Hong Kong Maths Olympiad Heat Individual (91-92)

1.If (log₁₀x)⁴-3(log₁₀x)²-4=0 and x=1, find x.

2.

3.An integer a lying between 0 and 9 inclusive is randomly selected. It is known that the probability that the equation x²-ax+3=0 has no real root is p/10, find p.

4,x^o is an acute angle satisfying

   0.5cosx^o¡Ù0.5(5-cosx^o)-2.

Determine the largest possible value of x.

5.Let f(x) be the hightest common factor of x⁴+64 and x³+6x²+16x+16, find f(2).

6.A fruit merchant divides a large lot of oranges into four classes : A, B, C, D. The number of oranges in class A and class B doubles that in class C while the number of oranges in class B and class D doubles that in class A. If 7 oranges from class B are upgraded to class A, class A will then contain twice as many oranges as class B. It is known that one of the four classes contain 54 oranges. Determine which one class it belongs to.

7.Given that n is a positive integer, find ALL the real roots of x^{2^n}-10^{2^n}=0.

8.If n is an integer randomly selected from 1 to 100, and the probability that the unit digit of 5678ⁿ is greater than 3 is 3/x, find x.

9.In£GABC, AB=8cm, BC=6cm, ¡çABC=90^o. If the bisector of ¡çACB cuts AB at R and CR=(3¡Ôa)cm, find a.

10.In figure 1, arc BD is 4 times the arc AC, ¡çDEB=80^o and ¡çADC=x^o, find x.

Figure1

11.In figure2, ABCD is a square. EDF is a straight line. M is the mid-point of AB. If the distances of A, M and C from the line Ef are 5cm, 11cm and xcm respectively, find x.

Figure2

12.In figure3, AB=AC=2BC and BC=20cm. If BF is perpendicular to AC and AF=xcm, find x.

Figure3

13.Figure 4 shows a figure obtained by producing the sides of a 13-sided polygon. If the sum of the marked angles is n^o, find n.

Figure4

14.In figure 5, PQ is a diagonal of the cube. If PQ is equal to 4 cm and the total surface area of the cube is xcm², find x.

Figure5

15.If (3x-1)⁷=a₁x⁷+a₂x⁶+a₃x⁵+¡K+a₈, find the value of a₁+a₂+a₃+¡K+a₈.

16.A(1,1), B(a,0), C(1,a) are the vertices of the triangle ABC. Find the value of a if the area of triangle ABC is 2 units and a>0.

17.If N=2¹²×5⁸, find the number of digits of N.

18.If a : b=3 : 4 and a : c= 2 : 5, find the value of ac/(a²+b²).

19.A rectangular piece of paper of width 6cm is folded such that one corner touches the opposite side as shown in figure 6. If £c=30^o and DE=xcam, find x.

Figure6

20.If sinx+cosx=(1/5) and 0¡Øx¡Ø£k,find tan x.