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A u s t rAl i An M At h e M At i c s c o M p e t i t i o n a n a c t i v i t y o f t h e a u s t r a l i a n m a t h e m a t i c s t r u s t thursday 4 august 2 011 senior Division Competition p aper instruCtions anD information GeneraL 1. Do not open the booklet until told to do so by your teacher. 2. NO calculators, slide rules, log tables, maths stencils, mobile phones or other calculating aids are permitted. Scribbling paper, graph paper, ruler and compasses are permitted, but are not essential. 3. Diagrams are NOT drawn to scale. They are intended only as aids. 4. There are 25 multiple-choice questions, each with 5 possible answers given and 5 questions that require a whole number answer between 0 and 999. The questions generally get harder as you work through the paper. There is no penalty for an incorrect response. 5. This is a competition not a test; do not expect to answer all questions. You are only competing against your own year in your own State or Region so different years doing the same paper are not compared. 6. Read the instructions on the answer sheet carefully. Ensure your name, school name and school year are entered. It is your responsibility to correctly code your answer sheet. 7. When your teacher gives the signal, begin working on the problems. tHe ansWer sHeet 1. Use only lead pencil. 2. Record your answers on the reverse of the answer sheet (not on the question paper) by FULLY colouring the circle matching your answer. 3. Your answer sheet will be scanned. The optical scanner will attempt to read all markings even if they are in the wrong places, so please be careful not to doodle or write anything extra on the answer sheet. If you want to change an answer or remove any marks, use a plastic eraser and be sure to remove all marks and smudges. inteGritY of tHe Competition The AMT reserves the right to re-examine students before deciding whether to grant official status to their score. australian school years 11 and 12 time allowed: 75 minutes ©amt P ublishing 2011 amtt limited acn 0 8 3 9 5 0 3 41

SeniorDivision Quest io ns 1to 10, 3 m arks each 1 . The expression 3x (x − 4) −2(5 −3x ) equals (A) 3x 2− 3x − 14 (B)3x 2− 6x − 10 (C)3x 2− 18x+ 10 (D) 3x 2− 18x− 10 (E)9x 2− 22x 2. A coac hnotices that2out of5pla yers inhis club arestudying atuniv ersity. If there are12univ ersitystuden tsin his club, how man ypla yers are there intotal? (A) 20 (B)24 (C)30 (D)36 (E)60 3. The value of14 ÷0.4 is (A) 3.5 (B)35 (C)5.6 (D)350 (E)0.14 4. In the diagram, ABCD isasquare. What is the value ofx? (A) 142 (B)128 (C)48 (D) 104 (E)52 .................................................................................................................................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................................................................................................................................................................................... ............................................................................................................................ 52◦ x◦ A B C D 5 . Whic hof the follo wing isthe largest? (A) 210 (B)2 10 (C) 10 2 (D) 20 1 (E)21 0 6. If m and nare positiv ewhole num bers and mn=100, then m+ncannot be equal to (A) 25 (B)29 (C)50 (D)52 (E)101

S2 7. PQ RS isasquare. Tisapoin ton RS suc hthat QT = 2R T. The value ofxis (A) 100 (B)110 (C)120 (D) 150 (E)160 ........................................................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P Q R S T x ◦ 8 . In my neigh bourho od, 90% ofthe prop erties arehouses and10% areshops. Toda y, 10% ofthe houses areforsale and30% ofthe shops areforsale. What percen tage of the prop erties forsale arehouses? (A) 9% (B)80% (C)331 3% (D)75% (E)25% 9. The value of 12+ 14+ 18 2+ 4+ 8 is (A) 16 (B)4 (C)1 (D)1 4 (E) 1 16 1 0. Anne’s morning exerciseconsistsofwalking adistance of1km atarate of5km/h, jogging adistance of3km at10km/h andfastwalking foradistance of2km at 6 km/h. Ho wlong does ittak eher tocomplete hermorning exercise? (A) 30min (B)35min (C)40min (D)45min (E)50min Quest io ns 11 to 20, 4 m arks each 1 1. The diagram shows asquare ofside length 12units divided into six triangles of equal area. What isthe distance, inunits, ofTfrom the side PQ ? (A) 4 (B)3 (C)2 (D) 1 (E)√ 5 . .......................................................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ... ... .. .... .. . ... .. .. . ... ... . .. ... ... ... ... .. .... ... . .. .. ... . .. .... .. ... ... .. . .. .... . .. ... .. . .. ... ... ... ... .. . . .. .. . . . . ... . .. .. . .. . .. .. . ... . . . . .. .. . . .. .. .. .. .. . ... .. .. . ... .. . . .. .. . . .. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................................................................................................................................................................................... . ................................................................................................................. ....................................................................................................................................................... P S Q R T

S3 12. Eac hof the first sixprime num bers iswritten onaseparate card.Thecards are sh uffled andtw o cards areselected. Theprobabilit ythat thesum ofthe num bers selected isprime is (A) 1 5 (B) 1 4 (C) 1 3 (D) 1 2 (E) 1 6 1 3. Tw otourists arewalking 12km apart alongaflat trackat aconstan tsp eed of 4 km/h. Wheneachtourist reaches theslopeof amoun tain,shebegins toclim b with aconstan tsp eed of3km/h. ✲ ✛ 12 km ✡✡ ✡ ✣ ✑ ✑ ✸ ✑ ✑ ✰ ? km What isthe distance, inkilometres, bet ween thetw o tourists duringtheclim b? (A) 16 (B)12 (C)10 (D)9 (E)8 14. Lines parallel tothe sides ofarectangle 56cm by 98 cm and joining itsopp osite edges aredrawn sothat they cutthis rectangle into squares. Thesmallest num ber of suc hlines is (A) 3 (B)9 (C)11 (D)20 (E)75 15. What isthe sum ofthe digits ofthe positiv ein teger nfor whic hn 2+ 2011 isthe square ofan integer? (A) 6 (B)7 (C)8 (D)9 (E)10 16. Of the staff inan office, 15rode apush biketo work onMonda y, 12 rode on Tuesda y and 9ro de on Wednesda y. If 22 staff rode apush biketo work atleast onceduring thesethreedays, what is the maxim umnum ber ofstaff whocould have ridden apush biketo work onall three days? (A) 4 (B)5 (C)6 (D)7 (E)8 12 km ? km

S4 17. Ho wman yinteger values ofnmak en 2− 6n + 8apositiv eprime num ber? (A) 1 (B)2 (C)3 (D)4 (E)aninfinite num ber 18. If x 2− 9x + 5= 0, then x 4−18x 3+ 81x 2+ 42 equals (A) 5 (B)25 (C)42 (D)67 (E)81 19. The centre ofasphere ofradius 1is one ofthe vertices ofacub eof side 1. What isthe volume ofthe com bined solid? (A) 7 π 6 + 1 (B)7 π 6 + 5 6 (C) 7 π 6 + 4 3 (D) 7 π 8 + 1 (E)π+ 1 20. In abest offivesets tennis match(where thefirst player to win three setswins the matc h),Chris hasaprobabilit yof 2 3of winning eachset. What isthe probabilit y of him winning thisparticular match? (A) 2 3 (B) 190 243 (C) 8 9 (D) 19 27 (E) 64 81 Quest io ns 21 to 25, 5 m arks each 2 1. Ho wman y3-digit num bers can be written asthe sum ofthree (notnecessarily differen t)2-digit num bers? (A) 194 (B)198 (C)204 (D)287 (E)296 22. A rectangular sheetofpap erisfolded alongasingle linesothat onecorner lies on top ofanother. Inthe resulting figure,60%ofthe area istw o sheets thickand 40% isone sheet thick.What isthe ratio ofthe length ofthe longer sideofthe rectangle tothe length ofthe shorter side? (A) 3:2 (B)5:3 (C)√ 2 :1 (D)2:1 (E)√ 3 :2

S5 23. An irrational spiderlives at one corner ofaclosed box whic his acub eof edge 1 metre. Thespider isnot prepared totravel more than√ 2 metres fromitshome (measured by the shortest routeacross thesurface ofthe box). Whic hof the follo wing isclosest tothe prop ortion (measured asapercen tage) ofthe surface of the box that thespider nevervisits? (A) 20% (B)25% (C)30% (D)35% (E)50% 24. Functions f, g and hare defined by f (x ) = x+ 2 g (0) =f(1) g (x ) = f(g (x − 1)) forx≥ 1 h (0) =g(1) h (x ) = g(h (x − 1)) forx≥ 1. Find h(4). (A) 61 (B)117 (C)123 (D)125 (E)313 25. A cone hasbase diameter 1unit andslantheigh t3 units. From apoin tA halfw ay up the side ofthe cone, astring ispassed twice around itto come toapoin tB on the circumference ofthe base, directly belo wA.The string isthen pulled until taut. . ....................................................................................................................................................................................................................................................................................... . .................................................. ..................................................................................................................................................................................................................................... . .. .. .. .. .. .. .. .. ... .. ... .. ... ... .... .... .... .... ..... ..... ..... ....... ........ .......... .......... .......................... .. ............................................ ........................................................................................................ . ....................................................................................................... . . . . . .. . . . .. . . .. . . . .. . . . .. . . . .. . .. . . .. . . .. . . .. . .. . . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. .. .. .. .. ... .. .. .. .. .. .. .. ... ... .... ... ... .... ... .... ......... .......... ........................................ .......................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . .. . . . . .. . . . .. .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . .. . . .. . . . . . .. . . . .. . . . .. . .. . .. .. . . . . . .. .. . . . .. . . .. .. . . . . . .. . . . .. . . . .. .. . . .. . . . . . . . . . . .. .. . .. .. . . . .. .. . . .. .. . .. .. . .. . .. . .. .. . . ... . .. .. .. . . ... .. . .. .. .. .. .. .. .. ... . ... .. . .. .. .. .. .. .. .. A B Ho wfar isitfrom Ato Balong thistaut string? (A) 3 8(√ 29 +√ 53 ) (B)3√ 7 2 (C) 3√ 3 2 (D) 9 4 (E) 3√ 108 8

S6 For quest io ns 26 to 30, shade the answ eras an in teger from 0to 999 in t he space provided on the answ ersheet. Quest io n 26 is 6 m arks, questio n 27 is 7 m arks, questio n 28 is 8 m arks, quest io n 29 is 9 m arks andquest io n 30 is 10 m arks. 26. Paul isone year older thanhiswife andthey have tw o children whoseagesarealso one year apart. Paul notices thatonhis birthda yin 2011, theproduct ofhis age and hiswife’s ageplus thesum ofhis children’s agesis2011. What would have been theresult ifhe had done thiscalculation thirteenyears b efore? 27. The diagram showsthe netofacub e.On eachface there isan integer: 1,w,2011, x , y and z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................................................................................................................................................................................................... .................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y z 2011w 1 If eac hof the num bers w,x, y and zequals theav erage ofthe num bers written on the four faces ofthe cubeadjacen tto it, find thevalue ofx. 2 8. Tw obeetles sitatthe vertices Aand Hofacub eAB CD EFGH with edgelength 40 √ 110 units. Thebeetles startmoving simultaneously alongACand HF with the speed ofthe first beetle twice thatofthe other one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................................................................................................................................................................... . ............................................................................................................................ ................. . .......................................................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................................................................................. A D C B E F G H ✈ s ✈ s . ..... ...... ...... ...... . ..... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... ........... ........... ...... ..... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What willbe the shortest distance bet ween thebeetles? 29. A family ofsix has sixChristmas crackers topull. Eachperson willpull tw o crac kers, eachwith adifferen tperson. Inhow man ydifferen tways can this be done?

S7 30. A 40 ×40 white square isdivided into 1× 1squares by lines parallel toits sides. Some ofthese 1× 1squares arecoloured redsothat eachof the 1× 1squares, regardless ofwhether itis coloured redornot, shares aside with atmost onered square (notcounting itself ).What isthe largest possible num ber ofred squares?

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