Hong Kong Maths Olympiad Heat Individual (90-91)

1.Find the value of log₃14-log₃12+log₃486-log₃7.

2.A scientist found that the popuation of a bacteria culture doubled everyhour. At four pm, he found that the number of bacteria was 3.2×10⁸. If the number of bacteria in that culture at noon on the same day was N×10⁷, find N.

3.If x+(1/x)=8, find the value of x³+(1/x³).

4.If the equations 2x+3y+a=0 and bx-2y+1=0 represent the same line, find the value of 6(a+b).

5.A boy walks from home to school at a speed of 2 metres per second and runs back at x metres per second. His average speed for the whole journey is 8/3 metres per second. Find x.

6.The straight line ax/3-(2by/5)=2a+b passes through a fixed point P. Find the x-coordinate of P.

7.If the daimeter of a sphere is increased by 20%, its volume will be increased by x%. Find x.

8.If log₇[log₅(log₃x)]=0, find x.

9.If for all real numbers x where x is not equal to 1 and 2, find A+B.

10.The marked price of an article is p% above its cost price. At a sale, the shopkeepers sells the article at 20% off the marked price. If he makes a profit of 20%, find p.

11.If a<0 and 2^2a+4-65×2^a+4=0, find a.

12.If one root of the equation (x²-11x-10)+k(x+2)=0 is zero, find the other root.

13.[x] denotes the greatest integer less than or equal to x.For example, [6]=6,[8.9]=8, etc. If

   [1^1/4]+[2^1/4]+[3^1/4]+¡K+[n^1/4]=b+2,

find n.

14.a, b are two different real numbers such that a²=6a+8 and b²=6b+8. Find the value of (4/a)²+(4/b)².

15.3¹²-1 is divisible by an integer which is greater than 70 and smaller than 80. Find the integer.

16.It is known that

2³-1³=3×1²+3×1+1,

3³-2³=3×2²+3×2+1,

4³-3³=3×3²+3×3+1,

     :  .

     :  .

101³-100³=3×100²+3×100+1.

Find the value of 1²+2²+3²+¡K¡K+100².

17.In figure1, PQ=PR=8cm and ¡çQPR=^o. A, D are the midpoints of PQ, PR respectively. If ABCD is a rectangle of area (¡Ôx)cm², find x.

Figure1

18.In figure2, XA=10cm, AB=2cm, XD=8cm and DC=x cm. Find the value of x.

Figure2

19.In figure3, AB=AC=6cm and BC=9.6cm. If the diameter of the circumcircle of triangle ABC is x cm, find x.

Figure3