(Please send in your solutions to different questions on different sheets of paper, with name and address on each.)

Normal Section

1.   Find, without direct summation, the value of

1-2+3-4+5-6+¼-998+999-1000+1001

2.   At the beginning of a game of cards, the Professor, Farai and Temba, had money in the ratio  11:8:5, and, at the end of the game, the same amount of money was distributed in the ratio  4:3:2. Who won?

3.   ``Boss, I have a complaint'', said the young investment counsellor. ``Farai and I were hired at the same time, and both of us have handled about the same number of assignments. They've all been worth about the same, and each required just a YES-or-NO decision. I've been right about 70% of the time, and Farai hasn't made the right recommendation more than 10% of the time. I know that you are as aware of this as I am, and yet you've given him a promotion and a raise while turning me down. How come?''
Could the boss have a valid reason for his action?

Abnormal Section

1.   Find the remainder, when 1³+2³+3³+4³+¼+100³ is divided by 7. (Do not use any known formula for the sum of cubes.)

2.   The Professor suspects that there is no ten-digit number in which all the digits are different and which is also a perfect tenth power. He asks a graduate student to check this conjecture, whereupon the student programmes a computer to test every ten-digit number and say whether or not it is a tenth power. What would be the best method of testing the conjecture, using the least amount of calculation?

3.   The manufacturers of Mazoe orange drink advertise their product as containing z % of pure orange juice. The quality control section tests a mixture of 100 kilolitres and finds that it contains w % of pure orange juice. By adding x kilolitres of a mixture containing y % of pure orange juice, they wish to produce a mixture containing z % of pure orange juice. Find x in terms of w, y, z.