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1 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Individual Events I a I a 6 I a 4 I4 a 8 I5 a 0 b b 60 b 4 b 9 b c c 00 c 50 c 4 c 57 d d 50 d 500 d 54 d 7 Group Events G6 p G7 a 6 G8 m G9 a 9 G0 a 50 m 8 b 8 d b b 0 r c n 96 x c 5 s d 6 s y 0 d 60 Individual Event I. Given that 7 x = 6 and 7 x = 6 a, find the value of a. 7 x = 6 7 x = 6 a = 6 a = I. Find the value of b if log {log [log (b) + a] + a} = a. log {log [log (b) + ] + } = log [log (b) + ] + = = 4 log [log (b) + ] = log (b) + = = 4 log (b) = b = = 4 b = I. If c is the total number of positive roots of the equation (x b) (x ) (x + ) = (x b) (x + ), find the value of c. (x ) (x ) (x + ) (x ) (x + ) = 0 (x ) (x + ) [(x ) ] = 0 (x ) (x + ) (x 5) = 0 x =, or 5 Number of positive roots = c = I.4 If c d, find the value of d. Reference: 999 HG, 00 FG., 0 HI7, 05 FI4., 05 FG. = d = = = = Page

2 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Individual Event I. If sin = 5 4, find a, the area of the quadrilateral. 4 Let the height be h. tan = 4 = 6 h h = 8 Area = 4 08 = 6 I. If b = 6 a, find b. b = 6 a = (6 6)(6 + 6) = 60 I. Dividing $(000 + b) in a ratio 5 : 6 : 8, the smallest part is $c. Find c. Sum of money = $(000 60) = $ c = 80 = 80 = I.4 In the figure, AP bisects BA. Given that AB c, BP d, P 75 and A 50, find d. Let BAP = = AP, AP =, BP = 80 d () and () sin sin80 sin sin () () d = 50 d c 0 P Page

3 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Individual Event I. If a is the remainder when is divided by, find a = = = a = 4 I. Let P(x, b) be a point on the straight line x + y = 0 such that slope of OP a (O is the origin). Determine b. (Reference: 994 FI.4) x + b = 0 x = 0 b b m OP = = 4 0 b b = 0 4b b = 4 I. Two cyclists, initially (b + 6) km apart travelling towards each other with speeds 40 km/h and 60 km/h respectively. A fly flies back and forth between their noses at 00 km/h. If the fly flied c km before crushed between the cyclists, find c. The velocity of one cyclist relative to the other cyclist is ( ) km/h = 00 km/h. Distance between the two cyclists = (4 + 6) km = 50 km 50 Time for the two cyclists meet = h = h 00 The distance the fly flied = 00 km = 50 km c = 50 I.4 In the figure, APK and BPH are straight lines. A If d area of triangle HPK, find d. BAP = KHP = 0 (given) 0 75 APB = KPH (vert. opp. s) 60 H c ABP ~ HKP (equiangular) 0 HK P 75 B HK = 40 K d = 50 40sin0 = 500 Individual Event 4 I4. Given that the means of x and y, y and z, z and x are respectively 5, 9, 0. If a is the mean of x, y, z, find the value of a. x y y z z x 5 (); 9 (); 0 () () + () + (): x + y + z = 4 a = 8 I4. The ratio of two numbers is 5 : a. If is added to each of them, the ratio becomes : 4. If b is the difference of the original numbers and b > 0, find the value of b. Let the two numbers be 5k, 8k. 5k 8k 4 0k + 48 = 4k + 6 4k = k = 5k = 5, 8k = 4 b = 4 5 = 9 Page

4 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 I4. PQRS is a rectangle. If c is the radius of the smaller circle, find the value of c. S Let the centres of the two circles be and D, with radius 9 and c respectively. Suppose the circles touch each other at E. J Further, assume that the circle with centre at touches SR, PS, PQ at I, J and G respectively. Let the circle with centre at D touches PQ, QR at K and H respectively. P Join I, J, G, ED, DF, DK, DH. I SR, J PS, G PQ, DK PQ, DH PR (tangent radius) DK // HQ (corr. s eq.) FDK = 90 (corr. s, DK // HQ) DFGK is a rectangle ( angles = 90) DFG = 90 (s sum of polygon) DF = 90 (adj. son st. line), E, D are collinear ( the two circles touch each other at E) I = J = G = E = 9 (radii of the circle with centre at ) DH = DK = DE = c (radii of the circle with centre at D) D = c + 9 FG = DK = c (opp. sides of rectangle DFGK) F = 9 c FD = GK (opp. sides of rectangle DFGK) = PD PG KQ = 5 9 c (opp. sides of rectangle) = 6 c F + DF = D (Pythagoras theorem) (9 c) + (6 c) = (9 + c) 8 8c + c + 56 c + c = 8 + 8c + c c 68c + 56 = 0 (c 4)(c 64) = 0 c = 4 or 64 (> 8, rejected) I4.4 ABD is a rectangle and EF is an equilateral triangle, ABD = 6c, find the value of d. Reference: HKEE M 98 Q5 ABD = 4 (given) AB = 4 (diagonals of rectangle) AEB = (s sum of ) ED = (vert. opp. s) EF = 60 ( of an equilateral triangle) DEF = 60 = 7 ED = E = EF (diagonals of rectangle, sides of equilateral ) DEF is isosceles ( sides equal) EFD = EDF (base s isos. ) d = (80 7) = 54 (s sum of isos. ) D A 9 9 I 9 - c F c G 5 d E 9 + c E D 6 - c c c F K 6c R 8 H Q B Page 4

5 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Individual Event 5 I5. Two opposite sides of a rectangle are increased by 50% while the other two are decreased by 0%. If the area of the rectangle is increased by a%, find a. Let the length and width be x and y respectively..5x 0.8y =.xy a = 0 I5. Let f (x) = x 0x + x a and g(x) = x 4 + x +. If h(x) is the highest common factor of f (x) and g(x), find b = h(). Reference: 99 HI5 f (x) = x 0x + x 0 = (x + )(x 0) g(x) = x 4 + x + = (x + )(x + ) h(x) = H..F. = x + b = h() = I5. It is known that b 6 has four distinct prime factors, determine the largest one, denoted by c 6 = ( )( + )( + )( 4 + )( 8 + ) = 5757 c = 57 I5.4 When c is represented in binary scale, there are d 0 s. Find d. 57 (x) = (ii) d = 7 Page 5

6 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Group Event 6 The following shows the graph of y = px + 5x + p. A (0, ), B,0, (, 0), O (0, 0). G6. Find the value of p. y = p x x p. It passes through A(0, ): = p = N O A y B R M x G6. If m 9 is the maximum value of y, find the value of m. y = x + 5x 9 = 4 m 4 m = 8 5 G6. Let R be a point on the curve such that OMRN is a square. If r is the x-coordinate of R, find the value of r. R(r, r) lies on y = x + 5x r = r + 5r r 4r + = 0 r = G6.4 A straight line with slope passes through the origin cutting the curve at two points E and 7 F. If is the y-coordinate of the midpoint of EF, find the value of s. s Sub. y = x into y = x + 5x x = x + 5x x 7x + = 0 Let E = (x, y ), F = (x, y ). 7 x + x = 7 y = s s = y = x x 7 = (x + x ) = Page 6

7 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Group Event 7 OAB is a tetrahedron with OA, OB and O being mutually perpendicular. Given that OA OB O 6x. G7. If the volume of OAB is ax, find a. ax = 6x 6x= 6x a = 6 G7. If the area of AB is b x, find b. A O B AB = B = A = 6 6x x = 6x AB is equilateral BA = 60 Area of AB = b x = 6 x sin 60 = 8 x b = 8 G7. If the distance from O to AB is c x, find c. By finding the volume of OAB in two different ways. 8 c = x c x 6x G7.4 If is the angle of depression from to the midpoint of AB and sin = d, find d. 8 x c x 6x Let the midpoint of AB be M. OM AB OAOB O = 6x, 6x OM 6x OM = x M = OM O = x 6x = 6x d O sin = = M 6x 6 = = 6x d = 6 Page 7

8 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Group Event 8 Given that the equation x + (m + )x = 0 has integral roots ( + ) and ( + ) with < and m 0. Let d =. G8. Find the value of m. ( + )( + ) = + =, + = or + =, + = (, ) = (,), (,0) When (, ) = (, 0), sum of roots = ( + ) + ( + ) = (m + ) m = 0 (rejected) When (, ) = (, ), sum of roots = ( + ) + ( + ) = (m + ) m = G8. Find the value of d. d = = ( ) = Let n be the total number of integers between and 000 such that each of them gives a remainder of when it is divided by or 7. Reference: 994 FG8.-, 998 HI6, 05 FI. G8. Find the value of n. These numbers give a remainder of when it is divided by. They are, +, +,, 95 + (= 996) n = 96 G8.4 If s is the sum of all these n integers, find the value of s. s = = = Page 8

9 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Group Event 9 B, A, AB are divided respectively by the points X, Y, Z in the ratio :. Let area of AZY : area of AB : a and area of AZY : area of XYZ : b. G9. Find the value of a. area of AZY = area of AZ (same height) = area of AB (same height) a = 9 G9. Find the value of b. Reference: 000 FI5. Similarly, area of BZX = area of AB; area of XY = area of AB 9 9 area of XYZ = area of AB area of AZY area of BZX area of XY = area of AB : b = area of AZY : area of XYZ = : 9 b = A die is thrown times. Let 6 x be the probability that the sum of numbers obtained is 7 or 8 and y 6 be the probability that the difference of numbers obtained is. G9. Find the value of x. Favourable outcomes are (, 6), (, 5), (, 4), (4, ), (5, ), (6,), (,6), (,5), (4,4), (5,), (6,) x P(7 or 8) = 6 x = G9.4 Find the value of y. Favourable outcomes are (, ), (, ), (, 4), (4, 5), (5, 6), (6,5), (5, 4), (4, ), (, ), (, ). y P(difference is ) = 6 y = 0 Page 9

10 Answers: (99-9 HKMO Final Events) reated by: Mr. Francis Hung Last updated: 7 December 05 Group Event 0 ABD is a square of side length and B respectively. G0. If AP ax, find a. AP = AD DP = x a = 50 G0. If PQ = b 0 x, find b. x = 50x 0 5x. P, Q are midpoints of D D A P Q B PQ = P Q = 0 0x b = 0 G0. If the distance from A to PQ is c 0 x, find c. c 0 x = A distance from to PQ = 0 5x 0 5x = 5 0x c = 5 d G0.4 If sin =, find d. 00 Area of APQ = AP AQsin = PQ c 0x 50x sin 0 0x 5 0x d sin = = 00 5 d = 60 Page 0

1983 FG8.1, 1991 HG9, 1996 HG9

1983 FG8.1, 1991 HG9, 1996 HG9 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 - Individual 1 11 70 6 1160 7 11 8 80 1 10 1 km 6 11-1 Group 6 7 7 6 8 70 10 Individual Events I1 X is a point on