Paper F - 2020 [PDF]

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4. Find the sum of all possible positive integers 𝑛 such that the expression below is an integer.



QUESTIONS 1 TO 10 ARE WORTH 3 MARKS EACH 1. Find the last two digits of 112020. (A) (B) (C) (D) (E)



4𝑛 3 − 16𝑛2 + 29𝑛 + 60 2𝑛 − 3



01 41 71 91 None of the above



(A) (B) (C) (D) (E)



2. The quadratic equation



42 69 75 81 None of the above



𝑥 2 − 52𝑥 + 𝑘 = 0 5. Evaluate the sum



has roots that are prime numbers.



𝑆=



Find the maximum value of 𝑘. (A) (B) (C) (D) (E)



520 576 640 667 None of the above



+⋯+



(A)



1 2



(B)



210 421



(C)



105 211



3. Let 𝑓(𝑥) = 𝑥 2 + 2020𝑥 + 20. How many ordered pairs of positive integers (𝑚, 𝑛) are there such that 𝑓(𝑚 + 𝑛) = 𝑓(𝑚) + 𝑓(𝑛)? (A) (B) (C) (D) (E)



20 1 + 202 + 204



(D) 1 (E) None of the above



1 2 3 4 None of the above



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1 2 3 + + 1 + 12 + 14 1 + 22 + 24 1 + 32 + 34



1



6. 2 white, 3 black and 4 grey marbles are shared equally among 9 students.



9. How many positive integers less than 2020 with the property that the sum of its digits equals 9?



Find the number of ways the marbles can be distributed so that Bran and Sansa gets the same colour and Arya gets a grey marble. (A) (B) (C) (D) (E)



(A) (B) (C) (D) (E)



120 130 140 150 None of the above



50 100 102 202 None of the above



10. The sequence {𝑎𝑛 } is defined by 𝑎𝑛+2 =



7. Given that 𝑎 is a real number such that 𝑎4 + 𝑎3 + 𝑎2 + 1 = 0.



with 𝑎1 = 1 and 𝑎2 = 2.



Evaluate 𝑎2020 + 2𝑎2010 + 3𝑎2000.



Evaluate 𝑎2020.



(A) (B) (C) (D) (E)



(A) (B) (C) (D) (E)



2 4 6 8 None of the above



11. Find the smallest prime factor of



1 1 1 =𝑏+ =𝑐+ 𝑏 𝑐 𝑎



1000 … 01 ⏟ 2020 𝑧𝑒𝑟𝑜𝑠



What is the largest possible value of 𝑎𝑏𝑐? (A)



(A) (B) (C) (D) (E)



1 2



(B) 2 (C)



1 2 3 4 None of the above



QUESTIONS 11 TO 20 ARE WORTH 4 MARKS EACH



8. Given that 𝑎. 𝑏 and 𝑐 are three distinct real numbers such that 𝑎+



1 + 𝑎𝑛+1 𝑎𝑛



3 5 7 11 None of the above



5 2



(D) 3 (E) None of the above



2



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12. In the expansion of



15. Given that 𝑎 and 𝑏 are real numbers satisfying



𝑓(𝑥) = (1 + 𝑎𝑥)4 (1 + 𝑏𝑥)5 2 2 { 6 − 5𝑎 + 4𝑏 − 3𝑎 + 2𝑎𝑏 − 𝑏 = 0 𝑎−𝑏=1



where 𝑎 and 𝑏 are positive integers, the coefficient of 𝑥 2 is 66.



Find the sum of all possible values of 30𝑎



Evaluate 𝑎 + 𝑏. (A) (B) (C) (D) (E)



𝑏



2 3 4 5 None of the above



(A) (B) (C) (D) (E)



13. The equation 𝑥 3 − 𝑎𝑥 2 + 𝑏𝑥 − 2020 has three positive integer roots.



. −15 −10 15 30 None of the above



16. The figure below shows a 5 × 6 rectangular board with a missing 1 × 2 rectangle in the center.



Find the least possible value of 𝑎. (A) (B) (C) (D) (E)



101 110 202 220 None of the above How many squares are there in the board?



14. Evaluate the sum 𝑆 = sin2 0° + sin2 2° + sin2 4° + ⋯ + sin2 180° (A) (B) (C) (D) (E)



(A) (B) (C) (D) (E)



80 81 88 90 None of the above



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3



14 30 54 56 None of the above



17. In ∆𝐴𝐵𝐶,



20. 𝐴, 𝐵, 𝐶 and 𝐷 are four distinct points lying on the circumference of a circle such that chords 𝐴𝐵 and 𝐶𝐷 are perpendicular at point 𝐸.



(sin 𝐴 + sin 𝐵) ∶ (sin 𝐵 + sin 𝐶) ∶ (sin 𝐶 + sin 𝐴) = 19 ∶ 20 ∶ 21



Given that 𝐸𝐴 = 4, 𝐸𝐵 = 2 and 𝐸𝐶 = 6 , find the radius of the circle.



Find the value of 99cos 𝐴. (A) (B) (C) (D) (E)



39 41 51 60 None of the above



18. Find the least positive integer 𝑛 such that



the



equation



10𝑛



⌊



𝑥



⌋ = 98 has



integer solution 𝑥. ⌊𝑘⌋ is the largest integer smaller than or equal to 𝑘. (A) (B) (C) (D) (E)



(A) (B) (C) (D) (E)



3 4 5 6 None of the above



√270 18 None of the above



QUESTIONS 21 TO 25 ARE WORTH 6 MARKS EACH 21. You need to tile a 10 × 1 hallway with a supply of 1 × 1 red, 2 × 1 red tiles and 2 × 1 blue tiles. Find the number of ways you can tile the 10 × 1 hallway.



19. How many positive integers 𝑘 < 100 such that 2(56𝑛 ) + 𝑘(23𝑛+2 ) − 1 is divisible by 7 for any positive integer 𝑛? (A) (B) (C) (D) (E)



√221 15



12 14 18 19 None of the above



22. 𝑥, 𝑦 and 𝑧 are real numbers such that 𝑥+𝑦+𝑧 = 7 {𝑥 + 𝑦 2 + 𝑧 2 = 19 𝑥 3 + 𝑦 3 + 𝑧 3 = 64 2



Evaluate 𝑥 4 + 𝑦 4 + 𝑧 4.



4



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23. In ∆𝐴𝐵𝐶 shown below, 𝐴𝐷, 𝐵𝐸 and 𝐶𝐹 intersect at 𝑃 . Suppose 𝐴𝑃 = 𝑎, 𝐵𝑃 = 𝑏, 𝐶𝑃 = 𝑐 and 𝐷𝑃 = 𝐸𝑃 = 𝐹𝑃 = 𝑥. Given that 𝑥 = 3 and 𝑎 + 𝑏 + 𝑐 = 20 , find 𝑎𝑏𝑐.



24. Positive integers 𝑎, 𝑏 and 𝑐 randomly selected from the {1,2,3, … ,2020} with replacement.



are set



Find the probability that 𝑎𝑏𝑐 + 𝑎𝑏 + 2𝑎 is divisible by 5.



25. 𝐴𝐵𝐶𝐷 is a convex quadrilateral such that 𝐴𝐶 intersects 𝐵𝐷 at 𝐸 . 𝐻 is a point lying in the segment 𝐷𝐸 such that 𝐴𝐻 is perpendicular to 𝐷𝐸. Suppose 𝐵𝐸 = 𝐸𝐷, 𝐶𝐸 = 9, 𝐸𝐻 = 12, 𝐴𝐻 = 32 and ∠𝐵𝐶𝐴 = 90° . Evaluate the length of 𝐶𝐷.



End of Paper



SEAMO 2020 Paper F © SEAMO PTE LTD



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SEAMO 2020 Paper F – Answers Multiple-Choice Questions Questions 1 to 10 carry 3 marks each. Q1 A



Q2 D



Q3 D



Q4 B



Q5 B



Q6 C



Q7 C



Q8 E



Q9 C



Q10 A



Questions 11 to 20 carry 4 marks each. Q11 D



Q12 B



Q13 B



Q14 E



Q15 A



Q16 D



Q17 C



Q18 B



Q19 B



Q20 A



Free-Response Questions Questions 21 to 25 carry 6 marks each.



© SEAMO 2020



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