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 1 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 39 6 2    a b . Find the value of a b  . 2) The sum of 2 positive integers is 97. Find the sum of all possible values of their Highest Common Factor. 3) Find the value of 3 3 3 3 3 3 20 21 22 38 39 40       . 4) Reform the recurring decimal 0.3927 into a simplified fraction. 5) The lengths of a triangle are 13, 14 and 15. Find the area. 6) It is known that 6 8 14 3 3 9 x y x y        . Find the value of x y  . 7) Find the number of first 2019 natural numbers that can be divisible by 23. 8) Find the value of 1 1 1 1 1 5 25 125 625 3125     . 9) It is known that x x     2 2 3 . How many possible values of x are there? 10) The interior angles of a triangle are x , y and z . x y z    7 360 . Find the value of z. Section B: 11th to 20th Question (Each carries 5 marks) 11) By solving the following system of equations 5 2 5 2 3 7 6 5 2 5 2 5 2 3 x y x y x y x y              , find the value of 3x. 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. 12) There are one 1g, one 3g, one 6g and one 10g mass. How many different weight can be measured? 13) It is known that real numbers x and y satisfy 2 x y     2019 ( 2017) 0 . Find the value of         2 3 19 20 x y x y x y x y x y           . 14) If x and y are positive integers, xy x y    7 6 6 . Find the number solutions of ( , ) x y . 15) It is known that prime numbers m and n satisfy 7 5 129 m n   . Find the sum of the possible values of m n  . 16) Find the sum of all prime factors of 2025023. 17) It is known that 6 13 a b ab       . Find the value of 3 3 a b  . 18) Refer to the figure below, AC  5, AB 12 , DEFG is a square. DF  4 and DF AB / / . Find the shortest distance between F and BC. (Please represent your answer in surd form) B Question 18 19) It is known that 2 2 a b   2465 and b a  . Given that ( , ) (8,49) a b  is one of the groups. Find another group of the solutions. G D A E C F 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. 20) Find the value of 2 3 4 17 18 19 ... 1 2 3 2 3 4 3 4 5 16 17 18 17 18 19 18 19 20                   . Section C: 21st to 25th Question (Each carries 7 marks) 21) The perimeter of a rectangle is x and the area of x  2019 . Find the sum of all positive values of x. 22) AB AC  , AB  5 and BC  8 . Find the radius. Question 22 23) Find the last 2 digits of 111 (11) . 24) When a 2019-digit number 777...777 is divided by 13, find the value of the remainder. 25) It is known that x and y are integers, 2017 2019 1 x y   . Find the solution of ( , ) x y . ~ End of Paper ~ A B C