Individual Events 5 I3 a 6 I4 a 8 I5 A Group Events

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1 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 SI a I a 6 I A Individual Events I3 a 6 I4 a 8 I A 6 b 4 b B 60 b *9 see the remarks b 0 B 36 c c 0 C c 6 c 3 C d d 0 D 68 d 48 d 03 D 0 SG a 9 G6 a 4, Grup Events G7 a 4 G8 a 6 G9 A G0 a 0 b 8 b 6 b 04 b 83 B b 37 c 0 c 3 7 c c 6 C 4 c d 00 d 86 d d 7040 D 9 d 4 Sample Individual Event (98 Final Sample Individual Event) SI. The sum f tw numbers is 40, their prduct is 0. If the sum f their reciprcals is a, find the value f a. Let the tw numbers be x, y. x + y = 40 ; xy = 0 a = + x y x + y 40 = = xy 0 = SI. If b cm is the ttal surface area f a cube f side ( a + ) cm, find the value f b. b = 6( + ) = 4 SI.3 One ball is taken at randm frm a bag cntaining (b 4) white balls and (b + 46) red balls. If 6 c is the prbability that the ball is white, find the value f c. There are 0 white balls and 00 red balls. P(white ball) = 0 = = = c c = SI.4 The length f a side f an equilateral triangle is c cm. If its area is d 3 cm, find the value f d. ( ) sin 60 = d 3 3 = d d = Page

2 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Individual Event I. The equatin x ax + (a + 3) = 0 has equal rts. Find a, if a is a psitive integer. = ( a) 4(a + 3) = 0 a 4a = 0 (a 6)(a + ) = 0 a = 6 r a = (rejected) I. In a test, there are 0 questins. a marks will be given t a crrect answer and 3 marks will be deducted fr each wrng answer. A student has dne all the 0 questins and scred 48 marks. Find b, the number f questins that he has answered crrectly. Reference: 998 HG0 6b 3(0 b) = 48 9b = 08 b = I.3 If x : y = : 3, x : z = 4 :, y : z = b : c, find the value f c. x : y : z = 4 : 6 : y : z = 6 : = : 0 c = 0 I.4 Let P(x, d) be a pint n the straight line x + y = such that the slpe f OP equals t c (O is the rigin). Determine the value f d. Reference: 993 FI3. x + d = x = d d m OP = = c x d = 0 d d = 0 0d d = 0 Page

3 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Individual Event 3 I. In square PQRS, Y is the mid-pint f the side QR and PX = PQ 4. P X Q If A is the rati f the area f the shaded triangle t the area f the square, find A. Let PQ = 4x, PX = 3x, QX = x, QY = YR = x Y A = ( ) ( ) ( ) 4x ( ) 4x 3x x x 4x x ( 4x) S R = 6 I. A man bught a number f ping-png balls where a 6A% sales tax is added. If he did nt have t pay tax he culd have bught 3 mre balls fr the same amunt f mney. If B is the ttal number f balls that he bught, find B. Let the price f ping-png ball be x. Sales tax = % Bx( + %) = (B + 3)x B = B B = 0B + 60 B = 60 I.3 Refer t the diagram, find C. R S C PQS = C ( s in the same segment) B C + 4 = B (ext. f ) 4 C = 60 4 = Q P I.4 The sum f C cnsecutive even numbers is 70. If D is the largest f them, find D. 30 [ D + ( 30 ) ( ) ] = 70 D = 68 Page 3

4 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Individual Event 3 I3. If 83a8 is a mutiple f 87, find the value f a. 87 = 7 4 and 64 7 = a8 898 = 0(a + ), a multiple f 7 a = 6 Methd The qutient The first digit f = 8368 a = 6 83a8 87 shuld be a tw digit number. 83 (apprximate value) is 6. The last digit must be 4 (Q 7 4 = 8) 3 I3. The number f psitive factrs f a is b, find the value f b. Reference 993 HI8, 997 HI3, 998 HI0, 998 FI.4, 00 FG4., 00 FI4.4 Remark: The riginal questin is: The number f factrs f a LL, which may include negative factrs. 6 = 3 Factrs f 36 are in the frm x 3 y, where 0 x, 0 y. The number f factrs = ( + )( + ) = 9 I3.3 In an urn, there are c balls, b f them are either black r red, (b + ) f them are either red r white and f them are either black r white. Find the value f c. Suppse there are x black balls, y red balls, z white balls. x + y = 9 () y + z = () z + x = (3) () + () + (3): (x + y + z) = 3 c = x + y + z = 6 I3.4 Given f(3 + x) = f(3 x) fr all values f x, and the equatin f(x) = 0 has exactly c distinct rts. Find d, the sum f these rts. Reference: 00 FG3.4 Let ne rt be 3 + α. f(3 + α) = 0 = f(3 α) 3 α is als a rt. 3 + α + 3 α = 6 Sum f a pair f rts = 6 There are 6 rts, i.e. 8 pairs f rts Sum f all rts = 8 6 = 48 Page 4

5 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Individual Event 4 I4. The remainder when x 6 8x is divided by (x )(x ) is 7x a, find a. Let f(x) = x 6 8x f() = = 7 a a = 8 I4. If x x + = 0 and b = x 3 3x + 3x + a, find b. b = x(x x + ) (x x + ) + 0 = 0 I4.3 Refer t the diagram, find c. E C Reference: 989 FG0. A b ADE ~ ABC, AD : AB = 0 : = : 3 D 9 c AD : DB = : G F B BD : AB = : 3 BDF ~ BAG, c : 9 = : 3 c = 3 I4.4 If c bys were all brn in June 990 and the prbability that their birthdays are all different is d, find the value f d. 9 8 P(3 bys were brn in different days) = d = 03 = d Page

6 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Individual Event 4 I. Given + 4 = 0. If A =, find the value f A. x x x Reference: 999 FI. x = 0 A = = x I. If B circular pipes each with an internal diameter f A cm carry the same amunt f water as a pipe with an internal diameter 6 cm, find the value f B. π() B = π(6) B = 36 I.3 If C is the area f the triangle frmed by x-axis, y-axis and the line Bx + 9y = 8, find the value f C. Reference: 990 FI3.3 36x + 9y = 8 x-intercept =, y-intercept = C = = I.4 Fifteen square tiles with side 0C units lng are arranged as shwn. An ant walks alng the edges f the tiles, always keeping a black tile n its left. Find the shrtest distance D that the ant wuld walk in ging frm P t Q. Length f a square = 0C = As shwn in the figure, D = 0(0C) = 0 P P Q Q Page 6

7 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Sample Grup Event (98 Sample Grup Event) SG. If x*y = xy + and a = (*4)*, find the value f a. *4 = (4) + = 9 ( * 4)* = 9* = 9() + = 9 SG. If the b th prime number is a, find the value f b. List the prime number in ascending rder:, 3,, 7,, 3, 7, 9. b = 8 SG.3 If c = L 3 4 0, find the value f c in the simplest fractinal frm. c = 3 L = 0 SG.4 If d is the area f a square inscribed in a circle f radius 0, find the value f d. Diameter = 0 = diagnal f the square Let the side f the square be x. By Pythagras Therem, x = 0 = 400 d = x = 00 Page 7

8 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Grup Event 6 G6. If lg a lg a =, find the value f a. lga lg = lg lg a (lg a) (lg ) = lg lg a (lg a) lg lg a (lg ) = 0 (lg a lg )(lg a + lg ) = 0 lg a = lg r lg a = 4 r G6. If b = lg 3 [(3 + )(3 + )(3 4 + )(3 8 + ) + ], find the value f b. Reference: 06 FG.4 b = lg 3 [(3 )(3 + )(3 + )(3 4 + )(3 8 + ) + ] = lg 3 [(3 )(3 + )(3 4 + )(3 8 + ) + ] = lg 3 [(3 4 )(3 4 + )(3 8 + ) + ] = lg 3 [(3 8 )(3 8 + ) + ] = lg 3 (3 6 + ) = 6 G6.3 If a 3-day mnth is taken at randm, find c, the prbability that there are Sundays in the mnth. st day = Sunday 9 th day = th Sunday st day = Saturday 30 th day = th Sunday st day = Friday 3 st day = th Sunday 3 Prbability = 7 G6.4 A grup f peple is t be selected frm 6 men and 4 wmen. Find d, the number f ways that there are always mre men than wmen men and wmen, number f cmbinatins = C = 0 3 C men and wman, number f cmbinatins = C = 60 men, number f cmbinatins = C 6 = 6 Ttal number f ways = = 86 4 C Page 8

9 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Grup Event 7 G7. There are a zers at the end f the prduct Find the value f a. Reference: 990 HG6, 996 HI3, 004 FG., 0 HG7, 0 FI.4, 0 FG.3 When each factr f is multiplied by, a trailing zer will appear in the prduct. The number f factrs f is clearly mre than the number f factrs f in 00! It is sufficient t find the number f factrs f., 0,,, 00; altgether 0 numbers, have at least ne factr f., 0, 7, 00; altgether 4 numbers, have tw factrs f. Ttal number f factrs f is = 4 There are 4 trailing zers f 00! a = 4 G7. Find b, if b is the remainder when is divided by = (000 ) 0 0 = = k= 0 9 k= 0 C C 0 k 0 k k 0 k k md 0 4 (QC 04 md 0 4 b = 04 k = 0000m, where m is an integer) G7.3 Find the largest value f c, if c = x + x and x >. (c + x ) = 4(x ) c + x cx 4x 4c = 4x 4 x + (c 4)x + (c 4c + 8) = 0 Fr real values f x, 0 4(c 4) 4(c 4c + 8) 0 c 8c + 6 c + 4c c c The largest value c = Methd Let y = x, then y = x x = y + c = (y + ) + y = ( y) The largest value f c =. 3 d G7.4 Find the least value f d, if d d 3 d 8 d 4 d The least value f d = Page 9

10 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Grup Event 8 G8. Frm t, there are a numbers which are multiplies f 3 r. Find the value f a. Reference: 993 FG8.3-4, 998 HI6, 0 FI3. Number f multiples f 3 = 40 (0 = 3 40) Number f multiples f = 4 (0 = 4) Number f multiples f = 8 (0 = 8) Number f multiples f 3 r = a = = 6 G8. Frm t, there are b numbers which are nt divisible by nr 7. Find the value f b. Number f multiples f = 4 (0 = 4) Number f multiples f 7 = 7 (9 = 7 7) Number f multiples f 3 = 3 (0 = 3 3) Number f multiples f r 7 = = 38 Number which are nt divisible by nr 7 = 38 = 83 Frm the digits,, 3, 4, when each digit can be used repeatedly, 4-digit numbers are frmed. Find G8.3 c, the number f 4-digit numbers that can be frmed. c = 4 4 = 6 G8.4 d, the sum f all these 4-digit numbers. Reference: 00 HI4 Q There are 6 different numbers,, 3, 4 each appears 64 times in the thusands, hundreds, tens and units digit. d = [000( ) + 00( ) + 0( ) ] 64 = (0) 64 = Page 0

11 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Grup Event 9 A, B, C, D are different integers ranging frm 0 t 9 and (ABA) = (CCDCC) < Find the values f A, B, C and D. (ABA) < < 36 A =, r 3 A B A A B A C C D C C When A =, then A = = C cntradict that A and C must be different rejected When A =, C = 4 (0 + 0B) = D B + 00B = D 4040B + 00B = D 404B + 0B = D B =, 44 = D D = 9 When A = 3, C = 9 ( B) = D B + 00B = D 606B + 0B = D B =, 66 = D n slutin fr D A =, B =, C = 4, D = 9 Page

12 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: 8 August 08 Grup Event 0 In rectangle ABCD, AD = 0, CD =, P is a pint inside the rectangle such that PB = 9, PA =. Find (Reference: 00 FG., 003 FI3.4, 08 HI7) G0. a, the length f PD and AP + BP = + 9 = = = = AB A θ APB = 90 (Cnverse, Pythagras therem) θ B Let ABP = θ, then cs θ = =, sin θ = 0 0 BAP = 90 θ ( s sum f ) P DAP = θ D C PBC = 90 θ a = PD = a = (Csine rule n ADP) G0. b, the length f PC. b = CP = = 37 sin 0 cs0 cs40 cs60 cs80 G0.3 It is given that sin θ = sin θ cs θ. Find c, if c = sin60 sin 0 C = = cs0 cs40 cs60 cs80 sin60 sin 80 cs60 cs80 8sin60 sin60 = 8sin60 6 = sin 40 cs40 cs60 cs80 4sin60 =. tan A + tan B G0.4 It is given that tan(a + B) =. Find d, if tan Atan B d = ( + tan )( + tan )( + tan 3 )( + tan 4 ). tan A + tan B If A + B = 4, = tan(a + B) = tan Atan B tan A tan B = tan A + tan B = + tan A + tan B + tan A tan B ( + tan A)( + tan B) = d = ( + tan ) ( + tan 4 )( + tan )( + tan 3 ) = = 4 Page