TERRY CHEW INSTITUTE OF MATHEMATICAL COLYMPIADS ân TCIÀACO A2 ) 3 S EA M O DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED Southeast Asian STUDENT''''S NAME Mathematical Olympiads m Read the instructions on th[.]
4 If ) and * are distinctive primes and QUESTIONS TO 10 ARE WORTH MARKS EACH Let ! be a number such that ! + Find the value of ! ' + # $( # $ ! + − )! + * = has distinctive positive integral roots, find + / = (A) (B) (A) 165 (C) 13 (B) 170 (D) 17 (C) 180 (E) None of the above (D) 198 (E) None of the above Evaluate # 23456 789 + # 2346: 789 Find the value of ! that satisfies + # 2345: 789 !+ − ! + !+1 + + =1 + ! +2 ! −!+1 (A) -1 (A) -2 (B) (B) -1 (C) (C) (D) (D) (E) (E) None of the above None of the above Given that ;= sin ; + cos + sin find the value of − cos + sin + sin (A) ; (B) 1−; (C) ; (D) 1+; (E) None of the above SEAMO 2018 Paper F © TCIMO Given that O+ + P+ + Q + = 1, find the Two circles with centres D and E are externally tangent at point F range of OP + PQ + QO (A) [1, 2] U0, V (C) U− , 1V (B) (D) [0, 1] (E) None of the above GH is tangent to the two circles Given that the radii of the two circles are cm and cm, respectively, find IJ KJ It is known that W(!Y ) = Y[# , \ ∈ ^ +Y[# Then W(!Y ) is (A) √3 (A) strictly decreasing (B) √5 (C) monotonic decreasing (C) √5 (E) None of the above (D) √6 (E) None of the above (B) non-strictly decreasing (D) strictly increasing 10 A network of domestic flights is set up in the country of Utopia A city is directly connected to at most others and there is at most transit flight from one city to another What is the When new classrooms were opened, maximum number of cities in Utopia? the average class size reduced by 10 When new classrooms were opened, (A) (B) (C) (D) (E) the average class size reduced by 16 Find the total number of students (A) (B) (C) (D) (E) 320 360 400 480 None of the above 10 11 12 None of the above SEAMO 2018 Paper F © TCIMO QUESTIONS 11 TO 20 ARE WORTH MARKS EACH 13 Given that OPQd ≠ 0, Q and d are roots of ! + + O! + P = and O and P are roots of ! + + Q! + d = 0, then (O + P + Q + d)+ 11 tan can also be expressed as (A) tan − 20 tan' + tana − 15 tan+ + 15 tanb − tanc (B) sin − 20 tan' + cos a − 15 sin+ + 15 cosb + tanc (C) cos − 20 tan' + sina − cos+ + 15 sinb + tanc (D) tan − 15 tan' + tana − 20 tan+ + 20 tanb − tanc (E) None of the above 12 In the figure below, ∠EDG = 90° and is (A) O+ P+ (B) O+ Q + (C) 2OPQ (D) 2OPQd (E) None of the above 14 Find the value of ! that satisfies ∠EDDi = ∠GDDi = 60° k! − + 2√! − + k! + + 4√! − = (A) (B) (C) (D) (E) Find the length of DF i (A) √15 (B) √17 (C) √23 (D) √31 (E) None of the above SEAMO 2018 Paper F © TCIMO None of the above 15 A biased coin is tossed times 17 Evaluate There is a non-zero probability of 2+ 3+ 4+ \+ + + + ' + b +⋯+ Y + ⋯ 2 2 getting heads or tails once The probability of getting heads once is the same getting as the heads probability twice Find of the probability of getting heads thrice 23 32 33 64 40 243 47 256 (A) (B) (C) (D) (E) (A) (B) (C) (D) (E) None of the above 18 It is known that the tangent from a point n to the circle ! + + ; + = is None of the above perpendicular to the tangent, also from the point n to ! + + ; + = Then the locus of n is a circle of radius 16 Find the remainder upon dividing the (A) following by (B) #op 10 (A) (B) (C) (D) (E) #oq + 10 #o( + 10 #opr + ⋯ + 10 (C) (D) 4 None of the above (E) None of the above SEAMO 2018 Paper F © TCIMO 19 Given Y sY = t vw# u (2\ − 2u + 1)(2\ − u + 1) QUESTIONS 21 TO 25 ARE WORTH MARKS EACH 21 A number sequence is such that Each term is divisible by and and each term is the difference between a Y xY = t u square number and 1, as shown below vw# 3, 15, 24, 48, 63, … then Find the remainder when the 2018}~ (A) sY > xY (B) sY ≥ xY (C) sY = 2xY (D) sY = xY (E) None of the above term is divided by 1000 22 Given that !# and !+ are real roots of 2! + − 4n! + 2n+ + 3n − = 0, find the value of n so that the value of !#+ + !++ is minimum 20 A circle of radius 17 is inscribed in triangle DEF such that it is tangent to side DE at point It is known D = 23 Find the value of \ in 19 , E = 21 , find the perimeter of 133a + 110a + 84a + 27a = \+ ∆DEF 24 Suppose O+ + P+ = 2019Q + , where a, b and c are sides of a triangle and a, b, g are its angles (A) (B) (C) (D) (E) Find the value of 183 184 185 186 None of the above SEAMO 2018 Paper F â TCIMO 3} ầ 3} ẫ[3} Ñ 25 In ∆DEF , it is known that DE = 12 , EF = 13 and DF = 15 is a point on DF such that the incircles of ∆DE and ∆EF have equal radii Let n and \ be positive prime integers such that ÖÜ JÜ relative = Y Find n + \ End of Paper SEAMO 2018 Paper F © TCIMO SEAMO 2018 Paper F – Answers Multiple-Choice Questions Questions to 10 carry marks each Q1 D Q2 D Q3 C Q4 A Q5 b Q6 D Q7 B Q8 C Q9 C Q10 B Questions 11 to 20 carry marks each Q11 A Q12 C Q13 A Q14 A Q15 C Q16 B Q17 A Q18 B Q19 C Q20 E Free-Response Questions Questions 21 to 25 carry marks each © SEAMO 2018 21 22 23 24 25 783 144 1009 45 Paper F ... students (A) (B) (C) (D) (E) 320 360 400 480 None of the above 10 11 12 None of the above SEAMO 2018 Paper F © TCIMO QUESTIONS 11 TO 20 ARE WORTH MARKS EACH 13 Given that OPQd ≠ 0, Q and d are... Paper SEAMO 2018 Paper F © TCIMO SEAMO 2018 Paper F – Answers Multiple-Choice Questions Questions to 10 carry marks each Q1 D Q2 D Q3 C Q4 A Q5 b Q6 D Q7 B Q8 C Q9 C Q10 B Questions 11 to 20... Questions 11 to 20 carry marks each Q11 A Q12 C Q13 A Q14 A Q15 C Q16 B Q17 A Q18 B Q19 C Q20 E Free-Response Questions Questions 21 to 25 carry marks each © SEAMO 2018 21 22 23 24 25 783 144 1009