SEAMO Paper C [PDF]

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SEAMO



SOUTHEAST ASIAN MATHEMATICAL OLYMPIADS



Exclusive Publisher& Distributor



DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED.



STUDENT'S NAME:



2010 ANSWER SHEET and lill in your NAME, SGHOOL and 0THER lNF0RMAT|0N. Use a 28 or B pencil Read the instructions 0n the



Do NOT use a pen Rub out any mistakes completely.



You MUST record your answers on the ANSWER SHEET.



UPPER PRIMARY



Mark only ONE answer for each question. Marks are N0T deducted for incorrect answers



SECTION A Use the information provided to choose the BEST answer from the five possible options



|lrilH



ANSWER SHEET fill in the oval that matches your



SECTION B



sHEET fill in your answer within the box



3l]fiilu*twER You are



N0T allowed to use a calculator.



1.



4. Mr. Wang met with heavy traffic on the



Find the value of



expressway As a result, the speed of his car



100-98+96-94+'..+ B- 6+ 4-2



was reduced by



What was the percentage increase in time taken for his



journey?



(A) 46 (B) 48



(c)



2.



(A) 70o/o (B) I5o/o



50



(D) (E)



52



(C) 20o/o (D) 25o/o (E) None of the above



54



The missing number in the sequence is



2Oo/o.



following



_.



4, 6, 10,'J,4,22,26,34,



(



), 46 ...



Find the number of ways in which you could



A to point B by passing through point X in the diagram shown travel from point



(A) 36 (B) 37



(c) (D) (E)



:



You can only move down and to the right.



38 3e



A



40



X



ln the pentagon below, AB : DE



below.



BC



:



CD



:



AE. Find zy.



*



(A) (B)



(c) (A)



106'



(B)



1080



(c)



110"



(D)



1.72"



(E)



None of the above



(D) (E)



6 2



16 17 18



1s None of the above



6. Jane forgot her pencil case when



she



8.



A bag contains 8 red, 7 white, 5 yellow,



walked to school, so her brother cycled to



blue and 2 black balls. Without looking Roy



give it to her. After receiving it, she took



takes out the balls one by one. \Nhat is the



4 more



minutes to reach school.



least number of balls he must take out so



Her brother reached home at the same time. lf cycling is 3 times faster than



that, for certain, he will have at least 4 balls



another



of the same colour?



walking, how many minutes did Jane take to



@



walk to school from home?



(A) (B)



16



(c)



18



(D) (E)



14



20 None of the above



(A) (B)



12



(c)



13



(D) (E) 7.



3



f, n



11



14



None of the above



Two dice are rolled at the same time. Wfrat is the chance that the sum of the 2 numbers



on the up faces is 8?



9.



The average age of m number of teachers



n number of engineers are 32 and 40, respectively. The average age of the same and



group of teachers and engineers is 35. Find (m + n).



(A) (B)



(c)



(A) ; (B) +



(c) (D) (E)



(D) (E)



i s 36



+,



2016) 3



4 6 8



10



None of the above



10. A fast food outlet sells chicken nuggets



in



13. Find the value of



boxes of either 4 or 7. \Mtat is the largest



771



number of nuggets that one cannot buy?



(A) (B)



-+-*-+...* 2 2xZ 2xzxz



2x2x2x2x2x2x2x2



11



17



(c)



56 (A) rn



1e



(B)



(D)



22



(E)



None of the above



727 728-



/^\ (t/



/n\ \" t



(E)



11. The figure below shows 3 squares and two



l2B



:LJO



255



2s6



None of the above



circles. Find the area of the smallest square in cm2.



In IABC, points D, E and F are midpoints of



14 cm



CE, AF and BD respectively. lt is known the



area of AABC is 56 cm2. Find the area of 14 cm



(A) (B)



(c) (D) (E) 12.



^DEF.



46 47 48 4e None of the above



On the number line shown below, 2a:3b -



8. Find the value of (2c + d).



(A) (B)



bc (A)



15



(B)



16



(c)



17



(D)



18



(E)



None of the above



5



6



(c) 7 (D) (E)



2016) 4



8



None ofthe above



15. Find the 2016th digit after the decimal in



|.



18. Evaluate 20L6



20762 (A)



(A) (B) (c) (D) (E)



(B)



(c) (D) (E)



16.



Let r, s, t, rr be



whole



(c) (D) (E) 17.



1



2015 2016 2017 None of the above



does



John cut away one sixth of a pizza. He realized the curved circumference of the remaining pizza is 15n inches. What was the diameter of the pizza in inches?



4 6 8



10



None of the above



Find the number of triangles that can be



formed by using any



3 points as their



vertices.



(A) (B)



(c) (A)



20



(B)



25



(c)



30



(D)



35



(E)



None of the above



2015 x2OI7



numbers. lf



2'x3s x 5t x 7u =252, then what r * 2s + 3t + 4u equal to?



(A) (B)



-



(D) (E)



10 12



14 16 18



20. There are 20O kilograms of oatmeal in a



supermarket. On



the 1"t



day,



oatmeal was sold. On the 2nd



j of the



day,iof



the



remaining oatmeal was sold. On the 3'o ,1 of the remaining oatmeal was sold. day,; This pattern went on until on the 199th day, 1



where



,*



of the remaining oatmeal was sold.



21. Aloysius, Barry, Carl, Dylan and Edward are participating in an international chess competition, where each contestant must play exactly one game against each other. Aloysius played 4 games.



Find the amount of oatmeal, in kilograms,



Barry played 3 games.



that was left.



Carl played 2 games. Dylan played 1 game. How many games has Edward played so far?



22. The digits below can form 24 dlflerent four-



digit numbers. Find the average of these 24 numbers.



2,5,7 and



I



23. The figure shows 2 blue circles and 3 blue (A)



T



(B)



2



(c)



3



(D)



4



(E)



None of the above



semicircles, all of identical radii, inscribed in



a big semicircle. Find the ratio of the area of the big semicircle to the area of the area of the blue regions.



24. The following four-digit numbers are similar to each other in some ways.



1383 1996 Firstly, they all start with



1231



1.



Secondly, there are 2 identical digits in each number.



How many such numbers are there?



25.



A



triangle can be formed with sides of



lengths 3, 4 and 5. lt is impossible, however,



to construct a triangle with sides of lengths



3, 4, and 7. Jane has 8 sticks, each stick having a different length, which are whole numbers. She observes that she cannot form a triangle using any 3 sticks as the sides.



What is the shortest possible length of the longest stick Jane has in cm?



2016t7



The answers in this publication are provided by TCIMO. Any queries or comments on the answers should be forwarded to TCIMO directly: [email protected]



- 98 = 96 - 94 =... = 4 L00+2+2=25pairsof2. 2x25=50 100



2=2



Divide the number sequence by 2, we have 2, 3, 5, 7, !1, 13, 17, 19,23 (Prime numbers) 79 x2=38



There are 3 As



3x1B0o:540o 540o+5:108o Ratio of speeds = 100 : 80



:5:4



Ratiooftime=4:5



5-4x



7o0o/o:25o/o



4



.t



I t2



& 4x4=16mins



=6x6



S-ample space



=36 7



D



Desired event



P(Sum is 8;



8



9



E



C



=



(2,6)(3,5) (6,2)(5,3)



:



t5



r



By the pigeonhole



theory



2+3+3+3+3+1=15



32nt+a0n;-ji.(m+n) :35m t 35n 3m=5n 7y1-zn=j:J m



ln:8



Ml: 10



B



Use formula a x b



:4x7 -(4+7) :28 - 7L



:L7



M2: Make a list



t1



D



1,4x14=796 196



+2:98



98+2=49-



2a:



S{q#4) - 8



i$a -! i.' -..



.. ''



i-"



:: e,3a



t2



c



(4,4)



r=



$'., 'l



..



-?



;



a:2 b:4 c:5 d:7 2c*d:2x5+7 =17



-



(a+b)



\\



I 21= L,22 :4,23 : B, ......28 :256 Theoriginalexpression: + + 1+* 248



:L-



*...* :zto



1



256



255 256



1t IJ



D



I



ns I 256



l4



D



C



There areT As of equal areas.



56+7:8cm2



l5



D



:0.7'l-4285 2076 + 6 :336 5



+



7



...



[these six numbers repeatJ



6) 10



1



For the product to end with "2", t must be "0"



252



l6



D



252:2 x2x3 x3 x7 =22X3zX7t



f:2 t=0 s=2



u:1 2+2x2+3xO+4x =21-4+0+4 =10 Iine



Iine 2 Using line 1 as base,



t7



C



sCz:3 aCt: 4



3x4=1,2Ls Using line 2 as base,



aCz=(4x3)/ QxZ)=6 3Cr=3



x 3 = 1BAs L8+L2=30 6



2076 20762 _ (2076 _



l8



C



r) (2076 +



2076 201,62_(20162_7) 2076



20!62-20762+1



:2076



2016t



11



1)



1



l9



E



300/360x2r.r=75n 5/62 x 2r*r = \5 x



r:9 d:2x9 =18 200



20



x



A



=



200



'1.rr't (1-) x (1-f *r1.23 ri * +,



x (1-7) x ...



t



...



x



(1--200)



199 ZOO



=l



A



27



Edward has played 2 games.



2+5+7l-B:22 22



6l10.5



22+4=5.5' 5.5 x 1000



*



5.5 x 100



+



5.5 x 10



*



5.5



5.5 x (1111)



= = 6110.5



Let the radius of the big semicircle be 21 cm, the radius of the small



circle is 7 cm



!x2L2 xn _2



-z3



9:7



;X /"Tt



27x27



=-7x7x7 9 7



are2" I's"



Case 1 : There



TT tt 79



T t B



There are 3 positions For 2"d" I" 24



432



9xBx3=2L6 Case 2: There is



only one " L"



There are 3 positions For the identical number



9xBx3=216 2L6+276=432



Smaller A that cannot be formed:



(7,2,3) 25



34



Next, (2, 3, 5)



Followed by (3, 5, 8)



7,2,3,.5,8,73,21,34



13



TC M O I



I



4tli'EflHs



bEffi,lEFx;rit'rurE



SEAMO



SOUT}IEAST ASIAN MATHEMATICAL OLYMPIADS



Exclusive Publisher& Distributor



DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED.



STUDENT'S NAME:



2817 Read the instructions on the ANSWER SHEET and



fill in your



NAME, SCHOOL and OTHER lNF0RMAT|ON. Use a 28 or B pencil. Do NOT use a pen. Rub out any mistakes completely. You MUST record your answers on the ANSWER SHEET.



UPPER PRIMARY



Mark only 0NE answer for each question. Marks are N0T deducted for incorrect answers



SECTION A Use the information provided to choose the BEST answer from the five possible options.



sHEET fill in the oval that matches your



|*il?t;:ANSWER SECTION B 0n your,ANSWER SHEET fill in your answer within the box provided. You are



N0T allowed to use a calculator.



4. A car travelled at 40 km/h for the first



2



hours. lt travelled at 60 km/h for the last 3 hours. What was its average speed?



1. A 3-digit number is such that it is equal to 19 times the sum of its digits. \Mtat is its



(A) s0 (B) 51



largest possible value?



(c)



(A) (B)



(D) (E)



114 133



(c)



152



(D) (E)



3ee None of the above



52 54 None of the above



Mark fills in each circle with a number from 1,2,3, ... 8, such that the sum of numbers at all corners of any triangle is 12. 2.



A rope 580 cm long is to be cut into 40 cm and 90 cm segments without any wastage.



Find(a+[+c+d).



How many ways are there to do this?



(A) (B) 2 1



(c) (D) (E)



3.



3 4 5



A new operation is defined as



(A) (B)



2@4:2*3+4*5:14, 5O3=5*6+7:18 Find the value of



(c)



minme- 7 = 49.



1



3



(D) (E)



4 s



11 L2 13



(D) (E) t4



(A) (B) 2



(c)



10



171 2



6.



Lines AC and BD divide the quadrilateral ABCD into 4 triangles of different areas. Given that BE:Pg=22 1 and AE: f,g = 1: 3, find the ratio of the areas AADE I ABCE.



9. < n > denotes



the smallest whole number that is not a factor of n. For example,



17 )= 2, is the midpoint of arc 12.



If Cindy spends $20 and Diane spends $10 a day, Cindy will have $3500 remaining when Diane finishes up her savings.



Given that the area of the shaded region 3 is 12 cmB , find the area of the shaded region C in D!B .



If Cindy spends $10 and Diane spends $20 a day, Cindy will have $3950 remaining when Diane finishes up her savings. How much money does Cindy have at first?



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



12 14 12E 14E None of the above



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



14. A bag contains 4 red and 4 black balls. When 3 balls are chosen at random, without replacement, the probability of getting 2 red balls and 1 black ball



12. 60% of the students at North Shore School were boys.



is



44% of the students took part in the sports meet, of which 52% were boys.



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



60% 64% 68% 72% None of the above



SEAMO 2019 Paper C © TCIMO



F . G



Find (! + #).



What percentage of the North Shore girls did not participate? (A) (B) (C) (D) (E)



$3900 $4100 $4300 $4500 None of the above



3



5 6 7 8 None of the above



15. At least how many numbers from 18. Find the value of H, when (H − I) is 1 to 30 must be chosen to ensure minimum. there always exists a number that is twice another? (A) (B) (C) (D) (E)



17 18 19 20 None of the above



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



573 575 577 579 None of the above



16. Find the integer part of 1 1 1 1 1 1 + + + + 71 72 73 74 75 (A) (B) (C) (D) (E)



19. Pipe A takes twice as long to fill a pool as compared with Pipe B.



14 13 12 15 None of the above



17. An equilateral triangle of side length 1 is rotated pivoting at >. Then it is turned at point 2. Find the distance travelled by the point 1.



It takes them 12 hours to fill a pool when turned on together. If Pipe A alone is turned on for ! hours and then turned off, Pipe B will take another 9 hours to fill the pool. Find !.



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



J E K L



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



E



K K E M N E M



14 16 18 20 None of the above



None of the above



4



SEAMO 2019 Paper C © TCIMO



20. Given that H = 5MO , I = 3KB , D = 2OP



24. The sum of a whole number # and 125 is a square number.



Which of the following statements is true?



The sum of # and 176 is also a square number.



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



Find the value of #.



H>I>D I>D>H I>H>D D>H>I None of the above



25. Evaluate 1 3 7 29 37 41 53 29 3 + + + + + + + + 7 8 36 56 63 72 77 84 88



QUESTIONS 21 TO 25 ARE WORTH 6 MARKS EACH 21. Find the remainder if the following expression is divided by 11.



2 TUUUUUUUVUUUUUUUW × 2 × 2 × …× 2 × 2 × 2 BXPO



YYYYY is a 3-digit number with no 22. HID repeated digits. Given that YYY + YYY YYY = HID YYYYY HI IH + HD YYY + DH YYY + YYY ID + DI the sum of digits of YYYYY HID are multiples of ____.



23. At 9:00 AM, Cars A and B left Towns X and Y, respectively, and travelled towards each other with their speeds in the ratio 5 : 4. After the two cars passed each other, Car A’s speed reduced by 20% while Car B’s speed increased by 20%. Given that Car B was still 10 km away from Town X when Car A reached Town Y, find the distance between the towns. End of Paper SEAMO 2019 Paper C © TCIMO



5



!



YOU MAY USE THIS PAGE FOR ROUGH WORKINGS







Section A For questions 1 to 10, each correct answer is awarded 6 marks. 1. How many squares does a straight-line cross in the 2019th figure?



1



© SEAMO X 2019







2. A pencil costs $5, while a pen costs $3. John paid $45 for ! number of pencils and " number of pens. Find 2 sets of values of ! and ".



3. A balancing machine has the following weights: 1g, 2g, 4g, 8g, 16g The weight(s) can only be placed on one end of the machine. How many measurements altogether can be taken?



2



© SEAMO X 2019







4. Maomao’s old watch is slower by 2 minutes in every hour. At 8 o’clock in the morning she adjusts it to the standard time. What is the actual time when her watch shows 12 noon?



3



© SEAMO X 2019







5. It is known that 5×2=5+6 = 11; 3×4=3+4+5+6 = 18; 6×3=6+7+8 = 21; Evaluate 1 × 9 + 2 × 9 + 3 × 9 + ⋯+ 9 × 9



4



© SEAMO X 2019







6. A circle of perimeter 1m rolls around the square of perimeter 4m.



How many turns does the circle make as it rolls around the square once without slipping?



5



© SEAMO X 2019







7. Alan, Ben and Charles took part in a test. The average test score of Alan and Ben is 2.5 marks higher than the average of the 3 children. The average mark of Ben and Charles is 1.5 marks less than the average of the 3 children. It is known that Ben scored 93. Find the score of Charles.



6



© SEAMO X 2019







8. Cindy has 12 sticks, each of different size. What is the minimum length of the longest stick such that no 3 sticks can form a triangle (in cm)?



9. The profit is 25% when a tennis racket is sold at its labeled price. The number of rackets sold is increased by another 1.5 times when the labeled price is discounted at 90%. Find the increased profit in percentage.



7



© SEAMO X 2019







10. The official date for SEAMO X is 19th January 2019, or 20190119. The sum of all digits of the date is 23. How many days are there in 2019 when the sum of all digits in the date is 23?



8



© SEAMO X 2019







Section B For questions 11 to 15, each correct answer is awarded 8 marks. 11. The figure below shows 2 semi-circles with the diameters lying on the same line 45. Given 67 ∥ 45, 67 = 12 cm, find the area of the shaded region. Take < = 3.14.



9



© SEAMO X 2019







12. 1 −



? @A



B



= B



CD B ED B FD C



? ?A



B



B GD B GD B HDH



Find the value of (J + K + L + M).



10



© SEAMO X 2019







13. 6745 is a square of side length 4 cm. O is the midpoint of 74, 54 = 45P. Find the area of the shaded region.



11



© SEAMO X 2019







14. Mark started jogging from Point 6 to Point 7, and then going back and forth between the two points at a speed of 300 m/min. At the same time, Nina started jogging from Point 7 to 6, and then going back and forth between the two points at 240 m/min. Both of them jogged for 30 minutes. It is known that the distance between Point 6 and Point 7 is 2400 m. a) How many times did they meet? b) Find the distance when their meeting point was nearest to Point 6.



12



© SEAMO X 2019







15. U, V, W are 3 consecutive whole numbers such that U is divisible by 15, V is divisible by 17, W is divisible by 19. Find the values of U, V, W.



13



© SEAMO X 2019







THIS IS A BLANK PAGE



QUESTION



SOLUTION



ANSWER



Fig. 1 : 3 squares à 2 × 2 + 1 Fig. 2 : 5 squares à 2 × 2 + 1 Fig. 3 : 7 squares à 3 × 2 + 1 Fig. 4 : 9 squares à 4 × 2 + 1 1



… … Fig. 2019 : 2019 × 2 + 1 = 4038 + 1 = 4039(Ans) 5> + 3? = 45 5> = 45 − 3?



2



>=



45 − 3? 5



When CDE } when CDJK } FDG FDL FDG FDL Ans: CDE }, CDJK }



M1 16 × 2 − 1 = 31 (Ans) M-



3



1 = 1



8 = 8



2 = 2



9 = 1 + 8



3 = 1 + 2



10 = 2 + 8



4 = 4



…



5 = 1 + 4



31 = 1 + 2 + 4 + 8 + 16



6 = 2 + 4



= 31



7 = 1 + 2 + 4



Ans: 31







Old watch: 58 min in 1 h. Standard time: 60 min in 1 h. 60 × 4 = 240 4



240 ×



60 58



8 min 29 8 = 4 h 8 min 29 = 248



1



© Terry Chew Academy 2018



Ans: 12 noon 8



8 min 29



1 × 9 = 1 + 2 + ⋯+ 8 + 9 = 45 2 × 5 = 2 + 3 + ⋯ + 9 + 10 = 54 3 × 9 = 3 + 4 + ⋯ + 10 + 11 = 63 … 5



… 9 × 9 = 9 + 10 + ⋯ + 16 + 17 = 117 45 + 54 + 63 + ⋯ + 117 =



(45 + 117) × 9 2



= 729 Ans: 729



6



At each corner, the ball tips 90° 90° × 4 = 360° is one round 4 + 1 = 5 rounds Ans: 5 rounds ] − ^ = (2.5 + 1.5) × 2 = 8 marks We have,



7



> + 8 + 93 + > 93 + > − = 1.5 3 2 Where > is the score of Charles Solving > = 86 marks Ans: 86 marks



8



2



In a ∆,







© Terry Chew Academy 2018







e + f > h (1, 2, 3) (2, 3, 5) (3, 5, 8) (5, 8, 13) (8, 13, 21) (13, 21, 34) (21, 34, 55) (34, 55, 89) (55, 89, 144) (89, 144, 233) Ans: 233 Let there be 10 rackets Assume each costs $100 Labeled price = Selling price = 100 × 125% = $125 $125 − $100 = $25 Profit = $25 × 10 = $250 9



Now $125 × 90% = $112.5 10 × (1+1.5) = 25 There were 25 rackets sold Profit = ($112.5 − $100) × 25 = $312.5 $(312.5 − 250) ÷ 250 × 100% = 25% 2+1+9 = 12; 23-12 = 11 The remaining 4 digits (mm/dd) should add up to 11. These dates are:



10



0128, 0209, 0218, 0227, 0308, 0317, 0326, 0407, 0416, 0425, 0506, 0515, 0524, 0605, 0614, 0623, 0704, 0713,



3



© Terry Chew Academy 2018



0722, 0803, 0812, 0821, 0830, 0902, 0911, 0920, 1019, 1028, 1109, 1118, 1127, 1208, 1217, 1236 Total: 34 (excluding example of 0119)



Area shaded



11



1 1 = stu − sv u 2 2 1 = s(tu − v u ) 2 now, 6u + v u = tu tu − v u = 36 =



1 × 3.14 × 36 2



= 56.52 cmu (Ans) Ans: 56.52 Consider right hand side. 1 10 = 3 3 3 23 2+ = 10 10 10 56 2+ = 23 23 23 79 1+ = 56 56 56 RHS = 79 56 LHS = 1 − 79 23 = 79 1 1 = yz = JK 3+ 3+



12



uL



4



uL







© Terry Chew Academy 2018







=



=



1 3+



3+



{| }~



1 3+



1



=



J







| u }~



1



=



J



J



3+



{ u {~ |







J u



{ ÄÅ



{ |



= ⋯ =



1 3+







J u



{ ÄÅ



}



}Å



} {



(v + Ç + É + Ñ) = 3 + 2 + 6 + 1 = 12 Ans: 12



Area ∆ ]ÖÜ =



1 × 4 × 4 2



= 8 cmu 13



Area ∆ ÖáÜ =



1 × 2 × 3 2



= 3 cmu ]à ∶ àá 8 ∶ 3 Area of shaded ∆ ]Öà =4×



8 8+3



32 cmu 11 32 Ans or 2s 11 =



2400 ÷ 300 = 8 min 14



5



2400 ÷ 240 = 10 min



© Terry Chew Academy 2018



As shown in the graph, ^, ä, á are 3 meeting points (a) Ans: 3 times (b) 2nd meeting at ä is the nearest to Point ]. At ä, together they have run 2400 × 3 = 7200 m Time = 7200 ÷ (300 + 240) 40 min 3 40 240 × å − 10ç 3 =



= 800 m Ans: 800 m Find the LCM of 15, 17, 19 15 × 17 × 19 = 4745 4745 + 15 = 4860 is divisible by 15 4745 + 17 = 4862 is divisible by 17 4745 + 19 = 4864 is divisible by 19 15



4860 ÷ 2 = 2430 4862 ÷ 2 = 2431 4864 ÷ 2 = 2432 Ans: e = 2430 f = 2431 h = 2432



6







© Terry Chew Academy 2018











Section A For questions 1 to 10, each correct answer is awarded 6 marks. 1. It is known that 4 ⋇ 2 = 14 5 ⋇ 3 = 22 3⋇5=4 Find the value of 6 ⋇ 9.



2. If * ∶ , = 5 ∶ 3, , ∶ - = 6 ∶ 7 and * ∶ - = / ∶ 0, find the value of (/ + 0).



1



Paper C



















© SEAMO X 2020







3. Ms. Nutcharut has a bag of sweets. 4 4 and another 20 of her sweets. She gave of the 5 6 4 remaining and another 6 sweets to Class Dauntless. She then gave the of the 7



She gave Class Courageous



remaining sweets to Class Elegance. In the end, there were 16 sweets left. How many sweets did Miss Nutcharut have at first?



© SEAMO X 2020



Paper C



2







4. Find the value of 8, where 8 is a whole number, for the inequality to work. 29 < 5; < 3
, ,> = 3,? and ,- = 3=-. Given that the area of ∆ *,- is 81 CDE , find the area of ∆ >?=, in CDE .



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Paper C



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HHHHHHHH is the largest 5-digit value divisible by 12. 6. It is known F724G Find the value of (F + G).



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Paper C



















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7. Given that the value of 1 1 1 1 1 1 1+ + + + + + ⋯+ 3 6 10 15 21 210 is



K , find the value of (D + M). L



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Paper C



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8. There are 3 Super Maths classes in a school. Classes A and B have the same number of students. The number of students in Class C is



N of the Super Maths students. There EO



are 3 more students in Class C than in Class A. How many Super Maths students are there in the school?



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Paper C



















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9. Find the 2020PQ number in 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, …



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Paper C



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10. A group of workers were working on a project outside City Hall. If an additional 8 workers were deployed, the project could be completed in 10 days. If an additional 3 workers were deployed, the project needed 20 days to complete. If an additional 2 workers were deployed, how many days did it take to complete the project?



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Paper C



















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Section B For questions 11 to 15, each correct answer is awarded 8 marks. 11. The figure shows a 3 × 2 CD rectangle. The two quadrants are drawn using vertices C and D as their respective centres. If the difference of area between shaded regions A and B is 0, find the value of 1000. Take U = 3.14.



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Paper C



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12. The road from Town A to Town B consists of an upslope, followed by a downslope as shown. The speed for each part of the journey is shown.



It is known that the time taken from Town A to Town B is 3 hours. The time taken E



from Town B to Town A is 2 5 hours. If the distance between the two towns is W, find 2W.



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Paper C



















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13. Mr. Smith has a 7-digit contact number. The sum of the two numbers formed by the first 4 and last 3 digits is 6861. The sum of the two numbers formed by the first 3 and last 4 digits is 4215. Find the first 3 digits of his contact number.



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Paper C



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X



14. At F Y minutes after 4:00 PM, the minute- and hour- hands form 90°. Given that F and G are both 1-digit numbers and C is a 2-digit number, find the value of (F + G + C).



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Paper C



















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15. In the set of odd numbers {1, 3, 5, 7, 9} , HHHHH FGC represents a 3-digit number where all HHHHH digits are distinctive and W]^ represents another 3-digit number where all digits are distinctive. HHHHH , W]^ HHHHH ` are there where FGC HHHHH − W]^ HHHHH = 198 ? How many pairs of _ FGC



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Paper C



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SEAMO X 2020 Paper C – Answers Section A Questions 1 to 10 carry 6 marks each. Q1



Q2



Q3



Q4



Q5



27



17



75



4



24



Q6



Q7



Q8



Q9



Q10



17



61



120



4



25



Section B Questions 11 to 15 carry 8 marks each. Q11



Q12



Q13



Q14



Q15



28



189



627



21



18



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Paper C



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Practice Set C Problems You are given 90 minutes to complete 15 open-ended questions. Write your answers in the space provided.



1. Find the last digit of 2018%&'( .



2. There are 10 points on the circumference of a circle. How many lines can be formed connecting any two of these points?



1



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3. In 2005, a father's age is twice the sum of his two sons' ages and is 7 times the difference of his two sons ages. The sum of the ages of the father and his two sons is 84. Which year was the father born?



4. What is the maximum number of regions that 10 straight lines can divide a plane into?



2



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5. A right-angled isosceles triangular cardboard was by joining the two furthest comers to form a smaller right-angled isosceles triangle. This process was repeated a 2nd time and then a 3rd time. If the shortest side of the final right-angled isosceles triangular cardboard is 2 cm, what is the area of the original cardboard?



6. Evaluate 1.1 + 3.3 + 5.5 + 7.7 + 9.9 + 11.11 + 13.13 + 15.15 + 17.17 + 19.19.



3



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'



'



%



7. Evaluate 100 − 3 3 ÷ 52 % − 0.6257 × 51.6 + 2 97.



4



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8.



At McDonald’s, nuggets are sold in boxes of 4 or 7 pieces. What is the largest number of nuggets one cannot buy?



9. In the figure below, 𝐴𝐵𝐶𝐷 is a square of side 6 cm. If 𝐵𝐸 = 2𝐶𝐸 and 𝐷𝐹 = 2𝐶𝐹, find the area of the shaded region.



5



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10. Find the value of 𝑎 + 𝑏 + 𝑐 + 𝑑 in 1−



6



1 7+



1



1 1+7



=



1 𝑎+



1 𝑏+



1



𝑐+



1 𝑑



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11. Evaluate 7 7 7 7 7 51 + 17 51 + 27 51 + 37 51 + 47 … 51 + 97 9 9 9 9 9 51 + 17 51 + 27 51 + 37 51 + 47 … 51 + 97



7



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12. The figure shows 4 semi-circles and a circle. A square of area 196 cm2 is inscribed within. Find the area of the shaded region. Take 𝜋 =



8



%% G



.



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13. The queue at SEAMO X 2019 Gala Dinner & Awarding Ceremony was growing at a constant rate before its opening at 5:30 PM. If 3 entrances were opened, all guests would admit in 9 minutes. If 5 entrances were opened, all guests would admit in 5 minutes. At what time did the first guest arrive?



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14. As shown in the figure, the journey from A to Town B consists of an up-slope from A to C and a down-slope from C to B. The speed going up-slope is 400 m/min while the speed going down-slope is 600 m/min. Given that the time taken to travel from A to B is 3.7 min while the time taken to travel from B to A was 2.5 min, find the difference in distances AC and CB.



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15. If Paul drove from Town A to Town B while, at the same time, May drove from Town B to Town A, without stopping, they would meet 6 hours later. If Paul stopped for 2.5 hours along the way, they would meet 7.5 hours later. What is the total time taken by Paul to drive from Town A to Town B if he decides to stop?



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QUESTION



ANSWER



SOLUTION First, take the last digit of the given number and calculate the number when this last digit is raised to the powers 1, 2, 3 and so on. Stop the calculation when the last digit of the given number repeats itself. The last digit of 2018 is 8. The powers of 8 are as follows: Power 1 2 3 4 5



1



Last Digit 8 4 2 6 8



2 8" = 512, hence the last digit is 2 8' = 4096, hence the last digit is 6 8, = 32768, hence the last digit is 8 The last digit of 8, is the same as 8/ and hence we stop the process. The pattern repeats in groups of four: 8, 4, 2, 6 Let us find out the remainder of the power 2019 when it is divided by 4. 2019 ÷ 4 = 504 𝑅3 The last digit is 2.



2



45



Take any one of the given points on the circumference of the circle and join it to any of the 9 other points and hence there are 9 such lines. Since there are 10 such points on the circumference, we will have 10 × 9 lines. Since joining a line from A to B is the same as from B to A, we divide by 2. Total number of lines =



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/4×5 6



Practice Set C



=



54 6



= 45



1



There are 3 unknowns in this question, the ages of the father and 2 sons. Let the father's age be 𝑥 and the sons' ages be 𝑎 and 𝑏. 𝑥 = (2𝑎 + 𝑏) ----------- (1) 𝑥 = 7(𝑎 − 𝑏) ----------- (2) 𝑥 + 𝑎 + 𝑏 = 84 -------- (3) Substituting (1) in (3) we get 2(𝑎 + 𝑏) +𝑎 + 𝑏 = 84 3



1949



→ 3(𝑎 + 𝑏) = 84 → (𝑎 + 𝑏) = 28 Substituting the value of (𝑎 + 𝑏) in (1) 𝑥 = 2(28) = 56 So the father's age is 56. Since the father is 56 years old in 2005, he was born in the year 2005 − 56 = 1949



No. of straight lines (𝑛)



4



Figures



No. of regions



1



𝑎/ = 2



2



𝑎6 = 𝑎/ + 2 = 4



3



𝑎" = 𝑎6 + 3 = 7



56 Now we can observe the pattern, i.e. 𝑎@ = 𝑎@A/ + 𝑛 Hence, 𝑎/4 = 𝑎5 + 10 = 𝑎B + 9 + 10 =⋯ = 𝑎/ + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 2 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 56 Therefore, there are 56 regions.



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Practice Set C



2



5



16



Area of cardboard =



1 (8)(4) = 16 cm6 2



1.1 + 3.3 + 5.5 + 7.7 + 9.9 + 11.11 + 13.13 + 15.15 + 17.17 + 19.19 = 1.1(1 + 3 + 5 + 7 + 9) + (11 + 13 + 15 + 17 + 19) 6



103.25



+ (0.11 + 0.13 + 0.15 + 0.17 + 0.19) = 1.1(25) + 75 + 0.75 = 27.5 + 75.75 = 103.25



7



8



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90



6 7



17



1 1 2 100 − 3 ÷ O2 − 0.625P × O1.6 + 2 P 8 2 3 25 25 5 64 = 100 − ÷O − P×O P 8 12 8 15 25 35 64 = 100 − ÷O P×O P 8 24 15 25 24 64 = 100 − ×O P×O P 8 35 15 25 24 64 = 100 − ×O P×O P 8 35 15 6 = 90 7



Guess and Check / Logic Any number lower than 4 are ruled out. Try 14, 15, 16, 17…18, 19, 20, … Anything larger than 17 will be possible.



Practice Set C



3



The blue lines divide the right-angled triangles into 4 smaller triangles. The 9



25.5 𝑐𝑚6



smaller triangles share the same heights and bases. Thus, their areas are the same. We refer to each smaller triangle as having an area of 1 unit. There are 6 right-angled triangles within the square. Thus, the area of the square is 4 × 6 = 24 units. 24 units à 6 × 6 = 36 𝑐𝑚6 "S



1 unit à 6' = 1.5 𝑐𝑚6 Area of shaded region = 24 − 7 = 17 units = 17 × 1.5 = 25.5 𝑐𝑚6



1



1−



10



15



7+



1



=1−



1 1+7



1 7 7+8



=1−



8 55 = 63 63



55 1 1 1 1 = = = = 1 1 63 63 1 + 8 1 + 1 + 7 1 55 55 6+8 6+ 1 1+7 𝑎 + 𝑏 + 𝑐 + 𝑑 = 1 + 6 + 1 + 7 = 15



B



5



Numerator à / × 6 × … × 11



8 34



Denominator à



/4 /



×



B×5×…×/S



// 6



/S 5



× …× B×5



/B 5 B



Simplify à /4×//×…×/B = /V×/B = "'



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Practice Set C



4



2



Diameter of semi-circle = √196 = 14 𝑐𝑚 Radius of semi-circle = 14 ÷ 2 = 7 𝑐𝑚 2 circles + 1 square à 2 × 12



196 𝑐𝑚6



66 V



× 76 + 196 = 504 𝑐𝑚6



2



2



Diameter of big circle à √146 + 146 = √392 Radius of big circle à Shaded à 504 −



66 V



2



√"56 6



×O



2



√"56 6



6



P = 504 −



66 V



×



"56 '



= 504 − 308 = 196



Rate of Growth of Queue à (3 × 9 − 5 × 5) ÷ (9 − 5) = 0.5 𝑢𝑛𝑖𝑡𝑠/𝑚𝑖𝑛 Original length of queue à 3 × 9 − 9 × 0.5 = 22.5 𝑢𝑛𝑖𝑡𝑠 13



4:45 PM



First guest arrived à 22.5 ÷ 0.5 = 45 minutes ago Queue started at 4:45 PM.



Draw a point D along AC such that AD = CB. Then A to D and B to C takes the same time. Difference in time from D to C and C to D à 3.7 − 2.5 = 1.2 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 14



1440 m



𝐷𝐶 𝐷𝐶 − = 1.2 400 600 𝐷𝐶 = 1440 𝑚 Scenario 1: Paul does not stop Each person travels for 6 hours. Scenario 2: Paul stops



15



12.5 hours



Paul travels for à 7.5 − 2.5 = 5 hours Ratio speeds à 𝑃𝑎𝑢𝑙 ∶ 𝑀𝑎𝑦 = 5 ∶ 7.5 Total time taken by Paul à 5 + 2.5 + 5 = 12.5 hours



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Practice Set C



5