Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit. O L Y M P I A D C H A M P I O N E D U C A T I O N C E N T R E Room 309-310, 8 Jordan Road, Yau Ma Tei, Kowloon, Hong Kong SAR, CHINA Tel (852) 3153 2028 / 9310 1240 Fax (852) 3153 2074 Website: www.olympiadchampion.com Email: olympiadchampion@gmail.com GUANGDONG-HONG KONG-MACAO GREATER BAY AREA MATHEMATICAL OLYMPIAD 2019 (GREATER BAY AREA REGION) Secondary 3 Question Paper Time allowed: 75 minutes Instructions to Contestants: 1. Each contestant should have ONE Question Book which CANNOT be taken away. 2. There are 3 sections in this exam. Section A consists of 10 questions. Each carries 4 marks. Section B consists of 10 questions. Each carries 5 marks. Section C consists of 5 questions. Each carries 7 marks. The total number of questions is 25. Total score is 125 marks. No points are deducted for incorrect answers or empty answers. 3. NO calculators can be used during the contest. All figures in the paper are not necessarily drawn to scale. 4. This Question Book will be collected at the end of the contest. THIS Question Book CANNOT BE TAKEN AWAY. DO NOT turn over this Question Book without approval of the examiner. Otherwise, contestant may be DISQUALIFIED. All answers should be written on the ANSWER SHEET. Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit. Section A: 1st to 10th Question (Each carries 4 marks) 1) It is known that x is rational, x  0 and 4 4 3 3 ... x    . Find the value of x. 2) If a, b are integers such that a b  and 45 4 10    a b , find the value of a b  . 3) If x  0 and 1 x 6 x   , find the value of 3 3 1 x x  . 4) Find the value of 1 1 1 1 1 1 289 288 288 287 287 286 4 3 3 2 2 1             . 5) It is known that tan 5 x  . Find the value of sin cos x x . 6) In ABC , it is known that   ABC 75 and   ACB 45 . Find the value of 2 ( ) BC AB . 7) If x, y are positive integers and xy x y    7 5 65 , find the number of solutions of ( , ) x y . 8) It is known that the ratio of 8 exterior angles is 1: 2:3:...:6:7:8 . Find the value of the smallest interior angle. 9) Find the minimum value of 2 x x   10 1000. 10) If x 17 (mod23) , find the minimum value of 3-digit odd number x. All answers should be written on the ANSWER SHEET. Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit. Section B: 11th to 20th Question (Each carries 5 marks) 11) It is known that x is a real number. Find the minimum value of xxx      4 7 11 . 12) The radius of a sector is 3 and its perimeter is 18. Find the area of this sector. 13) Find a natural number such that it adds 34 or minuses 34 to become a perfect square number. 14) Find the maximum value of x x    10 34 . 15) In right-angled triangle BAD,   BCA 60 and AC CD  . Find the value of sinADC . Question 15 16) Find the area of a quadrilateral with vertices A B C D ( 3,2), ( 1, 4), (3,1), (5,5)    in a rectangular coordinate system. 17) If  and  are the roots of 2 x x    15 23 0 , find the value of 3 3 2 2        . B A C D All answers should be written on the ANSWER SHEET. Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit. 18) It is known that x is real, find the maximum value of 2 2 23 6 20 x x  6  . 19) A, B, C, D and E are passing balls. Starting from A, 5th time pass will return back A. Within the pass, there is at least 1 time to A, how many passing methods are there? 20) There are 2019 balls numbered 1 to 2019 inside a box. How many balls are to be chosen such that there are at least 3 consecutive numbers? Section C: 21st to 25th Question (Each carries 7 marks) 21) In a rectangular coordinate system, we draw 2019 line segments. Each line is parallel to x-axis or y-axis. The quotient between those 2 numbers is a positive number. How many parts can at most be cut by these straight line segments? 22) There is at least 2 difference in each interior angle in a convex n polygon. When n reaches the maximum value, find the smallest value of the exterior angle. 23) Inside a regular 2019-sided polygon, the distance between each vertex and the center is 1. Find the perimeter of the polygon. (Answer to the nearest integer) 24) If ( 1) ( 2) ( 3) 3 2 1 ( ) ( ... )( ( 1)) x x x x f x x x x x x x x f x             , find the value of f (2019) . 25) Now there is a special calculator with only 4 buttons. “AC” is to make number to be 0. The other 3 buttons are 1, 3 and 5 . After pressing “AC”, how many times are required at least to show 2019? ~ End of Paper ~