1  Talent Search 2004 Round 2

Round 1 questions (see previous issue of Z Zimaths) were sent in May to all High Schools who participate in the Mathematical Olympiad, and Round 2 questions were sent in June to those who showed insight and mathematical talent. We were NOT looking for the well-drilled, who can very often not solve any problem they haven't met before, but for real creative problem-solving ability. Of the 600 scripts returned from Round 1, about 10% qualified to be sent Round 2 questions, although all were able to receive feedback and solutions through the Head of Mathematics at their school. In addition, 31 Upper 6th Formers qualified and were notified, but were not entered formally for Round 2, because they would not be eligible for future Olympiad competitions.

Participants were encouraged to give as much detail of their working as possible; they had to work on their own and at their own pace - i.e. not under strict examination conditions.

  1. The numbers in a telephone directory run from 10000 to 99999. If all the numbers containing the digit 9 are removed, how many numbers are left?

  2. Find the sum of all numbers between 0 and 300 which are multiples of 7 and 11.

  3. A committee of 7 is chosen from 10 women and 8 men. At least 2 men are to be included but there must be more women than men. In how many ways can the committee be chosen?

  4. Using as guides the asymptotes of this curve, sketch its general shape:
    y  =    (x-3)(x+2)
    (x-4)(x-1)
    		 ,

  5. Let [x] denote the greatest integer less than or equal to x. Sketch the following curves for real values of x:
    (i)   y = [x]     and     (ii)   y = x - [x].

  6. Prove that the following are all divisible by 6:
    (i)   n(n+1)(n+2),  (ii)   n(n+1)(2n+1)  and  (iii)   n(n2+5)

  7. It is claimed that the sum of all integers less than 10n which are not multiples of 2 or 5 is 20n2. Provide two proofs of the claim, one by induction on n and the other by direct deduction.

  8. Observe the pattern below and intelligently guess a general law. Provide a direct proof and an induction proof of your guess.
    1
    =
    0+1
    2+3+4
    =
    1+8
    5+6+7+8+9
    =
    8+27
    10+11+12+13+14+15+16
    =
    27+64

  9. Find all the real solutions of the following system of equations.
    x + 2y + 4z 
    =
     12
    4yz + 2zx + xy 
    =
     22
    xyz 
    =
     6

  10. Let f be a function satisfying this functional equation, for all real x, y:
    f(x+y)  =  f(x) + f(y) + xy + 1
    If f(1) = 1 find all the integers n  ¹ 1 for which f(n) = n.

2  Talent Search 2004 Round 3

Of the 30 scripts sent in response to Round 2, 15 received Round 3 questions, to be tackled under the same conditions described above. Responses are awaited. Those who come through all three rounds will receive certificates, and be further trained for possible participation in the Pan African Mathematical Olympiad. It cannot be emphasized too much that it is crucial to find and begin nurturing such talent as early as possible!

  1. Cylindrical rods of diameter 0.25 cm are bent to form circular links of a chain. If each link has an outer diameter of 2 cm, what is the length of a 100-link chain when pulled taut?

  2. If three of the angles in a pentagon are right angles and each of the other two angles is A degrees, find the value of A.

  3. The average of five consecutive integers is 10. What is the sum of the largest and the smallest of these five integers?

  4. A six-digit number is formed by repeating a three digit number, for example: 265265, or 345345. What is the largest common factor of all such numbers?

  5. Fifty-six people sign up for a singles tennis tournament in which one lost match eliminates a player. How many matches must be played in order to declare a champion?

  6. Two parallel lines intersect the x-axis, cutting off a line segment of length 3. The same lines also cut off a segment of the y-axis of length 4. What is the perpendicular distance between the lines?

  7. The inhabitants of Wonderland use two scales for measuring temperature. On the scale A, water freezes at zero degrees and boils at eighty degrees. On the scale B, water freezes at minus twenty degrees and boils at one hundred and twenty degrees. What is the equivalent on scale A of a temperature of 29 degrees on scale B?

  8. Let a,b,c and d be real numbers such that the sum of the squares of a and b is one and the sum of the squares of c and d is one. Prove that the sum of ac and bd is not greater than one.

  9. Positive integers such as 1287821 and 4554, in which the number is unchanged when the digits are reversed, are called palindromes. Determine the number of five-digit integers not less that 10 000 which are NOT palindromes.

  10. A circle of diameter 4 cm is inscribed in a triangle. The point of contact of the circle with one of the sides divides that side into segments of length 3 cm and 4 cm. Find the perimeter of the triangle.



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On 2 May 2005, 18:49.
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