ΔBCE and ΔCAD are equilateral triangles. AE, BD and CF are concurrent and they meet at G. Proof that AE = BD = CF.
2. If x + y +z = 100, where x, y and z are positive integers. Find the total number of solution sets for (x, y, z).
4. If x and y are positive integers, find all the solutions for (x + y) 2 = 10x + y.
5. Find 1 + 2 (1/3) + 3 (1/3) 2 + 4 (1/3) 3 + 5 (1/3) 4 + ….
6. Find the remainder when 11111111111111…111 (1000 digits in total) is divided by 12.
7. The following diagram shows a circuit formed by 7 identical resistors with resistant R. Find the equivalent resistance between A and B in terms of R.
8. At a certain instant, car B is at a point 10km from car A in the direction N60oW. Car A is moving at a speed of 15 km/h towards east. Car B is moving at a speed of 5 km/h. In order to obtain a shortest possible distance between the two cars, at what direction should car B move? What is the shortest possible distance between the two cars?
9. A 62 year old gentleman presented with malaise. CBC shows WBC 461.3, Hb 8.4, platelet 51. Peripheral blood smear as follow:
Flow cytometry shows that the abnormal cells are CD3, CD4, CD5, CD6, CD7 and CD8 positive; and are CD25, TdT negative. What is the diagnosis?
10. A 1 year old Chinese boy showed the following haemoglobin pattern:
What is the diagnosis?