After the APMOPS, here are some questions that I translated:
- A magician made a six-digit number \(A\), and the digit sum of \(A\) be \(B\). The magician called out a spectactor to evaluate \(A\)-\(B\). The spectactor said out 5 numbers, namely \(0,2,4,6\) and \(8\).The magician successfully revealed the last number. What number is it? Explain your reason.
*Bonus question: Find the minimum and maximum value for \(A\).
- The eight vertexes of an octagon are attached with circles, each required to fill up the numbers from 1 to 8. Can the sum of four consecutive attaching circles be:
a)larger than 16?
b) larger than 17?
If possible, find a way of doing so; if not, explain your reason.
- Wong cycled from station A to station B. Buses form station A and B each give out a bus at the same interval time (e.g. when station A gives out a bus every 30 minutes, station B does the same), but at different times. Every 6 minutes Wong meets up with a bus coming from the opposite, and every 9 minutes he is overtaken by a bus traveling at the same direction with him. It is known that all the buses from stations A & B take 50 minutes to travel to the other side ( it means that buses from station A travel 50 minutes to station B, and vice versa ). How long does it take for Wong to travel from station A to B?
4.In the figure below, 3 different heights of the triangle move from the bases to the vertexes of the triangle. P is a point in the triangle such that another 3 lines move from point P to the triangle, causing each of the lines to be parallel with AD, BE and CF respectively. If AD=2010 cm, BE= 2013 cm, CF = 2016 cm and PR = 1005 cm, PS= 671 cm, find the length of PT. (Oh, and I forgot, each line extending to the base is straight, or 90 degrees)
Feel free to discuss! Enjoy!