11 is the same as their sum, find the value of S.

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1 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Events I P 4 I a 8 I a 6 I4 a I5 a IS a Q 8 b 0 b 7 b b spare b 770 R c c c c 0 c 57 S 0 d 000 d 990 d 8 d 6 d 58 Group Events G a G a G a G4 a 4 G5 P 5 GS P 4 b 5 b 0 b 7 b 0 Q 6 spare Q 6 c 80 c c 0 c R R 5 d d 5 d *6 see the remark d S 50 S 8 Individual Event I. If the interior angles of a P-sided polygon form an Arithmetic Progression and the smallest and the largest angles are 0 and 60 respectively. Find the value of P. Sum of all interior angles = P (0 + 60) = 80(P ) 90P = 80P 60 P = 4 I. In ABC, AB = 5, AC = 6 and BC = P. If Q = cos A, find the value of Q. (Hint: cos A = cos A ) cos A = 65 4 cos A = cos A = 4 8 Q = 8 I. If log Q + log 4 Q + log 8 Q = R, find the value of R. R = log 8 + log log 8 8 = + + = R = I.4 If the product of the numbers R and S S 0 S S = 0 is the same as their sum, find the value of S. S Page

Answers: (998-99 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event I. If, y and z are positive real numbers such that a = y y z z, find the value of a.

2 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event I. If, y and z are positive real numbers such that a = y y z z, find the value of a. yz Reference: 99 HG y z y z y z Let = k, = k, = k. z y y z kz y z ky y z k () + () + (): + y + z = k( + y + z) k = From (), + y = z, (): + z = y, (): y + z = a = y y z z y z y z y z = = z y 8yz = = 8 yz yz I. Let u and t be positive integers such that u + t + ut = 4a +. If b = u + t, find the value of b. u + t + ut = 4 + u + t + ut = 5 ( + u)( + t) = 5 + u = 5, + t = 7 u = 4, t = 6 b = = 0 I. In Figure, OAB is a quadrant of a circle and semi-circles are drawn on OA and OB. If p, q denotes the areas of the shaded regions, where p = (b 9) cm and q = c cm, find the value of c. p =, let the area of each of two unshaded regions be cm Let the radius of each of the smaller semicircles be r. The radius of the quadrant is r. r + q = area of one semi-circle = ; + p + q = area of the quadrant = r r 4 () = (), + q = + p + q q = p; c = I.4 Let f 0 () = and f n () = f 0 (f n ()), n =,,,. If f 000 (000) = d, find the value of d. c Reference: 009 HI6 f 0 () =, f () = f 0 ( ) = f () = f 0 ( ) = =, which is an identity function. So f 5 () = f () =,, f 000 () = ; f 000 (000) = 000 = d and Page

3 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event (000 Sample Individual Event) I. For all integers m and n, m n is defined as: m n = m n + n m. If a = 00, find the value of a. Reference: 990 HI4 a + a = = = 00 a = 6 I. If b 6a b 6a, where b > 0, find the value of b. Reference: 005 FI., 06 FG. b 7 b 7 b + 7 b 7 b b b 7 (b 7) = 4 = b 7 b 7 b 7 4 = 7 7 b 7 b ; ( b b 7 ) 84 = [(b) 69] = 69 b b = 49 b = 7 Method b 7 b 7 We look for the difference of multiples of 8 b + 7 = 6, b 7 =, no solution b = 7 b + 7 = 54, b 7 = 6, no solution 7 b + 7 = 8, b 7 = 54 I. In figure, AB = AC and KL = LM. If LC = b 6 cm and KB = c cm, find the value of c. Reference: 99 HG6 Draw LN // AB on BM. BN = NM intercept theorem LNC = ABC = LCN (corr. s, AB // LN, base s, isos. ) LN = LC = b 6 cm = cm (sides opp. eq. s) c cm = KB = LN = cm (mid point theorem) B K A N L C M I.4 The sequence {a n } is defined as a = c, a n+ = a n + n (n ). If a 00 = d, find the value of d. a =, a = +, a = + + 4,, a 00 = = = 990 = d Page

4 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event 4 I4. Mr. Lee is a years old, a < 00. If the product of a and his month of birth is 5, find the value of a. 5 = = his month of birth and a = I4. Mr. Lee has a + b sweets. If he divides them equally among 0 persons, 5 sweets will be remained. If he divides them equally among 7 persons, more sweets are needed. Find the minimum value of b. 0m + 5 = 7n = + b 7n 0m = 8 By trial and error n = 4, m = + b = 74 = 5 b = I4. Let c be a positive real number. If + c + b = 0 has one real root only, find the value of c. + c + = 0 = 4(c ) = 0 c = I4.4 In figure, the area of the square ABCD is equal to d. If E, F, G, H are the mid-points of AB, BC, CD and DA respectively and EF = c, find the value of d. Area of EFGH = c = = 4 Area of ABCD = area of EFGH = 8 d = 8 Individual Event 5 I5. If 44 p = 0, 78 q =5 and a = p q, find the value of a. a = p q = 44 p 78 q = 0 5 = 4 4 I5. If + = 0, b = a, find b. Reference: 994 FI5. 0 ; =, b = = I5. If the number of real roots of the equation b + = 0 is c, find the value of c. + = 0 = 4 < 0 c = number of real roots = 0 I5.4 Let f() = c + and f(n) = (n ) f(n ), where n >. If d = f(4), find the value of d. Reference: 009 FI.4 f() = f() = f () = f() = f () = f(4) = f () = = 6 H D A E G B F C Page 4

5 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event (Spare) IS. If a is the smallest prime number which can divide the sum + 5, find the value of a. Reference: 00 FG. is an odd number 5 is also an odd number So + 5 is an even number, which is divisible by. IS. For all real number and y, y is defined as: y =. y If b = 4 (a 540), find the value of b. a 540 = b = 4 (a 540) = IS. W and F are two integers which are greater than 0. If the product of W and F is b and the sum of W and F is c, find the value of c. WF 770() W F c () 770 = 5 W =, F = 5 c = + 5 = 57 d IS.4 If, find the value of d. 4 c Reference: 986 FG0.4, 04 FG. = = = d = 58 Page 5

6 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event (000 Final Sample Group Event) G. Let * y = + y y, where, y are real numbers. If a = * (0 * ), find the value of a. 0 * = = a = * (0 * ) = * = + = G. In figure, AB is parallel to DC, ACB is a right angle, D C AC = CB and AB = BD. If CBD = b, find the value of b. ABC is a right angled isosceles triangle. BAC = 45 (s sum of, base s isos. ) b ACD = 45 (alt. s, AB // DC) A BCD = 5 Apply sine law on BCD, BD BC sin5 sin D ABsin 45 AB, given that AB = BD sin D sin D = ; D = 0 B CBD = = 5 (s sum of BCD) b = 5 G. Let, y be non-zero real numbers. If is 50% of y and y is c% of, find the value of c. =.5y () c y = () 00 c Sub. () into (): y =.5y 00 c = 80 G.4 If log p =, log q =, log r = 6 and log pqr = d, find the value of d. Reference: 00 FG.4, 05 HI7 log log log ; ; 6 log p log q log r log p log q log r ; ; log log log 6 log p log q log r log log log 6 log pqr log log log pqr d = log pqr = Page 6

7 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event G. If a = and + + = 0, find the value of a. 0 4 a = 4 G. If 6 b + 6 b+ = b + b+ + b+, find the value of b. 6 b ( + 6) = b ( + + 4) b = 0 G. Let c be a prime number. If c + is the square of a positive integer, find the value of c. c + = m m = c (m + )(m ) = c m = and m + = c m = G.4 Let d be an odd prime number. If 89 (d + ) is the square of an integer, find the value of d. d is odd, d + must be even, 89 (d + ) must be odd. 89 = (d + ) + m By trial and error, m = 5, 89 = d + = 8 d = 5 Page 7

8 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event G. Let a be the number of positive integers less than 00 such that they are both square and cubic numbers, find the value of a. 998 HG4 The positive integers less than 00 such that they are both square and cubic numbers are: and 6 = 64 only, so there are only numbers satisfying the condition. G. The sequence {a k } is defined as: a =, a = and a k = a k + a k (k > ). If a + a + + a 0 = a b, find the value of b. a =, a =, a =, a 4 =, a 5 = 5, a 6 = 8, a 7 =, a 8 =, a 9 = 4, a 0 = 55 a + a + + a 0 = = 4 = = a 7 b = 7 G. If c is the maimum value of log(sin ), where 0 < <, find the value of c. 0 < sin log(sin ) log = 0 c = 0 G.4 Let 0 and y 0. Given that + y = 8. If the maimum value of + y is d, find the value of d. (Reference: 999 FGS.) + y = y y y 8 6 y 8 y (GM AM) + y 6 = d (It is easy to get the answer by letting = y in + y = 8) Remark The original question is Given that + y = 8. If the maimum value of + y + y is undefined for < 0 or y < 0. Page 8

9 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event 4 G4. If a tiles of L-shape are used to form a larger similar figure (figure ) without overlapping, find the least possible value of a. From the figure, a = 4. G4. Let, be the roots of + b = 0. If > and <, and b is an integer, find the value of b. > 0 and + < 0 ( )( + ) < 0 + < 0 < ( ) < 9 ( + ) 4 < 9 b + 8 < 9 < b < b is an integer b = 0 G4. Given that m, c are positive integers less than 0. c 4 If m = c and 0. m c, find the value of c. m 5 0m c c 4 0. m c 99 m 5 0c c c 4 99 c 5 7c c 4 c 5 4c + 5c = c + 4c + c = 0 7c + c 66 = 0 (7c + )(c ) = 0 c = G4.4 A bag contains d balls of which are black, + are red and + are white. If the probability of drawing a black ball randomly from the bag is less than 6, find the value of d. 6 < + < = 0 d = + = Page 9

10 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event 5 G5. If the roots of P = 0 differ by, find the value of P. Reference: 999 FGS. + =, = P = ( ) = 44 ( + ) 4 = P = 44 P = 5 G5. Given that the roots of + a + b = 0 and + b + a = 0 are both real and a, b > 0. If the minimum value of a + b is Q, find the value of Q. a 8b 0 and 4b 4a 0 a 8b and b a a 4 64b 64a a 4 64a 0 a(a 64) 0 a 64 a 4 Minimum a = 4, b a b 4 minimum b = Q = minimum value of a + b = 4 + = 6 G5. If R 000 < 5 000, where R is a positive integer, find the largest value of R. Reference: 996 HI4, 008 FI4., 08 FG.4 (R ) 000 < (5 ) 000 R < 5 = 5 R < 5 < The largest integral value of R = G5.4 In figure, ABC is a right-angled triangle and BH AC. If AB = 5, HC = 6 and the area of ABC is S, find the A value of S. H Reference: 998 FG. 5 It is easy to show that ABH ~ BCH ~ ACB. 6 Let ABH = = BCH In ABH, BH = 5 cos B In BCH, CH = BH tan 6 tan = 5 cos 6 sin = 5 cos 6 sin = 5 5 sin 5 sin + 6 sin 5 = 0 ( sin + 5)(5 sin ) = 0 sin = ; tan = 5 4 C 4 BC = AB tan = 5 = 0 Area of ABC = 5 0= 50 = S Page 0

11 Answers: ( HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event (Spare) GS. If a number N is chosen randomly from the set of positive integers, the probability of the unit digit of N 4 P being unity is, find the value of P. 0 If the unit digit of N 4 4 is, then the unit digit of N may be,, 7, 9. So the probability = 0 P = 4 GS. Let 0 and y 0. Given that + y = 8. If the maimum value of + y is d, find the value of d. Reference: 999 FG.4 + y = y y y 8 6 y 8 y (G.M. A.M.) + y 6 = d GS. If the roots of R = 0 differs by, find the value of R. Reference: 999 FG5. + =, = R = ( ) = 44 ( + ) 4 = R = 44 R = 5 GS.4 If the product of a 4-digit number absd and 9 is equal to another 4-digit number dsba, find the value of S. Reference: 987 FG9, 994HI6 a =, d = 9, Let the carry digit in the hundred digit be. Then 9S + 8 = 0 + b...() 9b + = S...(); = S 9b...() Sub. () into (): 9s + 8 = 0(S 9b) + b 8 = S 89b S = 8, b = 0 Page

2 13b + 37 = 54, 13b 37 = 16, no solution

2 13b + 37 = 54, 13b 37 = 16, no solution Answers: (999-00 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 6 February 07 Individual Events SI P 6 I P 5 I P 6 I P I P I5 P Q 7 Q 8 Q 8 Q Q Q R R 7 R R 996 R R S 990 S 6 S S 666 S S