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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

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DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Hike and Hitch

Age 14 to 16 Challenge Level:

Fifteen students from Dr Challoner's Grammar School were told to go to a point 60 miles south. The only transport they could use was a car, which could only carry 5 students beside the driver (Thasan). Thasan took the students to their destination in 3 loads. As he left with the first 5, he ordered the remaining 10 to commence hiking along the road. He unloaded the first group some distance from the goal, with orders to hike the rest of the way. Then he returned until he met the remaining ten, picked up 5 of them, and took them part of the way along the route while the last 5 continued to walk. Finally he returned, picked up the last group and drove it the rest of the way. They all arrived at the same time. The students walked at a rate of 4 miles per hour without any pause for rest, while the car travelled at a speed of 40 miles per hour. How much time did the hiking save? (You can ignore the time spent in picking up and discharging passengers.)