Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 I a Individual Events I x 0 I3 a I r 3 I5 a b 3 y 3 b 8 s b c 3 z c t 5 c d w d 0 u d 6 3 6 G6 a 5 G7 a Group Events G8 V G9 A 9 G0 a b b V 0 B 6 b 3 c c 700 r 3 C 8 c 6 d 995 Individual Event I. Find a, if a log d 333 V 35 D d. a log log I. In the figure, AB AD DC, BD a. Find b, the length of BC. Let ADB, CDB 80 (adj. s on st. line) In ABD, cos a 8 A B a D Apply cosine formula on BCD. b (a) + (a)cos(80 ) b + 6 ( cos ) 7 + 8 8 8 b 3 I.3 It is given that f (x) px 3 + qx + 5 and f ( 7) b +. Find c, if c f (7). Reference: 006 FG. p( 7) 3 + q( 7) + 5 3 7 [p(7) 3 + q(7)] c f (7) p(7) 3 + q(7) + 5 + 5 3 I. Find the least positive integer d, such that d c + 000 is divisible by 0 + c. d 3 + 000 is divisible by 3 377 00 000 + 3 d 0 C http://www.hkedcity.net/ihouse/fh7878/ Page
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event x I. If x x x x 3 Reference: 998 HI3 x, find x. x 0 or (x )(x ) (x )(x 3) x 0 or x 5x + x 5x + 6 x 0 or 6 x 0 I. If f (t) 35 t and y f (x), find y. y f (0) 35 0 3 I.3 A can finish a job in y days, B can finish a job in (y + 3) days. If they worked together, they can finish the job in z days, find z. z z 3 6 I. The probability of throwing z dice to score 7 is w, find w. 6 P( sum of dice 7) P((,6), (,5), (3,), (,3), (5,), (6, )) 36 6 w 6 http://www.hkedcity.net/ihouse/fh7878/ Page
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event 3 I3. If a sin 30 + sin 300 + sin 3000, find a. 3 3 a + sin(3608 + 0) 3 I3. It is given that x + y k () z + x 3k () x y z x 3 y + z k (3) () + () + (3): (x + y + z) 9k (36)( ) 9k y z and x + y + z 36a. Find the value of b, if b x y. k b x + y k () 8 I3.3 It is given that the equation x + 6 + 8k k(x + b) has positive integral solution. Find c, the least value of k. x + 6 + 8k k(x + 8) (k )x 6 If k, the equation has no solution 6 If k, x k The positive integral solution, 6 must be divisible by k. The least positive factor of 6 is, c I3. A car has already travelled 0% of its journey at an average speed of 0c km/h. In order to make the average speed of the whole journey become 00 km/h, the speed of the car is adjusted to d km/h to complete the rest of the journey. Find d. Let the total distance be s. s 0.s 0.6s 0 d 00 3 5 d 0 00d d 0 00 00 00 http://www.hkedcity.net/ihouse/fh7878/ Page 3
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event I. In triangle ABC, B 90, BC 7 and AB. If r is the radius of the inscribed circle, find r. Let O be the centre of the inscribed circle, which touches BC, CA, AB at P, Q, R respectively. OP BC, OQ AC, OR AB (tangent radius) ORBP is a rectangle (it has 3 right angles) BR r, BP r (opp. sides of rectangle) CP 7 r, AR r AC AB + BC (Pythagoras Theorem) + 7 65 AC 5 CQ 7 r, AQ r (tangent from ext. point) CQ + AQ AC 7 r + r 5 r 3 I. If x + x 0 and s x 3 + x + r, find s. By division, s x 3 + x + 3 (x + )(x + x ) + C B C 7 - r Q 7 - r r r - r P O r r B r R - r I.3 It is given that F F and F n F n + F n, where n 3. If F t s +, find t. F t + 5 F 3 +, F + 3, F 5 3 + 5 t 5 A A I. If u t t t 3 t, find u. Reference: 993 HG6, 996 FG0., 000 FG3., 00 FG3., 0 FI.3 u 5 6 78 0 u http://www.hkedcity.net/ihouse/fh7878/ Page
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Individual Event 5 I5. It is given that log 7 (log 3 (log x)) 0. Find a, if a 3 x. log 3 (log x) log x 3 x 3 8 a 3 x I5. In the figure, PQ is a diagonal of the cube and PQ a. Find b, if b is the total surface area of the cube. Reference: 99 HI, 003 HI7 Let the length of the cube be x. PQ x + x + x (Pythagoras Theorem) 3x The total surface area b 6x I5.3 In the figure, L and L are tangents to the three circles. If the radius of the largest circle is 8 and the radius of the smallest circle is b, find c, where c is the radius of the circle W. Let the centres of the 3 circles be A, B, C as shown in the figure. L touches the circles at D, E, F as shown. AD L, WE L, BF L (tangent radius) Let AB intersects the circle W at P and Q. AD AP b 8, EW WQ PW c QB BF 8 (radii of the circle) D Draw AG // DE, WH // EF as shown 8 EW // FB (int. supp.) A AWG WBH (corr. s EW // FB) AG GW, WH HB (by construction) AGW ~ WHB (equiangular) GW c 8, BH c + 8 (opp. sides of rectangle) c 8 8 c (ratio of sides, ~ ) c 8 c 8 (c 8)(c + 8) (c + 8)(8 c) c + 0c c + 0c + c () c I5. Refer to the figure, ABCD is a rectangle. AE BD and BE EO 6 c. Find d, the area of the rectangle ABCD. E G c - 8 W Q 8 c c 8 A P W P F H B Q 8 - c L D L BO OD AO OC (diagonal of rectangle) AE OA OE (Pythagoras Theorem) AE 3 ABD CDB (R.H.S.) d area of ABD 3 6 3 B E O C http://www.hkedcity.net/ihouse/fh7878/ Page 5
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event 6 G6. a 9 b is a four digit number and its thousands digit is, its hundreds digit is a, its tens digit is 9 and its units digit is b, find a, b. a 9 b 000 + 00a + 90 + b If a 0, 9 b 090 + b 9 3 79, 9 656 No solution for a a > 0 and 0 b 3, 000 + 00a + 90 + b is divisible by b 0 or If b 0, a 090 + 00a 0 0, 08, 096 and 0 a 9 No solution for a b, 000 + 00a + 9 is divisible by 9 + a + 9 + 9m, where m is a positive integer a 5, b Check: 5 9 38 59 000 + 00(5) + 90 + G6. Find c, if c. 3 3 3 3 Reference: 006 FI. Let x, y, then c x(y ) y(x ) x + y 3 3 G6.3 Find d, if d 3 99 3 995 3 995 3 99 x, y 3 99 3 995 d x(y ) y(x ) x + y 995 http://www.hkedcity.net/ihouse/fh7878/ Page 6
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event 7 G7. Let p, q, r be the three sides of triangle PQR. If p + q + r r (p + q ), find a, where a cos R and R denotes the angle opposite r. p cos R a cos R q r pq p q r p q p q r p q p r p q r p q p q p r p q q r q r p q p q G7. Refer to the diagram, P is any point inside the square OABC and b is the minimum value of PO + PA + PB + PC, find b. PO + PA + PB + PC OB + AC (triangle inequality) OB A O P C B (, ) b G7.3 Identical matches of length l are used to arrange the following pattern, if c denotes the total length of matches used, find c. st row st row + nd row + 6 0 st + nd + 3 rd + 6 + 8 8... c st + + 5 th row + 6 + 8 + + [ + (5 )] na n d 5 700 G7. Find d, where d. Reference: 000 FI. (00 ) 999 3 333 d 333 } 5. rows..... http://www.hkedcity.net/ihouse/fh7878/ Page 7
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event 8 Rectangles of length and breadth b where, b are positive integers, are drawn on square grid paper. For each of these rectangles, a diagonal is drawn and the number of vertices V intersected (excluding the two end points) is counted (see figure). G8. Find V, when 6, b. Intersection point (3, ) V G.8. Find V, when 5, b 3 As 3 and 5 are relatively prime, there is no intersection V 0 b 3 V G8.3 When and < b <, find r, the number of different values of b that makes V 0? b 5, 7, are relatively prime to. The number of different values of b 3 G8. Find V, when 08, b 7. H.C.F. (08, 7) 36, 08 363, 7 36 Intersection points (3, ), (6, ), (9, 6),, (05, 70) V 35 http://www.hkedcity.net/ihouse/fh7878/ Page 8
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event 9 A, B, C, D are different integers ranging from 0 to 9 and A A B C Find A, B, C and D. B A C B If A 0, then B, (AABC) (BACB) < 0 rejected D A C D A > 0, consider the hundreds digit: If there is no borrow digit in the hundreds digit, then A A A A 0 rejected There is a borrow digit in the hundreds digit. Also, there is a borrow digit in the thousands digit 0 + A A A A 9 Consider the thousands digit: A B D B + D 8 () Consider the units digit: If C < B, then 0 + C B D 0 + C B + D 0 + C 8 by () C (rejected) C > B and there is no borrow digit in the tens digit Consider the tens digit: 0 + B C C 0 + B C () Consider the units digit, C > B C B D C B + D C 8 by () Sub. C 8 into () 0 + B 6 B 6 Sub. B 6 into (), 6 + D 8 D A 9, B 6, C 8, D http://www.hkedcity.net/ihouse/fh7878/ Page 9
Answers: (99-95 HKMO Final Events) Created by: Mr. Francis Hung Last updated: July 08 Group Event 0 Lattice points are points on a rectangular coordinate plane having both x- and y-coordinates being integers. A moving point P is initially located at (0, 0). It moves unit along the coordinate lines (in either directions) in a single step. G0. If P moves step then P can reach a different lattice points, find a. (, 0), (, 0), (0, ), (0, ) a G0. If P moves not more than steps then P can reach b different lattice points, find b. (, 0), (, 0), (0, ), (0, ), (, ), (, (, ), (, ), (, ) (, 0), (, 0), (0, ), (0, ), (0, 0) b 3 G0.3 If P moves 3 steps then P can reach c different lattice points, find c. (, 0), (, 0), (0, ), (0, ), (3, 0), (, ), (, ), (0, 3), (, ), (, ), ( 3, 0), (, ), (, ), (0, 3), (, ), (, ); c + 6 G0. If d is the probability that P lies on the straight line x + y 9 when P advances 9 steps, find d. Total number of outcomes + + 0 + 8 + 36 00 Favourable outcomes {(0,9), (,8), (,7),, (9,0)}, number 0 Probability 0 y x http://www.hkedcity.net/ihouse/fh7878/ Page 0