Zimaths competition

There are two sections in this competition: normal is open to any high school student, abnormal is open to anyone, teachers included. If you have any questions to send us, please do. Regrettably, no prizes are available to readers outside Zimbabwe. We know the problems are not mindbogglingly difficult - that is done on purpose, and hope this changes soon.

Click here for the previous contest.

1. How many square numbers are there between 1 and 1 000 001, including 1 itself?
2. What is the last digit of 7⁽⁷⁷⁷⁾.
3. Weah, Ayew, Fish and Kanu each own one of the following cars (listed in ascending order of price): Mazda, Hyundai, Mercedes, Ferrari. Each play a sport as a hobby. Weah does not play golf nor squash, while Fish plays tennis. Ayew's car is more expensive than Kanu's while the owner of the Mercedes plays golf. But the owner of the Mazda does not play ping pong. If Kanu owns the car everyone can own, who owns what?

1. Find, without trial and error, the least value of n such that (n+1)^(1/3) - n^(1/3) < 1/100.
2. Prove that there is no positive integer n such that 2ⁿ+2 is divisible by 7.
3. Consider a tetrahedron lying on one of its faces. It is possible that the vertical projection of its centre of gravity lies outside the face it is lying on. Is it possible for this to happen for every face of the tetrahedron?
4. A semicircle of diameter BC has centre O, ie O is the midpoint of BC. Project CB to A such that AB=BO(=OC). If E is a point on the circumference such that EÔC=60°, what is angle EÂC to two decimal places?