Intermediate Division Solutions
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1 Itermediate Divisi Slutis 1. Cmpute the largest 4-digit umber f the frm ABBA which is exactly divisible by 7. Sluti ABBA 1000A + 100B +10B+A 1001A + 110B 1001 is divisible by 7 ( ), s 1001A is exactly divisible by 7. Hece ABBA is divisible by 7 if ad ly if 110B is divisible by 7. As 110 is t divisible by 7 ad 7 is a prime, B must be divisible by 7 fr 110B t be divisible by 7. Thus ABBA is divisible by 7 wheever B 0 r 7. I particular, the largest such 4-digit umber ccurs whe A 9 ad B 7, s it is Remark: May studets assumed that A ad B had t be differet digits. This is t ecessarily the case, ad this pssibility als had t be dealt with. Remark: This questi culd als be slved briefly with trial ad errr, as 9779 is the third highest 4-digit umber f the frm ABBA.. Imagie a crridr with 100 light switches, all switched ff. 100 peple walk thrugh the crridr e after ather. The first pers flips every switch (startig with the secd), thus turig every secd switch ff, the third pers flips every third switch (startig with the third), etc. At the ed, state whether the 79 th switch ad the 100 th switch is r ff, givig reass. Sluti The first pers flips every switch i.e. every switch whse umber is a multiple f 1. The secd pers flips every secd switch thse switches whse umber is a multiple f. I geeral, the th pers flips every switch whse umber is a multiple f. S the m th switch is flipped by the th pers if ad ly if m is divisible by ifis a factr f m. 79 ly has factrs 1 ad 79. The 79 th switch is therefre switched twice ad at the ed is ff. 100 has 9 factrs 1,, 4, 5, 10, 0, 5, 50, 100. S the 100 th switch is flipped a dd umber f times ad at the ed is. (Shrtcut: e may tice that fr ay psitive iteger, its factrs may be gruped it pairs that multiply t. Fr example, Thus, every psitive iteger has a eve umber f factrs, except thse fr which a factr pairs up with itself t multiply t whe is a perfect square. Hece we may cclude that every switch whse umber is a perfect square is at the ed, ad every ther switch is ff.) Remark: May studets merely listed the factrs f 79 ad 100 ad wrte the aswer. remided that sme explaati is required t cstitute a prf! They are 3. Baktes i the mythical cutry Caalia are f fur demiatis: $1, $10, $100, $1000. Ca e have half a milli tes with ttal value $1 milli? Sluti N, e cat have 500, 000 tes f value $1, 000, 000. Suppse it were pssible t have such tes. The let a, b, c, d dete the umber f $1, $10, $100, $1000 tes respectively. We the have 1
2 a +10b+ 100c d 1,000, 000 (1) a + b + c + d 500, 000 () Subtractig () frm (1) gives 9b +99c+ 999d 500, 000 9(b +11c+ 111d) 500, 000 Nw 9(b +11c+ 111d) is a iteger divisible by 9, while 500, 000 is t. This is a ctradicti, s such a cmbiati f baktes is impssible. 4. A grup f N bys ad N girls are sittig arud a table. It is kw that sme f them always lie, while the rest always tell the truth. Further, the umber f liars amg the bys is the same as the umber f liars amg the girls. Whe asked the questi What geder is yur right-had eighbur? everye replies Girl. Prve that N is eve. Sluti Sice everye replies Girl, all thse sittig t the left f a girl tell the truth, while thse sittig t the left f bys lie. Thus the umber f liars is equal t the umber f bys, which is N. As there are equal umbers f male ad female liars, the ttal umber f liars, N, iseve. 5. Let S { , ,..., } be the set f all biary sequeces f legth 7. The distace f tw elemets s 1,s S is the umber f places i which s 1 ad s differ. Fr example, ad have distace, sice they differ i psitis 1 ad 7. Shw that if T is a subset f S havig mre tha 16 elemets (that is, take a sample f mre tha 16 such biary sequeces), the T ctais tw elemets whse distace is at mst. Sluti Let T be a subset f S ctaiig 17 r mre elemets. We must shw that there are tw biary sequeces i T which are sufficietly similar r clse tgether as t have distace. The idea is t iclude, alg with each biary sequece t i T, all biary sequeces that ly differ i e place frm it (i.e. all sequeces distace 1 away frm t). Thik f this as a ball arud t cverig everythig up t a radius f 1 away frm t, ad t as the cetre f the ball. S place a ball arud each sequece i T as shw i Figure??. If ay tw f these balls verlap (i.e. have a biary sequece i cmm), the there are tw elemets i T that differ by r less. This is because, if we let the cetres f the tw verlappig balls be u ad v (u ad v are biary sequeces i T ), ad bth balls ctai w, the w is 1 away frm u ad w is 1 away frm v. The the distace betwee u ad v is r less. Hwever, each ball ctais 8 thigs the cetre biary sequece ad 7 biary sequeces, each sequece differig frm the cetre i e place (differet fr each). There are 17 r mre balls, as there are at least 17 elemets i T, s the balls take tgether ctai at least sequeces. But S, the set f all biary sequeces f legth 7, ly ctais 7 18 sequeces. Thus the balls cat all be separate ad there must be tw balls which ctai cmm elemets they must verlap. Hece there must be tw sequeces i T whse distace is at mst. Remark: S here is a example f a mathematical bject kw as a metric space: a set with a distace defied betwee elemets.
3 + - sequece i T - ther sequece i S + u w + v + Figure 1: Balls arud sequeces i T Remark: This was a very hard prblem bdy fud the fficial sluti. Sme slutis were fud which csidered all pssible cases, makig liberal use f withut lss f geerality (WLOG) argumets ad the pige hle priciple (PHP). 6. Fid the smallest psitive iteger such that is the square f a iteger, 3 iteger, ad 5 is the fifth pwer f a iteger. is the third pwer f a Sluti Clearly is divisible by, 3, 5, s we ca express i a prime factrised frm: a 3 b 5 c m where a, b, c are psitive itegers ad m is t divisible by, 3 r 5. Recall that fr a iteger k i prime factrised frm, k is a perfect square if ad ly if the expets f all primes are eve. Similarly, k is a perfect cube if ad ly if the expets f all primes are multiples f 3 ad k is a perfect fifth pwer if ad ly if the expets f all primes are multiples f 5. Nw 3 5 a 1 3 b 5 c m (1) a 3 b 1 5 c m () a 3 b 5 c 1 m (3) These 3 expressis are respectively a perfect square, cube ad fifth pwer. S a is a multiple f 3, a multiple f 5 ad a 1 is a multiple f. The least such psitive iteger is 15. Similarly b is a multiple f ad 5 ad b 1 is a multiple f 3. The least such psitive iteger is 10. c is a multiple f ad 3 ad c 1 is a multiple f 5. The least such psitive iteger is 6. It fllws that a 3 b 5 c m But this value f has the required prperties, ad s is the smallest such iteger. 3
4 7. Pythagras therem relatig t a right triagle is well kw. Csider the fllwig three dimesial aalgue. The crer f a rectagular prism is cut ff by a diagal plae, resultig i a pyramidal shaped slid, 3 (triagular) faces f which derive frm the rigial cube, while the furth (triagular) face derives frm the diagal plae. Ca yu relate the area f this furth face t the area f the ther three faces, (ad prve yur claim)? Sluti (based the sluti preseted by Nichlas Sherida) Let the vertices f the slid be labelled O, A, B, C as shw i Figure??. Let the distaces frm O t A, B, C respectively be a, b, c. C c O b B a X A Figure : Right-agled tetrahedr fr Questi 7 First csider the base triagle OAB. Let X be the ft f the perpedicular frm O t AB. By Pythagras Therem, AB OA + OB a + b. Nw as AOB AXO 90, OAX BAO, wehave AOB AXO. Hece OX AO BO AB,s OX AO BO AB ab a + b Nw csider OCX, which is right-agled as OX lies i plae OAB ad OC is perpedicular t this plae. By Pythagras Therem, CX OX + OC CX a b a + b + c ab b + a c + b c a + b a b + a c + b c 4 a + b
5 OC is perpedicular t plae OAB, soc AB. Als OX AB by defiiti. Thus plae OCX is perpedicular t AB, ad CX AB. S we ca csider AB a base, ad CX a height, f triagle ABC. The area f ABC ca w be calculated: ABC 1 AB CX 1 a a + b b + a c + b c a + b (ab ) ( ac ) ( ) bc + + OAB + OAC + OBC Hece ABC OAB + OAC + OBC. S the square f the area f the furth face is equal t the sum f the squares f the areas f the ther three sides a three dimesial aalgue f Pythagras therem. Remark: This was a difficult questi. Very few slved it, ad (disappitigly) very few eve maaged t guess the aswer despite a strg hit i the statemet f the prblem ( Csider the fllwig three dimesial aalgue f Pythagras therem ). A alterative apprach is t use Her s frmula fr the area f a triagle give its side legths. 5
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