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1 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Individual Events I a 6 I P I3 a I a IS P 8 b + Q 6 b 9 b Q 8 c 7 R 6 c d S 3 d c 6 R 7 8 d *33 3 see the remark S + Group Events G a 7 G a * see the remark G3 a 0 G P 00 GS a 6 b 0 b. b 0 Q b c 3 c c 00 R c d 0 d d 00 S d 9 Individual Event I. Given that there are a positive integers less than 00 and each of them has eactly three positive factors, find the value of a. If = rs, where r and s are positive integers, then the positive factors of may be, r, s and. In order to have eactly three positive factors, r = s = a prime number. Possible =, 9,, 9,, 69. a = 6. I. If a copies of a right-angled isosceles triangle with hypotenuse cm can be assembled to form a trapezium with perimeter equal to b cm, find the least possible value of b. (give the answer in surd form). or The perimeter = or The least possible value of b = + I.3 If sin(c 3c + 7) =, where 0 < c 3c + 7 < 90 and c > 0, find the value of c. b sin(c 3c + 7) = c 3c + 7 = c 3c 8 = 0 (c 7)(c + ) = 0 c = 7 or (rejected) I. Given that the difference between two 3-digit numbers yz and zy is 700 c, where > z. If d is the greatest value of + z, find the value of d. yz zy = 700 c y + z (00z + 0y + ) = z = 693 z = 7 Possible answers: = 8, z = or = 9, z = d is the greatest value of + z = 9 + = Page

2 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Individual Event I. In Figure, ABCD is a square, M is the mid-point of AD and N is the mid-point of MD. If CBN : MBA = P :, find the value of P. Let ABM =, CBM = P. Let AB =, AM =, MN = = ND. tan tan tan = = tan P, P = tan 3 I. Given that ABCD is a rhombus on a Cartesian plane, and the co-ordinates of its vertices are A(0, 0), B(P, ), C(u, v) and D(, P) respectively. If u + v = Q, find the value of Q. ABCD is a rhombus, It is also a parallelogram By the property of parallelogram, the diagonals bisect each other Mid point of B, D = mid point of AC 0 u 0 v,, u = 3, v = 3 Q = u + v = 6 I.3 If + ( + ) + ( + + 3) + + ( Q) = R, find the value of R. R = + ( + ) + ( + + 3) + + ( ) R = = 6 I. In the figure, EBC is an equilateral triangle, and A, D lie on EB and EC respectively. Given that AD//BC, AB = CD = R and AC BD. If the area of the trapezium ABCD is S, find the value of S. A ABC = BCD = 60, AC intersects BD at J, AC BD. ACD DBA (S.A.S.) AC = BD (corr. sides, s) y ABD = DCA (corr. s, s) JBC = 60 ABD = JCB JBC is a right-angled isosceles triangle. B JBC = JCB = BJ = CJ = y, AJ = AC y = BD y = DJ = JAD = JDA =, ADC = 0 A 6 y Apply sine formula on ACD, sin sin0 y sin y = 8 6, area = 3 Method EBC = 60 = ECB Draw AF BC and DG BC, cutting BC at F and G. BF = CG = 6 cos 60 = 8 AF = DG = 6 sin 60 = 8 3 AC intersects BD at J, AC BD. ACD DBA (S.A.S.) ABD = DCA (corr. s, s) JBC = 60 ABD = JCB JBC is a right-angled isosceles triangle. JBC = JCB = CF = AF cot = 8 3 FG = CF CG = 3 8 = AD Area of the trapezium ABCD = S = = 3 B E D J 6 6 E J D 8 F G y 8 C C Page

3 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Individual Event 3 I3. Let and 3.If a is the real root of the equation,find the value of a = 3 = = a a I3. If b >,f (b) = and g(b) = +. If b satisfies the equation log b log 3 b f(b) g(b) + f(b) + g(b) = 3, find the value of b. Similar question: 007 HI9 log log3 log log3 log 3 3 b log b log log log b log b 3b b log log 3b b log log 3b or log log 3b b b 3 = 6 or b = 9 When b 3 = 6, (b 3 ) = 6 < 78 = 3 b < b log log 3b < 0 rejected. When b = 9, log = log b 8 = log 3b > 0 accepted. Method y y Define the maimum function of, y as: Ma(, y) = y y Similarly, the minimum function of, y is: Min(, y) = a log log log b log 3 log 3b f(b) =, g(b) =+ = log b log 3 b log b log b log b The given equation is equivalent to Ma(f(b), g(b)) = 3 log log3b If f(b) > g(b), i.e. b <, then the equation is f(b) = 3 3 log 3 log 6 = log b 3 b 3 = 6 < < <, which is a contradiction; rejected If f(b) g(b), the equation is equivalent to g(b) = 3 log3b i.e. 3 log 9b = log b 3 9b = b 3 b = 9 I3.3 Given that 0 satisfies the equation 0 + (b 8) = 0. If c = 0 + = 0, + =, + + =, + + = c = = 0 = 3 0, find the value of c. Page 3

4 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 I3. If and 6c are the roots of the equation p + d =, find the value of d. and 9 are roots of p + d = 0 Product of roots = p = 9 p = 8 d 7 Sum of roots = = +9 8d = 7 d = p 8 Individual Event I. Let a be a real number. If a satisfies the equation log ( + ) = + log ( + 3), find the value of a. log ( + ) = log + log ( + 3) + = ( + 3) ( ) + = ( ) 3 0 = ( ) 3 ( )( + ) = 0 =, = = a I. Given that n is a natural number. If b = n 3 an n + is a prime number, find the value of b. (Reference: 0 FI3.3) Let f (n) = n 3 8n n + f (6) = = = 0 f (6) is a factor By division, f (n) = (n 6)(n 6)(n + ) b = n 3 8n n +, it is a prime n 6 =, n = 7, b = I.3 In Figure, S and S are two different inscribed squares of the right-angled triangle ABC. If the area of S is 0b +, the area of S is 0b and AC + CB = c, find the value of c. Reference: American Invitation Mathematics Eamination 987 Q A A R F b E c b S y h 0 Q c C D a B C P a B Page

5 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Add the label D, E, F, P, Q, R, S as shown. CDEF, PQRS are squares. Let DE =, SR = y, then = =, y = 0. Let BC = a, AC = b, AB = c = a b. Let the height of the triangle drawn from C onto AB be h, then ab = ch = area of...(*) b ab AFE ~ ACB: = =...() b a a b height of CSP from C h h y h ch CSP ~ CAB: y = SP c y c c h = 0 ab ab a b By (*), = () ab a ab b c c From () ab = (a + b)... (3), sub. (3) into (): 0 a b a b a b ab ab a b a b a b a b a b a b a b a b Cross multiplying and squaring both sides: 0[(a + b) (a + b) + ] = [(a + b) (a + b)] (a + b) (a + b) 0 = 0 (a + b 6)(a + b + 0) = 0 AC + CB = a + b = 6 I. Given that c + = d, find the positive value of d. d = 6 + = 6 d = 6 Reference: 昌爸工作坊圖解直式開平方 Divide 6 into 3 groups of numbers,, 6. Find the maimum integer p such that p p = 3 p = = 6 Find the maimum integer q such that (60 + q)q q = = q + q = 66 Find the maimum integer r such that (660 + r)r 66 r = d = 33 Method : d = 6 + = = < 6 < = < d < = < 6 < 600 = < d < 30 The unit digit of d is 6 the unit digit of d is or 6 33 = d = 33 or 6 is not divisible by 3, but 336 is divisible by 3 d = 33 Remark: Original question:, find the value of d. d = 33 Method 3 Observe the number patterns: 3 = 6 33 = = 6... d = 33 Also, 33 = = = = 333 = = =... Page

6 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Individual Event (Spare) IS. In figure, ABC is an acute triangle, AB =, AC = 3, and its altitude AD =. If the area of the ABC is P, find the value of P. BD = CD = = 9 3 = P = area of = 9 = 8 IS. Given that and y are positive integers. If + y is divided by + y, the quotient is P + 3 and the remainder is Q, find the value of Q. + y = 97( + y) + Q, 0 Q < + y Without loss of generality, assume y, < + y = 97( + y) + Q < 98( + y) 98() = 96 3 < 96 On the other hand, + y = 97( + y) + Q = 3 ( + y) y( 3 y 3 ) ( 3 97)( + y) = y( 3 y 3 ) + Q RHS 0 LHS = Net, + y = 97( + y) + Q = y 3 ( + y) + ( 3 y 3 ) (97 y 3 )( + y) = ( 3 y 3 ) Q (97 y 3 )( + y) > ( 3 y 3 ) ( + y) (98 y 3 )( + y) > 0 98 > y 3 y... () 0 Q 0 + y 97( + y) 97( + y) 6 + y 97y 0 + y... () By trial and error, y = and y = satisfies () So there are two possible pairs (, y) = (, ) or (,). + = 66 = ( + )0 +, the quotient is not = 88 = ( + )97 + 8, Q = 8 IS.3 Given that the perimeter of an equilateral triangle equals to that of a circle with radius cm. Q If the area of the triangle is R cm, find the value of R. Radius of circle = cm =. cm circumference =. cm = 3 cm 8 Side of the equilateral triangle = cm Area of the triangle = sin 60 cm 3 = cm 3 R = 3 IS. Let W =, S = W +, find the value of S. R W W W W =, S = + S S = 0, S =, S > 0, S = + only S Page 6

7 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Group Event G. Given that a is an integer. If 0! is divisible by a, find the largest possible value of a.,, 6, 8,, 0 are divisible by, there are even integers. Reference: 990 HG6, 99 FG7., 996 HI3, 00 FG., 0 HG7, 0 FI., 0 FG.3, 8,, 8 are divisible by, there are multiples of. 8,, 8 are divisible by 8, there are 6 multiples of 8. 6,, 8 are divisible by 6, there are 3 multiples of 6. 3 is the only multiple of 3. a = = 7 G. Let [] be the largest integer not greater than. For eample, [.] = If b = , find the value of b = = = < <( ) , b = G.3 If there are c multiples of 7 between 00 and 00, find the value of c. 00 = 8.6, the least multiple of 7 is 79 = = 7., the greatest multiple of 7 is 77 = = a + (c )d = 03 + (n )7; c = 3 G. Given that 0 0 and 0 satisfies the equation find the value of d. sin sin sin cos + sin + sin sin ( cos ) = cos cos = 0 = 0 d = tan 0 = 0 sin sin sin. If d = tan 0, Page 7

8 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Group Event G. If the tens digit of is a, find the value of a. =, the tens digit = a = Remark Original question: If the tenth-place digit, this is the position of the first digit to the right of the decimal point. G. In Figure, ABC is a right-angled triangle, AB = 3 cm, AC = cm and BC = cm. If BCDE is a square and the area of F A ABE is b cm, find the value of b. cos B = AB 3 3 BC B Height of ABE from A 3 9 = AF = AB sin(90 B) = 3 b = BE C = 9 9 G.3 Given that there are c prime numbers less than 00 such that their unit digits are not square numbers, find the values of c. The prime are: {, 3,, 7, 3, 7, 3, 37, 3, 7, 3, 67, 73, 83, 97} c = G. If the lines y = + d and = y + d intersect at the point (d, d), find the value of d. = ( + d) + d = 0 = d d = E D Page 8

9 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Group Event 3 G3. If a is the smallest real root of the equation 3 7, find the value of a. Reference: 993 HG6, 99 FI., 996 FG0., 000 FG3., 0 FI.3 Let t = +., then the equation becomes t.t 0.t 0.t. 7 9 t t 7 9 t t 7 6 t t 7 6 t t 7t t = or t = 7 3 = t. = = 7 or 0 a = the smallest root = 0 G3. Given that p and q are prime numbers satisfying the equation 8p + 30q = 86. p If log 8 = b 0, find the value of b. 3q 8p + 30q = 86 3p + q = 3 Note that the number is the only prime number which is even. 3p + q = 3 = odd number either p = or q = p If p =, then q = ; b = log 8 = log8 log8 < 0 (rejected) 3q 3 6 p 7 If q =, then p = 7; b = log 8 = log 8 0 3q 3 G3.3 Given that for any real numbers, y and z, is an operation satisfying (i) 0 =, and (ii) ( y) z = (z y) + z. If 00 = c, find the value of c. c = 00 = (0) 00 = (000)+ 00 = + 00 = 00 f 3 = d, find the value of d. G3. Given that f () = ( ) If 3 ( + ) = 3 + = 0 By division, = ( + )( + ) = d = f 3 = f() = ( ) = ( ) = 00 Page 9

10 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Group Event 000 G. If f and P = f f f, find the value of P Reference: 0 HG, 0 FI. f f 000 P = f f f = f f f f f f = G. Let f () = a + + a, where a and 0 < a <. If Q is the smallest value of f (), find the value of Q. Reference: 99 HG, 00 HG9, 008 HI8, 008 FI.3, 00 HG6, 0 FGS., 0 Final G.3 f () = a a = = = Q G.3 If m = 3 n = 36 and R =, find the value of R. m n Reference: 00 HI, 003 FG., 00 FG.3, 00 HI9, 006 FG.3 log m = log 3 n = log 36 m log = n log 3 = log 36 log36 log36 m = ; n = log log3 log log3 = = m n log36 log36 log6 log36 = log6 log6 G. Let [] be the largest integer not greater than, for eample, [.] =. If a and S = [a], find the value of a < a < = = < 00 S = [a] = = Page 0

11 Answers: (003-0 HKMO Final Events) Created by: Mr. Francis Hung Last updated: 9 September 07 Group Event (Spare) GS. For all integers n, F n is defined by F n = F n + F n, F 0 = 0 and F =. If a = F + F + + F + F, find the value of a. F = 0 + =, F 3 = + =, F = + = 3, F = + 3 = F + 0 = F =, F + = 0 F =, F 3 + ( ) = F 3 =, F + = F = 3, F + ( 3) = F = a = F + F + + F + F = = 6 GS. Given that 0 satisfies the equation + + = 0. If b = , find the value of b. By division, b = = ( )( ) + = GS.3 Figure shows a tile. If C is the minimum number of tiles required to tile a square, find the value of C. C = GS. If the line + y 00 = 0 has d points whose and y coordinates are both positive integers, find the value of d. (, y) = (8, ), (6, 0), (, ), (, 0), (0, ), (8, 30), (6, 3), (, 0), (, ) d = 9 Page

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

11 is the same as their sum, find the value of S. Answers: (998-99 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