MTA-IAPT Pre-Regional Mathematical Olympiad(PRMO),2018 ...

1 MTA-IAPT Pre-Regional Mathematical Olympiad(PRMO),2018 Date: August 19, 2018 Time: 10 AM to 1 PMNumber of Questions 30:Max Marks: of mobile phones, smartphones, ipads, calculators, programmable wrist watches isSTRICTLY ordinary pens and pencils are allowed inside the correction is done by machines through scanning. On the OMR Sheet, darken bubblescompletely with a black pencil or a black or blue ball pen. Darken the bubbles completely onlyafter you are sure of your answer; else, erasing may lead to the OMR sheet getting damagedand the machine may not be able to read the name, email address, and date of birth entered on the OMR sheet will be your logincredentials for accessing your PRMO and carelessly filled information may disqualify your question has a one or two digit number as answer.

2 The first diagram below showsimproper and proper way of darkening the bubbles with detailed instructions. The seconddiagram shows how to mark a 2-digit number and a 1-digit answer you write on OMR sheet is irrelevant. The darkened bubble will be cosidered asyour final 1 to 6 carry 2 marks each; questions 7 to 21 carry 3 marks each; questions 22 to30 carry 5 marks questions are are no negative all rough work in the space provided below for it. You also have blank pages at the endof the question paper to continue with rough the exam, you may take away the Candidate s copy of the OMR your copy of OMR sheet till the end of current olympiad season. You will need itlater for verification may take away the question paper after the examination.

3 ( ) book is published in three volumes, the pages being numbered from 1 onwards. The pagenumbers are continued from the first volume to the second volume to the third. The numberof pages in the second volume is 50 more than that in the first volume, and the number pagesin the third volume is one and a half times that in the second. The sum of the page numberson the first pages of the three volumes is 1709. Ifnis the last page number, what is the largestprime factor ofn? a quadrilateralABCD, it is given thatAB=AD= 13,BC=CD= 20,BD= 24. Ifris theradius of the circle inscribable in the quadrilateral, then what is the integer closest tor?

4 All 6-digit numbers of the formabccbawherebis odd. Determine the number of allsuch 6-digit numbers that are divisible by ,The equation166 56 = 8590is valid in some baseb 10(that is,1,6,5,8,9,0are digits in basebin the above equation). Find the sum of all possible values ofb 10satisfying the a trapezium in whichABkCDandAD?AB. SupposeABCDhas an incirclewhich touchesABatQandCDatP. Given thatPC= 36andQB= 49, ,b,csatisfya+b c=1anda2+b2 c2= 1. What is the sum of all possiblevalues ofa2+b2+c2?SPACE FOR ROUGH WORK21. , т 1 т 50 т , т т т 1709 n т , n ?

5 2. ABCD AB=AD= 13,BC=CD= 20 BD= 24 r а , r ?3. 6- abccba b 6- 7 ?4. 166 56 = 8590 (base)b 10 ( 1,6,5,8,9,0 (base)b ) b 10 ? AB CD AD AB а аAB Q CD P PC= 36 QB= 49 PQ ? , b, c a+b c=1 a2+b2 c2= +b2+c2 , ? pointPin the interior of a regular hexagon is at distances 8,8,16 units from three con-secutive vertices of the hexagon, respectively.

6 Ifris radius of the circumscribed circle of thehexagon, what is the integer closest tor? a chord of a circle with centreO. LetCbe a point on the circle such that\ABC=30 andOlies inside the triangleABC. LetDbe a point onABsuch that\DCO=\OCB= 20 .Find the measure of\CDOin , bare integers anda+bis a root ofx2+ax+b=0. What is the maximum possiblevalue ofb2? a triangleABC, the median fromBtoCAis perpendicular to the median the median fromAtoBCis 30, determine(BC2+CA2+AB2) are several tea cups in the kitchen, some with handles and the others withouthandles. The number of ways of selecting two cups without a handle and three with a handleis exactly 1200. What is the maximum possible number of cups in the kitchen?

7 SPACE FOR ROUGH WORK37. P - 8,8 16 r а r ? а O C а ABC= 30 O ABC D аAB DCO= OCB= 20 CDO 9. a+b x2+ax+b=0 b2 ?10. ABC B CA C AB A BC 30 (BC2+CA2+AB2)/100 11. , , 1200 ?

8 The number of8-tuples( 1, 2, , 8)such that 1, 2, 82{1, 1}and 1+2 2+3 3+ +8 8is a multiple a triangleABC, right-angled atA, the altitude throughAand the internal bisector of\Ahave lengths3and4, respectively. Find the length of the median cos 1 cos 2 cos 3 cos 89 andy= cos 2 cos 6 cos 10 cos 86 , then what is the integernearest to27log2(y/x)? natural numbers such that2a b, a 2banda+bare all distinct is the smallest possible value ofb? is the value ofX1 i<j 10i+j=odd(i+j) X1 i<j 10i+j=even(i+j)?SPACE FOR ROUGH WORK412. - ( 1, 2, , 8) 1, 2, 8 { 1,+1} 1+2 2+3 3+ +8 8 3 ?13. ABC, A , A A - л 3 4 A ?

9 14. x=cos1 cos2 cos3 cos89 y=cos2 cos6 cos10 cos86 27log2(y/x) ?15. a b 2a b, a 2b a+b - b ?16. : 1 i<j 10i+j=odd(i+j) 1 i<j 10i+j=even(i+j)? such that\A=\D,AB=DE= 17,BC=EF= 10andAC DF= 12. What isAC+DF? , b, c 4are integers, not all equal, and4abc=(a+ 3)(b+ 3)(c+ 3), then what is the valueofa+b+c? 6 + 66 + 666 + + 666 66, where there are hundred 6 s in the last term in thesum. How many times does the digit 7 occur in the numberN? the sum of all possible positive integersn, the product of whose digits equalsn2 15n an acute-angled triangle and letHbe its orthocentre.

10 LetG1,G2andG3bethe centroids of the trianglesHBC,HCAandHAB, respectively. If the area of triangleG1G2G3is7units, what is the area of triangleABC?SPACE FOR ROUGH DEF A= D, AB=DE= 17,BC=EF= 10 AC DF= 12 AC+DF ?18. a, b, c 4 , , 4abc=(a+ 3)(b+ 3)(c+ 3) a+b+c ?19. N= 6 + 66 + 666 + + 666 66 6 N 7 ?20. n2 15n 27 21. ABC H G1,G2 G3 HBC,HCA HAB G1G2G3 7 , ABC ? positive integerkis said to begoodif there exists a partition of{1,2,3.}