IOQM QUESTION PAPER

IOQM PART 1

PRMO + RMO

' - - - - - -...~---e-=~=J Time: 75 llllllltell Number or Que■tlon11: 12 Max Marita: 40 INSTRUCTIONS 1. Use of mobile phones, smnrlplloncs. lpads, cnlcu lat.ors, progrumnmble wrist watches ts STRICTLY PROHIBITED. Only ordlllnt)' pens nnd pencils are allowed inside the ex- nmlnalton hall. 2 . The correction Is clone by machines through scanning. On Lhc OMR Sheet, darken bubbles completely with a black or blue ball pen. Please DO NOT use a pencil or a gel pen. Darken the bubbles completely. only after you are s ure of your answer: else, erasing may lend lo lhc OMR sheel getting damaged and the machine may not be able to read the answer. . 3. The name, email address, and dale of birth entered on the OMR sheet will be your login credentials for accessing your score. 4. Incompletely, incorrectly or carelessly filled information may disqualify your candi- dature. 5. Each question has a one or two digit number as answer. The first diagram below shows improper and proper way of darkening the bubbles with detailed instructions. The second diagram shows how to mark a 2-digit number and a ! -digit number. INSTRUC TIONS Q. 1 ·®Q. 2 1. \"Think before yo:,r ink\". [1][z] [Q)~ 2. Marking should be done with 131ue/Black B:'.1ll Poinl Pen only. / 3. Darken only one ©® 00 Examplo Solow . circle 1or each question as shown In 00 00 WRONG ,:;;-1:-=r:--1oos) CORRECT METHOD CDG G)(D ® -i$- ~:@ ® ® ©@ • 0(DG) 0m•0 4 . If moro th:in one circ:ln is dnrkonod or ii the ro:;ponso Is C)G) ©© marked in :my ciah1:r way us shown \"WRONG\" obcwe, II sholl be lre oled 11s wrong way ol n•or>:ln9. 0o0• <D CD 5 , M ake the murks o nly in tho spaces prov ldod. ®<D ©CD 6 . Corofulty teAr off tho duplicote copy of tho OMR without 00 ©© tamporlno 1110 Original. 7 . Please cJo not moku :m y s trily m;;rks on the answor shi.el. 6 . The answer you write on OMR sheet is irrelevant. The darkened bubble will be con- sidered as your final answer. 7. Questions 1 and 2 cany 2 marks each; questions 3 to 9 carry 3 marks each; ques- tions 10 to 12 cany 5 m arks each. 8. All questions are compulsory. 9 . There are no negative marks. 10. Do all rough work in the space provided below for it. You also have blank pages at the end of the question paper to continue with rough work. 11. After the exam. you may take away the Candidate's copy of the OMR sheet. 12. Preserve your copy of OMR sheet till the end of current olyrnpiad season. You will need it later for verification purposes. 13. You may take away the question paper after the examination.

/.1urt· p.,r.illd llm'\" t,1• 1~.1,.1 art• llrawn 111 1hr plaric such lhal lht· J><:rpernJfrular dlslance t1<·twc·t·n 1,1 amt /,,J 1-. :11111«1 th•• prrpc·mllc11l11r <11-.1mw<' between /,,i and /,_• 1-. :tl-;o 3. A ~,,t1.11t· t\\llCI) ,~ t·o11-.lrul\"lr1l stwh lhal ;\\ lies 011 t., . II lies 011 /,,J nml C lies on l✓i. Find the• arr,, of tfu· s11u11rr. :z. w., w111rsth,w11 tht• 1111111lu-r!i l.:l.. .. . IOI 111 n·II 111111 IJl11r 1,1•11c;. ·n,r lar~csl blue number . 4,.. c,111,1110 thc· 1111111lwr 111' 1111111lwrN wrlllt'll 111 111111· 111111 tlw smallrsl red 1111111t,cr •~ c<1unl lo C ,half thr n11ml\"·r 11f 1111111l11•r!i wrlfll'II 111 rl'd. I luw 1rn111y m11111>ers did l<Ja wrllc wllh red pen? ·r.....~ /4111~1tkr 1hr ~,·t of nil lrla11glcs whose sldcs arc dlsll11cl prime mrrrrhe rs which .ire nlso rIn ;1rttl1111dll' progrc·sslon. I.c l 6 E be the triangle with the lec1st pcrlmel<:r. If an Is the a l.1.1·.~1·!'-I a11gk of £1 and ff f., Is Its fJcrlrnclcr. clctcrmlnc the value of -I . _.)1 onsl<lc•r tlw set of all G-dlgll 1111mucrs co11sls ll11g of only 3 digits. a...b , c, where a. b, c arc cllsllnc-t. Supposc lhc sum ofall lhcsc numbers Is 593999406. Whal Is the largest remainder Whl'll the three dlgll number olJc Is divided by I 00? SPACE FOR ROUGH WORK 1J' C1. il - -3 Fr,_,- ...('-;:,(-;. ,·..,..-~:l ,.-, ( \\ cg ,- l\\ 3 r;~:::-:~-- ,._\\ / ~~ ~ -r-- SL\\2- ~ - ( \"r,\":..~ \" \\-r- (0_ ' • •·;- ft. r.-.. '1 &. f:)? 72- . ,i l 2 /.

✓.:,:,mlldogrnm IsAllCD lhc longer side lwlce lhe shorlcr side. Lei XYZW he lhe or<11mdrllat<'ral funned by the lnh:nml blscclnrs the nr1glcs of AIJCD . If lhc ar<':t of XYZW l~~llw ar('a of AIJCIJ. = tr =~ c l x. !/, z he positI\\'(' l\"l':1I lltllllhl·rH kl 1d1 I hul -'<' + !f\" ,m , + !J~ + zl 36 nm.I .e· v:iux+ V3.\\7. + :,,::.! = 25 , If llw vahu• nf 'l.X!J + + 7-\\' t'llll ht! wrlllc 11 as f) ..fi'I where p. q arc lnh•gcrs a11<1 ,, IH 11111 dlvlsllJhi hy sc1rn1rc of a11y prime numbt!r. flml ,, + <J . 7 . Ftn<I the 1111mbcr of maps J: 11 . 2,31 - t I 1, 2 , 3, 4, 5) such lhal j(l) $:JU) whcnc..-vcr I < J . ~n·atcsl Integer such thal LLJ denote lhc largest Integer ::; L• Suppose that N Is the l ✓l~Jl= 4 Find U1c sum of digits of N . SPACE FO:R ROUGH WORK 1,2.. 1 -·· r ' <:>\\ abc. b·bc._ d c, be.ho~ y /_,,, '~.•.•J~- _- I ~Cl. 10 o. l•)c ' -.. I 3

nllll drlhu· I',., 1 ::. ( ,·,,. !f.,l f11r 11 2 0 liy .\\',, -~,,,, 1. /,, . I = - :l . the· .,rr.1 nf thr q11,11ltll,1ln11I l11tt1ft'tl l1y lltl' 111•l11l <i l',111 , 1'111. / 1,,,,. J•,~, . \"'ro Sltf'j)H'•I' lh,11 ,, ,.• ,, ... p11ly1111111i.,1 111 l,•,1•,I tlq(ll' I' WIiii 11111·1.!n (·odfi1·1,•111~ such th.ti L\\,r:-,) .,, ( ,,/;__,,. , ·:-,) ::-: ~! ( ,:; - 1-\"1111! I 'I~ 1.,---;) / \"' ..JJ.--111 h,,w 111.111y w.,y ·, 1'.111 f1111r 111arrh·tl c·rn1plr!-l !-ill 111 11 t1ll'rry·/.!O •m1111d with lth•11ll<•;il s(·at.-. •,111·.h fh.11 11w11 .111d w111111·11 mTupy allc.•nmlc ~ea.ls 11ml 110 husha11d s1·11ls 111·x1 Ir, hl<i wlfr? Li'.! , I:.! ho.ml I~ dl\\'ldc<I 11110 1·H u111t squares by drawing lines pnralkl lo I he side~. Two 111111,,, pl,H·,·<I n11 two 1111II squares are snld lo lit: 11011 nllac·klr1.(.! If lllcy ;;1n• 1101 111 Ilic: s,irrw n1h111m or ~;1111l' row. Ft11d the 1t-a~l 11u111her N s ud, lhnl If N rooks nr<: placed on the unit NJ11an·s. Olll' rook pc-r sciuan·. wt• cnn nlways find 7 rooks such lhnl no two nrc altackin_ct g'2...-l',1C-h otlwr. ~ SPACE FOR ROUGH WORK

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ROUGH WORK .. • :

IOQM PART 2 OM

MR Carbon copy

·-. IOOM 202 1·22 orNAME Ttl( CM IOl::lATI: P;• .-•.e rt\", J c :1r c futf1 tll• 1n, 1111c t ,c;r. ~ a n 1110 fl'JPS:lon P-'f\"' ' 1 1G2 JO .. I I , / i .,_ ,\\ I ..' , 11!1) • l !' • • •• t •• : ': -' • '' I W!l l ' t: A... • I I ., ., ,. ~,. C, ..I.. } ,::, '• !) . ..., •, • ' ~ - . , !'\" 1 • - I I t :· _,. I t i ! .... .. ... : • 1 ..., • . , tit , ~J -f , ' ~ f •:t \\ ·I , I 1_1 ( 1 J n, . .._ ..,■ J ' · . •• ••• • .. 0 • ~ -:-r ; • '~ 'L ·: :~·· ,. '\"1 • , ·• C\"' I t • ,, ' , • • • , ,\\ \\ :,! ..,_~ :-:- ._ • -.. '.'.'..._ !.• •-•,-. : • • \\ It\\ .... :ot,•, • J • 1 11 1;,:: L •t • •• o o o e( I ,,o,-r: r t.\" : H •\"JU •. ■ !) ..... , •, ~ ' -,• a . ' \\ · J,•n,... tJ C- :'- 11,\\(! ' ft ,!.:J,- Jf,\")' ~J ,t\" ·r .. ■ th :, •..:,,~ \\ ••-, t,l ....t\"': ' ,•·•·t•• \\ n l u• , ;. !tu t .1-.r·,· , 1c•,,..r • :,: •j •f ■ t~:.,,t t ...•.!: .... ~... .l\"\"·· \\ ' ., r , .....!i ~ t..... ., ••, ...-~a•-, t ,: ..,J ,' ■ ~ t.! ! ·. - . - :· 1 .... ! ;... •~ ,._, ~ t-i..,-;.n .:-• -.r'\"\"' -;\"' • --- - - 111 IlllillllH'.l • • •• li •J. l30 !• ■I ■I I ■I •' •■ 'I •! Ml\\!•'< YOUR A\"ISW!oRS HERE • 1 0 .01 I . l 0 .02 I J_QTI fo i1lr0 .03 0 .04 0.05 b-~1] ~1Q.06 '~0 1 1]_'.t i 0( a , ■, 0 (a) (a) (o) ( u) ( ' ) 0 '~.:'; (,_·; <.:•) { I J l:) ■I (1) ( :) (: l ( ,) r :) ( : I ,...,.~ (_: J ( -j ( l) 0 (;' ( >~I : ' ■i ( ,) ' ,) ■I (,) CJ r j 1I I : ( J) ' ~ (.·) ! i ..' (~ ( ,) ,,,o, ·.• 1:1 ( •) ( i) ~'•t...~) \\ ,\"\" ( •) ( •) ( .,, f :; ( ·) ( ,) (i) (_ ,) <__,) (0 l -' · ~' ( , ) ( ,; ( l) ( ' l (,)( ,) ·~·'! , .') ( •.) ( •,) I (,,1 ( ·-) (,I ( ,J ( ' ) (.,) ,,\\.'' '• 0 (~) • I ( ,,) ( ,,) ( •,) ( , ) (,) (·) ( ~J 0 l.;, \\.,' (1) (,) (_1) (,) • II ti ( ,) (,; 0 ( ,) (~) 0 .12 11...., 17. 1 . (!) (•) (ai <•1 (,) (,, (;) ( ,'I • ( ,) ' •) ( 1) ( ,) ~ (~) ( 9) 80 rn •• l ( ;1 t ,; ~'-~.l'1 ,,) rn0 .08 0 .09 ~ 0.10 [BJ 0.11 •l<. C•l • 0.07 <•·)• ( :) \\ :) • ( ,j (u\"J 0 G <•) ·-- { 1) ,_., oEJ• • t 1) ■ ( o) (11 I ( () ( 1) ( 1) \\ 1) • t,:, l. ,) (, ) (:,) (,) 1.: ) l >) ■ . (1) (1) ( 1) (: ) ( ,) i.__,) ~·' \\ ,') ( i) ( ,) • (•1 l. I) ( I) ■ (9 (,) l •' (.•) i•h·' ( • ) ( <) r • l ( •) ( ,) . ( () ( •) 1.•) • '-~) ( >) ('.) (,) l ·,) t, ) v,) 1.·,) •• ( •) (. ) l •) l•) (~) (,) (o)(,) ( I ) (. ,) (,) ( :\"} ■ ( >) ( >) v) \\ ,) l ' ) \\ '.) l ,)( , } ■ ( .1 ( 1,) (1,) (•' 1....-1 ( ,) ( ,) (u) ( •) \\..) l..) l.•) ■ (' ) ( ') ( •) ( 'I ( ~) ( ' ) ( ') l •1) {!) C•. J '.) ( . ' ••■ L_______ ( 'I h I ·- - - -

10 M PAR

RT 2 INMO

PARTB Time: 2.5 hours This QP Contains Both English & Hindi Versions

f',\\ ll~I' IJ Ti11w: :.?.:-, hour:.. I11-.1 r11.-t i, ,11,.: • <'ak11lat111 .. (in :111y f11r111) :111d prul r11durs an• 1101. 11ll,,wr·rl. • Hul,·r~ a111I ,·11111pa-.-.c•s arn allowetl. • ,\\ II q11,,,1 io11~ n,rry <•qual mark.'1. l\\lmd111111r1 mnrks: 5 J. • ~o 111:irks will l,e awardt•d for slali11~ nu a11swt:r will1011t j11stifi<:r1tiori. • A ll!'-1\\\\'(' r all the! q11t•st.io11s. • PLEASE HEAD TIIE INSTllUCTIONS ON TJIE ANS\\VEll BOOJ<LBT VEHY CAHEFULLY DEFORE ANS\\VERING TIJE QUESTIONS. /i.,., I) 1,., \"\" i111,:1·i\"r poi111. on the s ide BC of a11 ac11tc-a11glcd trin11glc ABC. Ll'I t lie· circ.:11111<:ircle of l.ria11~l,! AD[] intersect J\\C agai11 at E(-:/- ;\\) and the cir<·1111wird,! of lri:111µ, le 1\\DC intersect/\\/] again at F(# J\\). Let ilD, DE and CF' i111 c-rsc•c:t t.11<~circ11uH·ird c of t riangle ABC again at D1 (i J\\), E1 (i fJ) :md ' F 1( f C ), n~sp<•C'I ively. Let, [ and / 1 be the inccntrcs of tria11gle~ DEF and r,:·:t., ,/ D- 1£ 1F 1, r<·spl'c.:Livdy. Prove Lhat E, F, I, /1 arc conc·yclic. (,' \\Js·•r-:-Ju_\\ (c. •' ,..,I~... <.J r~' /2. Find all 1rnL11rnl rn1mben; n for which there exbts a pcrmutatio11 <1 of I, 2, ... in :;11d1 I liat. L11 a(-i)(-2i- l = 0. i=l Nole: A vcrm-ufolion of 1, 2, . .. , n is a. bijective function from {1, 2, . .. , n} to OHsdf. (_--t1;•11 ·-•r cc•:,,~ fo,1 positive integer N, let T(N) denote t he 1111mhcr of arrangements of th\" i11tegen; 1, 2, ... , N i11to a sequence tq, a2, .. . , ftJV such that a; > a2; for all i, I < i < 2i < N all(I a; > a2;-1-1, for all ·i, 1 < i < 2i + 1 < N . For l'Xamplc. T{aV~ 2, :iucc the posi;ihlc an~angemcnts arc 321 and ~H2. {/J Frnd T(7). ( f ;yrr1\\vJ0 ...~t) jr,(;r /( is tlu: la.rw!sl. 11011-11cµ;al.ivc i11l,eµ;er 80 I.hat 21\\· clivid<~H T(2\" - 1), sho,,·(:r.,·<.t,e\\iOT\"J ;9a1. /( = 2\" - u - I . ('1'°f'iud t.hc large:-;!. 11011-rJC•~al.ivu iut.,iµ;nr K :m thnl. 2K divicll's T(2\" + 1). { ,;,..:~••. ~...~) .. - - 00000- --

lfl1T 6f ~: 2.St'R ~~r: • c\\il\"lg,Mc-< (ffl>mlft\"~-q) m~~cm ~ ~ t 1 . ~~~crfRcm~t, • ~)!'~;riRTiRm~t, ~Jf<p: 51 Jjq;m~,• Jr~~q;r~~~~urmci~'IR\"~ • ~ >f~;ff cliT urcITTr ~ I • -sr~;ffcpTJlR~~tlffl3tH-~ffi1cf>I ~~ITT~furR~~I 1. llR\"ffi~ D~~-cmur~ ABC ~ ~ BC1Rw:@~3icr:~5 llfRfil\"ffl,~ ADBq;r~ ~ AC~~: E (# A)1Rfurrcn-t,-am~ ADCcpl\"lffiilo~ AB*~= F (# A)\"tR Ml\"lctl 61 lfR'ffi ~ A D ,BE cf C F ~ ABC\"$~*F= wll~f: D1 (# A), E1 (# B) cf F1 (# C) 1Rfm;:@-f1 lfR\"ffi ffl;~ DEF c f ~ D 1E1F1 \"$ 3IB:~wll~f: I cf Ii -g, ~~ffl, E,F,I,I1 ~4icfillll 61 w ~2. !,llcp@4> fi&-llq n ~elm. ~fc;w 1,2, ...n cpl~ w'iilll ~ 0-6 Mfl~ fc1,- n I>T(i)(-2l-1 = 0 i=l 1le.: 1, 2, ..., n 'qi\"J wife.Ill 1lA' {1, 2, ..., n} ~ f f l l t ~ ~ .slli8icJl5 ~ I 1lR ftp~ tl11\"14> ~ N \"$ ~ ~1fq,)' 1, 2, ..., N ~ T(N) ~ ~ ~ ~ JW1TI° a1, a2, ...., a N ~ > a2i ~lfWcf'6~3\";JWTI' i \"$~, ~~ l $ i < 2·i $ N ID', ai 321 'q t; q3';f'fl'lfi ·i \"$~, ~~ 1 $ ·i < 2i + 1 $ N ID' U-i > a2i+l 61 ~ffl;' T(3) <ITTlfR 2 m i •3. 312 m~~ ~ f l (a) T(7) q;r-i:rr,:r~~I s(b) WR K cf6 ~ ffi'-;JtUT ~ Mfl~ ~ T(2n - 1) ~ 21< ~ ~ \"6. m~ ~ \"ffl; I< = 2n - n - 1 61 (c) ci6~m-~~ K~cim Mfl~ ~T(211 +1)~2/(~~61 --000000--

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