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The best of the 1996 Zimbabwe Maths Olympiad

In 1984, the first Zimbabwe Maths Olympiad was held, thanks to sponsorship from the largest insurance company in Zimbabwe, Old Mutual. Old Mutual also sets the rules and organizes the contest. Mrs Erica Keogh has set questions since 1984, while Dinoj Surendran joined her in 1996.

The contest is open to Sixth Form students only. Nearly all A'level schools in the country participate. As each school is only allowed 15 entrants, just over a thousand students compete annually. There are three rounds. Round 1, a multiple choice paper, eliminates about 80% of students. Round 2, an answers-only paper, leaves around 30 students, who then write a full-working-required three-hour paper. The top three candidates receive their medals at a luncheon in one of the five-star hotels in the capital.

Here are a selection of questions from the three rounds of 1996. If you wish to see the list of those who qualified for the Final Round, click here

  1. If log2(log3 a) = 2, what is the value of a?

  2. In triangle ABD, the angular bisector of angle DAB meets BD at C. If CD=6, BD=10 and AD=9, what is the length of AB?

  3. Tendai received a $100 gift voucher to be spent at a large stationery store. He aims to purchase 20 items, using all the money. The items he is interested in are pens at \$5 each, notebooks at \$7 each and rubbers at $2 each. In how many ways can he spend his money if he buys at least one of these three items?

    [for question 4]
  4. The diagram to the right shows a circle with centre O and AB parallel to CD. find angle DOA.

  5. The number a is randomly selected from the set {0,1,2,3,...,98,99}. The number b is selected from the same set. What is the probabilty that the number 3a+7b has a units digit equal to 8?

  6. Distinct 3-digit numbers are formed using only the digits 1,2,3 and 4, with each digit used at most once in each number formed. What is the sum of all possible numbers so formed?

  7. All pupils in two schools, Zamunya and Wakiti, take an exam. The table below shows the average scores for boys and girls at each school. Also shown is the overall average for each school and the overall average for all boys at the two schools. What is the overall average for all the girls at the two schools? 
    			Zamunya		Wakiti		Both Schools
		Boys	   71		  81		     79
		Girls	   76		  90		     ?
	Boys and Girls	   74		  84


  8. Write down one factor, other than one and itself, of the number 1000100010001.

  9. A common three letter word has the following properties:
    1. LOG has exactly one letter, in the same position, in common with the word.
    2. AIR has exactly one letter, in the same position, in common with the word.
    3. LAP has exactly one letter, in a different position, in common with the word.
    4. ARK has no letter in common with the word.
    What is the word?

  10. What is the smallest number by which 10! must be multiplied to give a perfect square?

  11. A triangle has one side equal to 8cm whilst the other two sides are in the ratio 5:3. What is the largest possible area of the triangle?

    [for question 12]
  12. In an infinite pattern, a square is placed inside a square, inside a square, ... as shown, such that each square is at a constant angle theta to its predecessor. The largest, outermost square is of side 1cm. Find the sum of the areas of all the squares in the infinite pattern in terms of theta.

  13. In a chess tournament, Rodney, Ephraim, Sesedzai and innocent play three games each, with each one playing each of the others. Points are scored as 2 points for a win, 1 for a draw and 0 for a loss. The winner is the person with the highest points; a winning tie would occur if two or more people scored the same highest number of points.

    By the end of this tournament, no-one had won all of his/her games, though Sesedzai had lost all of hers. In the toughest match of the tournament, Innocent beat Ephraim and emerged as the sole winner of the tournament. If only one of the six games played was a draw, who was playing in this game?

  14. A plonky polygon is a regular polygon whose interior angles have integral values. What is the largest possible number of sides for a plonky polygon?

    [for question 15]
  15. Shown is a semicircle with diameter AB = 2r and centre O. An arc OM with centre A is drawn. Find in terms of r the circumference of the circle inscribed in triangle OMB.

  16. Suppose I randomly choose two numbers in the range 0 to 1. What is the probability that the sum of these two numbers, when rounded to the nearest integer, is 1?

  17. Solve the equation 2x4 + 7x3 - 44x2 - 49x + 210 = 0

  18. The Belch function B(x) acts on any non-negative integer 0,1,2,3,... and produces a single digit 0,1,2,...,9. It does this by summing the digits of x to produce a new number, say x1. If x1 has more than one digit, these are summed to give a new number x2, and so on, until a single digit is obtained. For example,
    B(12) = 1 + 2 = 3,
    B(19) = B(1+9) = B(10) = 1 + 0 = 1
    B(1996) = B(1+9+9+6) = B(24) = 6

    1. Find all values of x such that B(x)=0.
    2. Find all values of x such that x is prime and B(x) = 3.
    3. Prove that B(xy) = B(B(x)B(y)) for any x,y.


  19. An octagon with four consecutive sides of length two and the remaining sides of length three, is inscribed in a circle. Find the area of the octagon.

  20. A rare fungus has the property that at the end of the first phase of its existence, it either dies, splits into two similar fungi, or splits into three similar fungi, with probabilities 3/7, 3/7, 1/7 respectively. Let K > 0 be the probability that a colony of fungus that lives forever will develop from one such fungus. Find K.

The questions are certainly not IMO standard! But Zimbabwean kids are simply not mathematically mature enough to handle harder problems. We hope that Zimaths will help fix this by steadily introducing new problem solving techniques and viewponts. For the school syllabus is getting weaker and weaker --- it is now possible for a student to get to university without ever having met the notion of proof (even induction)!

There is talk of Lower Sixes being permitted to compete in the ZMO in future. and an O'level Olympiad is being planned for 1999.

Yet a lot more can and should be done. But funds are hard to come by, and really enthusiastic maths boffins even harder. There is hope, though, since neighbouring South Africa has a well established youth maths program which they have offered to extend to Zimbabwe. 

Erica Keogh			Dinoj Surendran
UZ Statistics Department	Kutama College
P.O.Box MP167			P. Bag 909
Mt Pleasant			Norton
Harare				Zimbabwe
Zimbabwe

keogh@stats.uz.zw 		951832P@maths.uz.zw




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