1983 FG8.1, 1991 HG9, 1996 HG9

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1 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February Individual km Group Individual Events I1 X is a point on the line segment as shown in figure 1. If 7, and 0, find the minimum value of X + X. Reference: 18 FG8.1, 11 HG, 16 HG 7 Reflect point along to. y the property of reflection. and = 7 Join, which cuts at X. X X (S..S.) X 0 (Figure 1) I X + X = X + X This is the minimum when, X, are collinear. raw E // which intersects produced at E. Then E E (corr. s, // E) E = 0 and E = 7 (opp. sides, rectangle) = 0 + (7 + ) = 116 = The minimum value of X + X = In quadrilateral, //, and, intersect at O (as shown in figure ). Given that area of O 6, area of O, determine the area of the quadrilateral. Reference: 17 HG, 000 FI., 00 FI1., 00 HG7, 010HG, 01 HG O ~ O (equiangular) O area of O O area of O 6 O O 6 O (Figure ) rea of O = 6 area of O = 6 6 = 0 rea of O = 6 area of O = 6 = 0 rea of quadrilateral = = 11 Page 1

2 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 I In figure, is a square of side 8( 1). Find the radius of the small circle at the centre of the square. Let and intersect at O.. Let H, K be the centres of two adjacent circles touch each other at E. The small circle touches one of the other circles at P. (Figure ) HE = EK = = 1= KP, HK = 1 OH = OK = HK cos = OP = OK KP = 1= The radius = I Thirty cards are marked from 1 to 0 and one is drawn at random. Find the probability of getting a multiple of or a multiple of. Let be the event that the number drawn is a multiple of. be the event that the number drawn is a multiple of. is the event that the number drawn is a multiple of 10. P( ) = P() + P() P( ) 1 6 = = = 0 I The areas of three different faces of a rectangular box are 10, 7 and 60 respectively. Find its volume. Let the lengths of sides of the box be a, b, c, where a > b > c. ab = 10 (1) bc = 60 () ca = 7 () (1)() (): (abc) = (606) abc = 70 The volume is 70. I6 For any positive integer n, it is known that n n( n 1)(n 1) =. Find the value 6 of (Reference: 18 HG) = = = 6 6 = (870 ) = 1160 I7 If x and y are prime numbers such that x y = 117, find the value of x. Reference: 1 HG, 17 HI1 (x + y)(x y) = 117 = 1 Without loss of generality, assume x y. x + y = 117, x y = 1 (1) or x + y =, x y = () or x + y = 1, x y = () From (1), x =, y = 8, not a prime, rejected From (), x = 1, y = 18, not a prime, rejected From (), x = 11, y = x = 11 Page H O E P K

3 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 I8 If m is the total number of positive divisors of 000, find the value of m. Reference 1 FI., 17 HI, 18 HI10, 18 FI1., 00 FG.1, 00 FI. 000 = Positive divisors are in the form x y z where x, y, z are integers and 0 x, 0y, 0z Total number of positive factors = = 80 I If a is a real number such that a a 1 0, find the value of a a a a 10. Reference: 000 HG1, 001 FG.1, 007 HG, 00 HG a a y division algorithm, a a 1 a a a a 10 a a a a 10 a a a = (a a 1)(a a + ) + 1 a a a = 1 a a a I10 In figure, E and E are straight lines,,, 60, E : E 1 :. If : E : 1 and area of a =, find the value of a. 0 1 rea of = sin 60 = = 1 rea of E = area of = rea of = area of E = 10 0 a = E a a (Figure ) a 10 a 1 Page

4 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 Group Events G1 car P is 10 km north of another car Q. The two cars start to move at the same time with P moving south-east at km/h and Q moving north-east at km/h. Find their smallest distance of separation in km. onsider the relative velocity. Keep Q fixed, the velocity of P relative to Q is km/h in the direction of P, where PQ =. Let P =, PQ = sin =, cos = P km/h km/h km/h 10 km 10 km km/h Q G km/h Q distance diagram velocity diagram P sin = sin( ) = sin cos cos sin = = When the course of P is nearest to Q (i.e at G), 1 The shortest distance is GQ = PQ sin = 10 = km G If, are the roots of the equation x x = 0, find + 1. Reference: 010 HI, 01 HG = 0 = + = ( + ) + = =, = + 1 = = 1 + = G s shown in figure 1, the area of is 10., E, F are points on, and respectively such that : :, and area of E area of quadrilateral EF. Find the area of E. F E Join E. rea of E = area of EF E and EF have the same base and the same height (Figure 1) E // E : E = : = : (theorem of equal ratio) E rea of EF = rea of = 10 = 6 G What is the maximum number of regions produced by drawing 0 straight lines on a plane? lines: maximum number of regions = = + lines: maximum number of regions = 7 = + + lines: maximum number of regions = 11 = lines, maximum number of regions = = 1 0 = 11 Page

5 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 G The product of consecutive positive integers is 0. Find the largest integer among the four. Let the four integers be x, x + 1, x +, x +. Reference 1 HG6, 1 FI., 16 FG10.1, 000 FG.1, 00 FG.1, 01 FI., 01HI x(x + 1)(x + )(x + ) = 0 (x + x)(x + x + ) = 0 (x + x) + (x + x) + 1 = 0 (x + x + 1) = x + x + 1 = or x + x + 1 = x + x = 0 or x + x + 6 = 0 (x 6)(x + ) = 0 or no real solution x > 0 x = 6 The largest integer = G6 Find the sum of all real roots of the equation (x + )(x + )(x + )(x + ) =. Reference 1 HG, 1 FI., 16 FG10.1, 000 FG.1, 00 FG.1, 01 FI., 01 HI Let t = x +. (t 1.)(t 0.)(t + 0.)(t + 1.) = t t 0 16 t t 0 t 0 t = 1 1 t = 7 1 x = t. = Sum of real roots = 7 G7 If a is an integer and a 7 = , find the value of a. G = 0 7 < < 0 7 = learly a is an even integer. 7 8, 7, 6 7 6, 8 7 (mod 10) a = 6 x xy y x y 1 If x and y are real numbers satisfying xy 1 find the value of x. Reference: 010 FI1., 01 FI. Let t = x + y, (1) becomes (x + y) (x + y) = 1 t t = 0 (t + 1)(t ) = 0 t = 1 (rejected) or t = x + y = and xy = 1 x and y are the roots of u u + 1 = 0 x = Method = 0 = 0 = 1 = ( 1)( + 1) 0 = 6 = 678 The largest integer is. Method (x + )(x + )(x + )(x + ) = (x + 7x + 10)(x + 7x + 1) = Let y = x + 7x (y + 10)(y + 1) = y + y = 0 (y + )(y + 1) = 0 When y = = x + 7x x + 7x + = 0 When y = 1 = x + 7x x + 7x + 1 = 0 = < 0, no solution Sum of roots = 7 and x > y > 0, Page

6 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 G Each side of a square is divided into four equal parts and straight lines are joined as shown in figure. Find the number of rectangles which are not squares. (Reference: 01 FI1.1) Number of rectangles including squares = = 100 Number of squares = = 0 Total number of rectangles which are not squares = = 70 (Figure ) G10 If 0 0 and cos sin =, find the value of cos + sin. Reference: 1 HI0, 1 HI, 007 HI7, 007 FI1., 01 HG cos sin 1 sin cos = sin cos = 0 sin cos = 0 (sin + cos ) sin cos = 0 tan tan + = tan = or 6 6 When tan = 1, sin =, cos = 1, 6 Original equation LHS = cos sin = = 1 (reject) 6 When tan = 6, sin =, cos 1 = 1 Original equation LHS = cos sin = cos + sin = Method cos > sin < < 0 cos sin 1 sin cos = + = 1, sin = cos = 6 < 0 cos sin cos sin cos sin Page 6

7 nswers: (1- HKMO Heat Events) reated by: Mr. Francis Hung Last updated: 6 February 017 cos sin = cos sin 6 = cos = cos sin cos + sin = Method cos sin 1 sin cos = sin cos = 6 1 = sin cos = cos + sin cos + sin 1 = cos sin cos + sin = or 0 0 cos + sin > 0 1 cos + sin = Page 7