Pre Regional Mathematical Olympiad, 2016 Delhi Region Set C

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Lindsay Osborne
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1 Pre Regionl Mthemticl Olympid, 06 Delhi Region Set C Mimum Mrks: 50 Importnt Note: The nswer to ech question is n integer between 0 nd 06. Ech Cndidte must write the finl nswer (in the spce provided) s, Finl nswer = Correct nswer + sum of ll the digits of their roll number. Only the finl nswer shll be considered. Problem. The five-digit number 9b is perfect squre. Find the vlue of Problem. Let sn nd b b. pn denote the sum of ll digits of n nd the product of ll digits of n (when written in deciml form), respectively. Find the sum of ll two-digit nturl numbers n such n s n p n. tht Problem 3. Let AD be n ltitude in right tringle ABC with A 90 nd D on BC. Suppose tht the rdii of the incircles of the tringles ABD nd ACD re 33 nd 56 respectively. Let r be the rdius 3 r 7. of the incircle of tringle ABC. Find the vlue of Or Problem 3. Find the sum of digits in deciml form of the number (There re nines) Problem 4. Between 5 pm nd 6 pm, I looked t my wtch. Mistking the hour hnd for the minute hnd nd the minute hnd for the hour hnd, I mistook the time to be 57 minutes erlier thn the ctul time. Find the number of minute pst 5 when I looked t my wtch. Problem 5. In tringle ABC right ngled t verte B, point O is chosen on the side BC such tht the circle centered t O of rdius OB touches the side AC. Let AB = 63 nd BC = 6, nd the rdius of be of the form m n where m, n re reltively prime positive integers. Find the vlue of m n. Problem 6. There re three kinds of fruits in the mrket. How mny wys re there to purchse 5 fruits from mong them if ech kind hs t lest 5 of its fruit vilble? Problem 7. The dte inde of dte is defined s ( month number + dy number). Three events ech with frequency of once in dys, 3 dys nd 9 dys, respectively, occurred simultneously for the first time on July 3, 96 (Irelnd joining the Europen Economic Community). Find the dte inde of the dte when they occur simultneously for the eleventh time. Problem 8. term. Consider the sequence, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, nd evlute its 06 th Problem 9. In school there re 500 students. Two-thirds of the students who do not wer glsses, do not bring lunch. Three-qurters of the students who do not bring lunch do not wer glsses. Altogether, 60 students who wer glsses bring lunch. How mny students do not wer glsses nd do not bring lunch? Problem 0. Suppose tht nd b re rel numbers such tht b d the equtions nd b 0b 0 0 hold. Find the vlue of b b.

2 Problem. The hegon OLYMPI hs refle ngle t O nd conve t every other verte. Suppose tht LP 3 units nd the condition O 0L Y 5M P 0 I holds. Find the re (in sq. units) of the hegon. [4 mrks] Problem. Points G nd O denote the centroid nd the circumcenter of the tringle ABC. Suppose tht AGO 90 nd AB 7, AC 9 Problem 3. Consider the 50 term sums: S T Find the vlue of BC. [4 mrks] The rtio S T is written in the lowest form m where m, n re reltively prime nturl numbers. Find n the vlue of m n. [4 mrks] Problem 4. Find the vlue of the epression when written in lowest form. [4 mrks] Problem 5. A nturl number hs four digits nd ends with the sme four digits s tht of. Find the vlue of 0, 080. [4 mrks]

3 Answers Solutions. Squre of 0000 to 9000 must lie b/w 4 & 73. Check for the squre 49, 5, 59, 6, 7, & 69 s the squres of these numbers will end with b 9 is 59 5 b b 4 b 5'. 0 b b b 9 b b 9 possible number re 9,9, Correct nswer = Ans Obviously the hour hnd ws between 5 & 6 Lets sy it ws minutes pst 5. He mistook the hour hnd to be minute hnd. So he thought it is 5 minutes. He mde mistke of 57 minutes so he thought the time to be 4 hour & 5 minutes

4 So A.T.Q So it ws 4 minutes pst 5. C LCM of, 3 & 9 = 06 It will hppen fter every 06 dys. th event will hppen on 76 th dy fter July 3, 96. Every 4 yers the number of dys will be 46 dys. 5 sets of 4 yers will pss for 95 dys. More 6 dys to go fter 60 yers. So it will 6 th dy fter 3 July 0. Additionl 43 dys for 3 mrch 0. The dy is 8 April 0. Inde will be (Ans) 8. n continues till n 05 is will continue to become terms from 937 th to 05 th terms. So Ans. is do not bring Lunch y do not wer glsses. Z= no lunch no glss 3 y Z 3 4 8y=9

5 Also y 440 z y 760 solving the eqution we get 30 so 3 z Roots of Eq& Eq re reciprocls. Let roots of Eq be A & B Roots of Eq, will be & If A, b B b b s b b b b b A B. A B AB AB = AB Sum of roots + Product of roots. 40 /0 The bove is mde ccording to the question. Her LYPI form rectngle with sme re s OLYMPI. So Are of hegon = Are of rectngle = y. M We know y 8 To get integer solution of re we consider y 3 Y P Required re 9 O y L I

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h

3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h PAKTURK 8 th Ntionl Interschool Mths Olmpid,.9. Q: Evlute 6.9. 6 6 6... 8 8...... Q: Evlute bc bc. b. c bc.9.9b.9.9bc Q: Find the vlue of h in the eqution h 7 9 7.. bc. bc bc. b. c bc bc bc bc......9 h