State 2003 Mathcounts

8/19/2019 State 2003 Mathcounts

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2003 STATE COMPETITION

SPRINT ROUND QUESTIONS

1. Row A shows the integers 4 and 10.Row B shows the integers 3, 5, 9, 11.

Row C shows the integers 2, 6, 8, 12,14.Row D shows the integers 1, 7, 13.14 will aear in Row C, 15 in RowB and 16 in Row A. !o Row A nowas 4, 10 and 16. "he atterne#erges. $a%h integer in Row A is 6greater than the re&io's integer inRow A. And the (irst integer is 2less than 6. !o the &al'e %an )e

e*ressed as 6* + 2. "here(ore, theeighth &al'e in Row A is

6 × 8 + 2 - 48 + 2 - 46. Ans.

2. irst, ass'#e there are no lea /ears.ater, we will see whether it #attersor not. "here are 365 da/s in a /ear.

3657

7365= with no re#ainder. t

does not #atter whether there is a

lea /ear or not within the 7 /earsor e&en two or none at all )e%a'sethe additional da/s wo'ld not %reatean additional wee. 365 Ans.

3. e are ased to (ind the er(e%t+s'are integer that is %losest to 273.102 - 100 and 202 - 400. )&io'sl/,the &al'e is in )etween. 152 - 225,162 - 256 and 172 - 289. t #'st )eeither 16 or 17.

273 + 256 - 17289 + 273 - 16289 is %loser )/ 1. 289 Ans.

4. "here are 300 st'dents that too thetest. "o (ind a &al'e that is )etterthan 75 o( the s%ores, we #'st (indthe 225th &al'e sin%e 225 st'dents are

75 o( the total. "hat is the 76th

s%ore i( we start (ro# the highest.

7 16 - 23 with a s%ore ≥ 11.

23 37 - 60 with a s%ore ≥ 10.

60 45 - 105 with a s%ore ≥ 9.

"he 225th

st'dent #'st ha&e s%ored a9. "o ha&e a s%ore that is higher thanthat, add 1. 10 Ans.

5. 17, a, ), %, 41 is an arith#eti%se'en%e. "his #eans that thedi((eren%e )etween two &al'es in these'en%e is alwa/s the sa#e.41 + 17 - 24"his in%re#ent o( 24 is rea%hed in 4stages so the di((eren%e )etween

&al'es is41 × 24 - 6.

17 6 - 2323 6 - 29"here(ore, ) - 29. Ans.

6. "wo %o#le#entar/ angles are inthe ratio o( 7 to 23. "he %o#le#ento( angle A is angle B and the%o#le#ent o( angle B is angle A."h's, the ratio o( angle B to angle A

is :'st the oosite o( the ratio o(angle A to angle B or

7

23. Ans.

7. n 7 /ears, Ri%h;s #one/ do')led(ro#


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/ -4

17

2

1 x

"he sloe, or #, is +2

1. "he sloe o(

a line arallel to this line will ha&e

this e*a%t sloe.

2

1 Ans.

9. >ow #an/ distin%t ositi&e integers%an )e reresented as the di((eren%eo( two n'#)ers in the set1, 3, 5, 7, 9, 11, 13Clearl/, ea%h integer s')tra%ted (ro#the integer dire%tl/ on its right gi&esa di((eren%e o( 2. $a%h integers')tra%ted (ro# the integer, two&al'es to the right, gi&es a di((eren%eo( 4. !i#ilarl/, we %an getdi((eren%es o( 6, 8, 10 and 12 (or atotal o( 6 integers. 6 Ans.

10. oint is lo%ated at 1,3 and ointR is lo%ated at 7,15. oint = is the#idoint o( seg#ent R. "he *+%oordinate o( the = is7 + 1 - 6

6 ÷ 2 - 31 3 - 4"he /+%oordinate o( = is15 + 3 - 12

12 ÷ 2 - 63 6 - 9!o = is 4, 9.!eg#ent R is re(le%ted o&er the *+a*is as in the (ig're )elow.

$a%h oint has the sa#e * and /%oordinates except the /+%oordinateis negated. !o the re(le%ted oint (or

= is4, +9.4 +9 - +5 Ans.

11. n a three+digit n'#)er, the h'ndreds

digit is greater than 5, the tens digitis greater than 4 )'t less than 8 andthe 'nits digit is the s#allest ri#en'#)er. >ow #an/ three+digitn'#)ers satis(/ all o( these%onditions"he h'ndreds digit is greater than 5whi%h #eans 6, 7, 8, or 9 (or a totalo( 4 ossi)le &al'es. "he tens digitis greater than 4 )'t less than 8 or 5,6, 7 (or a total o( 3 di((erent &al'es.

"he 'nits digit is the s#allest ri#en'#)er or 2. "his is a total o( onl/one digit.

4 × 3 × 1 - 12 Ans.

12. "he tri (ro# Car&ille to @iath

re'ires 42

1ho'rs when tra&eling at

an a&erage seed o( 70 #iles erho'r. "his #eans the total distan%e

is 42

1 × 70 -

2

9 × 70 - 9 × 35 - 315

#iles. ( tra&eling at a seed o( 60#iles er ho'r, the tri #'st tae

4

15

60

155

60

315= - 5.25 Ans.

13. Re%tangle ABCD has the &erti%esA0,0, B6,0, C6,10, and D0,10."he oint $ is on seg#ent CD at2,10. A line %onne%ting $ with Ais drawn as in the (ig're )elow.

"his %reates a right triangle, AD$,with two o( the sides )eing 10 and 2."he area o( the triangle is

2003 =A">C?@"! !tate Co#etition sol'tions


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2

1 × 10 × 2 - 10

"he area o( 'adrilateral ABC$ isthe area o( triangle AD$ s')tra%ted(ro# the area o( re%tangle ABCD.

"he area o( ABCD is 10×

6 - 60."he area o( 'adrilateral ABC$ is60 + 10 - 50"he ratio o( the area o( triangle AD$to the area o( 'adrilateral ABC$ is

5

1

50

10= Ans.

14. "he digits 2, 4, 6 and 9 are la%eds'%h that two are in the n'#eratorand two are in the deno#inator. e

are ased to (ind the s#allest o( allthe %o##on (ra%tions that %an )e(or#ed. "he s#allest &al'e willo%%'r when the s#allest &al'e is onto and the largest &al'e is on the )otto#. Choose 24 (or then'#erator and 96 (or thedeno#inator.

4

1

96

24= Ans.

15. At 10 A=, Boon "ee is the 225

th

erson in line to ride the roller%oatser. $a%h roller %oaster trainholds 36 eole and a ('ll trainlea&es e&er/ (o'r #in'tes. "he (irst36 eole in line lea&e on the1001train. irst deter#ine whi%h gro'Boon "ee will )e in.

36

225- 6 R 9.

Boon "ee will )e in the 7th gro'.

"he (irst gro' lea&es at 1001. twill tae 6 × 4 - 24 #ore #in'tes'ntil Boon "ee %an lea&e.1001 24 - 1025 Ans.

16. A triangle has &erti%es A6,1, B4,1and C4,4 as shown in the i%t're.

"he triangle is a right triangle withhoriEontal length, 2, and &erti%al

length, 3. "he triangle is rotated 90degrees %o'nter%lo%wise aro'nd B."his #eans the triangle will ha&e&erti%al length, 2, and horiEontallength 3, as in this i%t're.

!in%e B is at 4,1, A #'st )e 2 'nitshigher, or 4,3. C is now 3 'nits tothe le(t o( B or 1,1. Ans.

17. hat is the least nat'ral n'#)er that%an )e added to 40,305 to %reate a alindro#e A alindro#e is thesa#e n'#)er when read (orwards or )a%wards. "he nearest alindro#ewo'ld )e 40,304 )'t we #'st (indthe ne*t alindro#e greater than40,305. "here %an )e no #ore alindro#es in the 40,300 range. !owe #'st loo at 40,400 and larger.

"he ne*t alindro#e is 40,404.40404 + 40305 - 99 Ans.

18. A F B - B A B

A×

4 F 2 -2

4 4 × 2 - 2 8 - 10

20 F 10 -10

20 20 × 10

- 2 200 - 202 Ans.

19. @'#)ers on a standard si*+(a%ed dieare arranged s'%h that n'#)ers onoosite (a%es alwa/s add to se&en.e need to (ind the largest rod'%to( all (a%es e*%et the to and )otto#. Clearl/ the larger then'#)er the larger the rod'%t will )e. B't we are li#ited )/ oosing



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&al'es ha&ing to add to 7. Certainl/the ideal is to get rid o( a 1 in the rod'%t. !o we re#o&e 6 and 1.

5 × 4 × 3 × 2 - 120 Ans.

20. Angle GR is a right angle. "hethree 'adrilaterals shown ares'ares.

"he s'# o( the areas o( the threes'ares is 338 s'are %enti#eters.e #'st (ind the area o( the largests'are. hat we are looing (or,then, is the addition o( three er(e%ts'ares to e'al 338. n additiontriangle GR is a right triangle sothese &al'es are ro)a)l/ one o( o'r(a&orite right triangles whi%h are3, 4, 56, 8, 105, 12, 138, 15, 17"he s'are o( 5 is 25. "his will )etoo s#all."he s'are o( 10 is 100. "he s'#will )e too s#all."he s'are o( 13 is 169, 12 is 144and 5 is 25.169 144 25 - 338 BingoH169 Ans.

21. Container holds 8 red )alls and 4green )alls.

Container holds 2 red )alls and 4green )alls.Container also holds 2 red )allsand 4 green )alls.e sele%ted a %ontainer and a )all atrando# and #'st (ind the ro)a)ilit/that the sele%ted )all is green."he ro)a)ilit/ o( Container )eing

sele%ted is3

1. "he ro)a)ilit/ o(

i%ing a green )all (ro# Container

is3

1

12

4

48

4=

"h's, the

ro)a)ilit/ o( sele%ting a green )all(ro# Container is

9

1

3

1

3

1= .

"he ro)a)ilit/ o( sele%ting

Container is3

1. "he ro)a)ilit/ o(

sele%ting a green )all is3

2

6

4

42

4=

.

"h's, the ro)a)ilit/ o( sele%ting agreen )all (ro# Container is

.

9

2

3

2

3

1=

"he ro)a)ilit/ o( sele%ting a green )all (ro# Container is the sa#e asthe ro)a)ilit/ o( sele%ting a green

)all (ro# Container or9

2.

9

5

9

2

9

2

9

1= Ans.

22. "he arith#eti% #ean o( ninen'#)ers is 54. ( two n'#)ers ' and

& are added to the list, the #ean o(the ele&en+#e#)er list )e%o#es 66.!in%e the #ean o( the original ninen'#)ers is 54, the s'# o( all 9

&al'es #'st )e 54 × 9 - 486.Adding two n'#)ers gi&es a #ean o(66, so the s'# o( all 11 n'#)ers is726.726 + 486 - 240240 is the s'# o( ' and &. "h's, the

#ean o( ' and & is2

240 - 120. Ans.

23. A gasoline ga'ge originall/ read8

1

('ll. "hen, 15 gallons o( gasoline

were added and the ga'ge read4

3

('ll. "h's,



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8

5

8

1

8

6

8

1

4

3= o( the tan holds 15

gallons o( gas. !o8

1 o( the tan

holds5

15 - 3 gallons. !in%e the tan

is4

3 ('ll it is a%t'all/

8

2

4

1= e#t/.

2 × 3 - 6 Ans.

24. "hree (air, standard si*+(a%ed di%e o(di((erent %olors are rolled. n how#an/ wa/s %an the di%e )e rolleds'%h that the s'# o( the n'#)ersrolled is 10 Below is a ta)le. @o%o#)ination will )e reeated.

1 2 31 1 8 #ossi)leH1 2 7 #ossi)leH1 3 6 6 wa/s1 4 5 6 wa/s2 2 6 3 wa/s2 3 5 6 wa/s2 4 4 3 wa/s3 3 4 3 wa/sAn/ other %o#)ination has alread/ )een entered.

6 6 3 6 3 3 - 27 Ans.

25. Arlene sis all #'ltiles o( 3 andall n'#)ers that %ontain the digit 3.n the (irst 10 n'#)ers there is onen'#)er %ontaining 3 and 3 #'ltileso( 3 one o( whi%h is 3 so 3 n'#)ersare dis%o'nted. "here(ore there are 7n'#)ers )etween 1 and 10.n the ne*t 10 n'#)ers, i.e., 11+20,there are 3 #'ltiles di&isi)le )/ 3

and one n'#)er %ontaining 3 that isnot di&isi)le )/ 3, so 4 n'#)ers aredis%o'nted, lea&ing 6. 7 6 - 13.n the ne*t 10 n'#)ers, i.e., 21+30there are 4 n'#)ers di&isi)le )/ 3and one n'#)er %ontaining 3 that isnot di&isi)le )/ 3, i.e., 23 and onen'#)er %ontaining 3 that is di&isi)le

)/ 3 and so is alread/ %o'nted, i.e.,30 so 5 n'#)ers are dis%o'nted. 13 5 - 18n the ne*t 10 n'#)ers, i.e., 31+40onl/ one n'#)er has no 3 in it and it

40 is not di&isi)le )/ 3 so onl/ 1n'#)er is dis%o'nted. 18 1 - 19!tarting with 41+50 the atterne#erges + 7, 6, 5 (or e&er/ thirt/n'#)ers.19 6 - 25 41+5025 5 - 30 51+6030 7 - 37 61+70!in%e we are looing (or the 40th

n'#)er, there are onl/ 3 to go. 71 isthe 38th n'#)er. 72 is di&isi)le )/ 3

and 73 ends in 3. "here(ore, the 39

th

n'#)er is 74. 75 is di&isi)le )/ 3 sothe 40th n'#)er is 76. Ans.

26. n2 gi&es a re#ainder o( 4 whendi&ided )/ 5 and n3 gi&es are#ainder o( 2 when di&ided )/ 5.( n2 gi&es a re#ainder o( 4 whendi&ided )/ 5, the n'#)er #'st ha&ea s'are whi%h ends in either 4 or 9."hen, i( n3 gi&es a re#ainder o( 2when di&ided )/ 5, the n'#)er #'stalso ha&e a %')e whi%h ends in either2 or 7.Case @'#)ers that, when s'ared,end in 4, #'st end in 2 or 8. 23 - 8so a n'#)er ending in 2, when %')edwo'ld end in 8. !o n %annot end in2. 83 - 512 so the n'#)er %o'ld endin 8.Case @'#)ers that, whens'ared, end in 9, #'st end in 3 or 7.73 - 343 so a n'#)er ending in 7,when %')ed wo'ld end in 3. !o n%annot end in 7. 33 - 27 so then'#)er %o'ld end in 3.hen an/ n'#)er ending in 8 or 3 isdi&ided )/ 5, the re#ainder will )e3. Ans.



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27.7

4=

y

x

3

14=

z

y

3

8

3

24

3

14

7

4==

z

y

y

x

z

x

3

22

3

14

3

8==

z

y x

z

y

z

x Ans.

28. "wo di((erent n'#)ers are rando#l/sele%ted (ro# the set! - 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11"he ro)a)ilit/ that their s'# is 12wo'ld )e greater i( the n'#)er n had(irst )een re#o&ed (ro# the set.( we re#o&e 1, the airs whose s'#

wo'ld res'lt in 12 are 2,10, 3, 9,4, 8, 5, 7 or 4 %hoi%es.( 2 is re#o&ed (ro# the set, the%hoi%es whose s'# wo'ld res'lt in12 are 1, 11, 3, 9, 4, 8, 5, 7 or4 %hoi%es.( 3 is re#o&ed (ro# the set, the airs are 1, 11, 2, 10, 4, 8, 5, 7or 4 %hoi%es. !i#ilarl/ (or 4 and 5.hat a)o't 6"he %hoi%es are 1, 11, 2, 10, 3,

9, 4, 8, 5, 7 (or 5 airs. "his is )e%a'se 6 6 - 12 and we %an;t%hoose the sa#e &al'e twi%e.!i#ilarl/, re#o&ing 7, 8, 9, 10, or 11wo'ld gi&e 's 8 %hoi%es ea%h ti#e.6 Ans.

29. "he %lo% reads 420 and the se%ondhand #aes one %o#lete %ir%lee&er/ 4 se%onds. "he #in'te handand ho'r hand )eha&e nor#all/

within this %onte*t. "o#as %a#einto the roo# at 900 a.#. and willlea&e at 950 a.#.

50 #in'tes - 50 × 60 - 3000se%onds.$a%h 4 se%onds are re(le%ted )/ 60se%onds o( #o&e#ent in the %lo%."his #eans that the %lo% #o&es

4

3000 × 60 - 45000 se%onds or

75060

45000= #in'tes

750 #in'tes is 12 ho'rs and 30

#in'tes. !in%e the %lo% was at 420at 9, the %lo% will )e at 450 when"o#as lea&es. 450 Ans.

30. 41

=

x x

161

21

2

22

=

x x

x x

=

6411

21

2

23

x x

x x

x x

=

3

2

23 122

x x x x

x x x x

=

3

3 1212

x x x

x x x

=

x x

x x

x x

12

11

3

3

=

x x

x x

13

1

3

364

3

3 1

x x - 64 + 3 × 4 - 64 +12 -

52 Ans.

TARGET ROUND QUESTIONS

1. A sta% o( 100 ni%els is 6.25 in%heshigh.

9681225.6

100 x x=

×

=

6.25* - 96 × 100 - 9600

* -25.6

9600- 1536 ni%els

201536 -


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"hese lines are / - *, / - * 1, / -* 2, et%. and / - +* 2, / - +* 3,et%. As is e&ident, there are 8interse%tion oints within the shadedregion.8 Ans.

3. "he s'# o( 9 %onse%'ti&e integers is9. et * )e the (i(th, or #iddle one.* + 4 * + 3 * + 2 * + 1 * * 1 * 2 * 3 * 4 - 9

9* - 9* - 1* + 4 - 1 + 4 - +3 Ans.

4. A %')e is sli%ed )/ a lane whi%hgoes thro'gh two oosite %ornersand the #idoint o( two edges asshown.

"he edge o( the %')e has length one'nit. !o ea%h side o( the rho#)'s isthe h/oten'se o( a triangle whose

other sides are 1 and2

1.

"he area o( a rho#)'s is

212

1d d where 1d and 2d are the

diagonals o( the rho#)'s. "he

shorter diagonal, 1d , is :'st the

e'i&alent o( a diagonal o( a (a%e o(the %')e.

21111 222

1 =d

21 =d

"he longer diagonal, 2d , is the

diagonal (ro# the )otto# o( the %')eon one side to the to o( the %')e on

the other side. "his diagonal is theh/oten'se o( a right triangle whosesides are an edge o( the %')e and adiagonal o( the (a%e whi%h is the

sa#e as 1d .

321212

222 =d

32 =d

2

632

2

1

2

121 =d d Ans.

5. 310 #illion o( the 6.25 )illion eoleli&e in @orth A#eri%a.

=

000,000,250,6

000,000,3100.0496

0.0496 × 100 - 4.96 ≈ 5 Ans.

6. a%o 'ses a sinner to sele%t an'#)er (ro# 1 thro'gh 5, ea%h withe'al ro)a)ilit/. =an' 'ses adi((erent sinner to sele%t a n'#)er(ro# 1 thro'gh 10, ea%h with e'al ro)a)ilit/. e #'st (ind then'#)er o( sele%tions whose rod'%t

is less than 30. Clearl/, there are 5 ×10 - 50 %o#)inations.( a%o %hooses a 1, then =an' %an

%hoose an/thing (or 10%o#)inations.( a%o %hooses a 2, then =an' %an%hoose an/thing (or 10 #ore%o#)inations.( a%o %hooses a 3, then =an' %an%hoose 1 thro'gh 9 (or 9 #ore%o#)inations.( a%o %hooses a 4, then =an' %an%hoose 1 thro'gh 7 (or 7 #ore%o#)inations.

( a%o %hooses a 5, then =an' %an%hoose 1 thro'gh 5 (or 5 #ore%o#)inations.10 10 9 7 5 - 41

50

41 Ans.



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7. hat is the s#allest ositi&e integer

@ s'%h that the &al'e 7 30 × @ isnot a ri#e n'#)er.

7 30 × 1 - 7 30 - 37 is ri#e

7 30 × 2 - 7 60 - 67 is ri#e

7 30 × 3 - 7 90 - 97 is ri#e7 30 × 4 - 7 120 - 127 is ri#e

7 30 × 5 - 7 150 - 157 is ri#e

7 30 × 6 - 7 180 - 187 - 11 × 17187 is not ri#e.6 Ans.

8. An ant starts at the to &erte* o( areg'lar o%tahedron and wals alongthe edges o( the triangles witho'te&er tra&ersing the sa#e edge twi%e.

"he ant %o'ld a%t'all/ tra&erse alledges e*%et (or the (a%t that the ant#'st sto when it ret'rns to thestarting &erte*. "he i%t're )elown'#)ers the &erti%es.

"he ant %an onl/ tra&erse one &erte*(ro# 1 down to an/ o( 2, 3, 4, or 5."he ant will )e a)le to #o&e aro'nd(ro# 2 to 3 to 4 to 5 (or a total o( 4edges and down to the )otto#&erte*, 6, as #an/ ti#es as it taes(or 4 #ore edges. 1 4 4 - 9ne e*a#le is 1, 2, 6, 3, 2, 5, 6, 4,3, 1.

9 Ans.

TEAM ROUND QUESTIONS

1.128

987670- 7716.17185

7716 × 128 - 987648

987670 + 987648 - 22 Ans.

2. e are ased to (ind all re%tangles o(area 8 s'are 'nits that %an )e(or#ed 'sing onl/ the line seg#ents

o( the grid as the sides o( there%tangles. "his #eans 'singintegral (a%tors o( 8. "hese are1 * 8 and2 * 41 * 8 or 8 * 1 is i#ossi)le )e%a'sethe grid is 6 * 6. 2 * 4 or 4 * 2 is%ertainl/ ossi)le. "he i%t're )elow shows the horiEontal and&erti%al t/es o( re%tangles that are ossi)le.

"he 2 * 4 is (eat'red at the to le(tand the 4 * 2 is (eat'red at the )otto# right. irst, let;s deter#inehow #an/ 2 * 4 re%tangles we %anget. ( /o' #o&e the 2 * 4 re%tangleone %')e to the right /o' get anotherlegal 2 * 4 re%tangle. Do it on%e

#ore and the re%tangle will a)'t theedge o( the %')e. !o, there are 3re%tangles on that line. !i#ilarl/,#o&e the re%tangle one row down.Io' %an do that (or a total o( 5 ti#es

so /o' %an ha&e 5 × 3 - 15 o( the 2 *4 re%tangles. !i#ilarl/, tae the 4 *2 re%tangle and #o&e it to the le(tone %')e and /o' get another &alid 4* 2 re%tangle. Io' %an %ontin'e#o&ing le(t (or a total o( 5

re%tangles. Io' %an also #o&e 'towards the to o( the %')e (or a total

o( 3 re%tangles. Again, 5 × 3 - 15.15 15 - 30 Ans.

3. ( t is an odd ositi&e integer, thenthe &al'e o( the (ollowing ter# is3t + 9



9/11

( the &al'e o( a gi&en ter# is ane&en ositi&e integer, then the &al'eo( the (ollowing ter# is2t + 7"he ter#s o( the se'en%e alternate

)etween two ositi&e integers a, ),a, ), ... .!'ose a is odd. "hen the ne*tter# is 3a + 9 - ). 3a is alwa/s oddand i( /o' s')tra%t an odd (ro# anodd, /o' get an e&en &al'e so ) ise&en. "here(ore the ne*t &al'e #'st )e 2t + 7 or 23a + 9 + 7 - 6a + 18 + 76a + 25 - a5a - 25a - 5

) - 6"his wors.Co'ld a ha&e )een e&en "hen )wo'ld )e 2a + 7 whi%h wo'ld )e anodd n'#)er. "his wo'ld #ae thene*t ter# 32a + 7 + 9 - 6a + 21 + 96a + 30 - a5a - 30a - 6"hen, as we alread/ now, ) wo'ldha&e to )e 5.5 6 - 11 Ans.

4. A, B, C, D, and $ are all di((erentdigits in the (ollowing addition C A E

+ C D

A B B

$ither, $ D - B or $ D - B 10!o $ D + B - 0 or $ D + B - B 10 + B - 1010 0 - 10 Ans.

5. 3* / - 175/ E - 143* 5E - 416* 6/ 6E - 726* / E - 72* / E - 12 Ans.

6. e ha&e 6 di((erent !ider#an%o#i%s, 5 di((erent Ar%hie %o#i%s,and 4 di((erent Jar(ield %o#i%s.hen sta%ed, ea%h ind o( %o#i% isgro'ed together with others o( the

sa#e ind. "his #eans there are6H di((erent %o#)inations (or the!ider#an %o#i%s,5H di((erent %o#)inations (or theAr%hie %o#i%s, and4H di((erent %o#)inations (or theJar(ield %o#i%s.

6H × 5H × 4H -

720 × 120 × 24 -2,073,600 Are we done ell, not'ite. "here are 6 di((erent wa/s

that /o' %an la%e the sets o( %o#i% )oos so

6 × 2,073,600 - 12,441,600 Ans.

7. n is the ro)a)ilit/ than an KnK isrolled on a die.1 - 23 - 4 - 54 - 325 - 26hat is 6

4 - 323 - 325 - 32

6 -2

23

2

5 P P =

1 - 21 2 3 4 5 6 - 1

23232322 P P P P P

1

2

23=

P

12

23211 =

P P

12

225=

P

2 -25

2



10/11

6 -25

3

25

2

2

3

2

23=

P Ans.

8. An isos%eles traeEoid is ins%ri)ed ina se#i%ir%le as shown )elow.

"he three shaded regions are%ongr'ent. "he radi's o( the %ir%le is1. !in%e the traeEoid is an isos%elestraeEoid and the three shadedregions are %ongr'ent, we %an )reathe traeEoid ' into 3 %ongr'ente'ilateral triangles, ea%h o( side 1."here(ore, the height o( ea%h triangle

is

h2 2

2

1 - 12 - 1

h2 - 1 +4

3

4

1=

h -2

3

"he area o( the triangle is

4

3

2

3

2

1=

"here are three triangles so the totalarea is

4

33- 1.299038106 ≈ 1.3 Ans.

9. hat is the greatest ositi&e integern s'%h that 3n is a (a%tor o( 200H "oanswer this we need to now how#an/ #'ltiles o( 3 e*ist in 200H.

3

200- 66 R 2 so there are at least 66

3;s that %o#e (ro# (a%toring 200HB't now we ha&e to %onsider how#an/ n'#)ers are #'ltiles o( 9whi%h has two 3;s in it.

9

200- 22 R 2 so there are an

additional 22 #'ltiles o( 3. @ow%onsider how #an/ n'#)ers are

#'ltiles o( 27 whi%h has three 3;s init.

27

200- 7 R 11 so there are an

additional 7 #'ltiles o( 3. @e*t,

%onsider how #an/ n'#)ers are#'ltiles o( 81 whi%h has 4 three;s init.

81

200- 2 R 38 so we ha&e 2 #ore

#'ltiles o( 3. "he ne*t ower o( 3is 243 whi%h is too large.n - 66 22 7 2 - 97 Ans.

10. An a)'ndant n'#)er is a ositi&einteger, the s'# o( whose distin%t

roer (a%tors is greater than then'#)er. "he roer (a%tors are allthe (a%tors e*%et the n'#)er itsel(.>ow #an/ n'#)ers less than 25 area)'ndant n'#)ersRe#o&e 1 sin%e it has no (a%torsother than 1. Re#o&e all ri#essin%e the/ ha&e no (a%tors other than1 and the#sel&es. "his lea&es :'stthe e&en n'#)ers e*%et (or 2.a%tors o( 4 are 1 and 2. @o.

a%tors o( 6 are 1 2 and 3. @o.a%tors o( 8 are 1, 2, and 4. @o.a%tors o( 10 are 1, 2, and 5. @o.G'ite %learl/ we ha&e to ha&e so#eo( the (a&orite n'#)ers whi%h ha&e#an/ (a%tors )'t we;ll ee going.a%tors o( 12 are 1, 2, 3, 4, 6. IesHHHa%tors o( 14 are 1, 2, and 7. @o.a%tors o( 16 are 1, 2, 4, and 8er(e%t s'ares aren;t good%andidates, are the/a%tors o( 18 are 1, 2, 3, 6, and 9.IesHHHa%tors o( 20 are 1, 2, 4, 5, and 10.IesHHHa%tors o( 22 are 1, 2, and 11. @o.a%tors o( 24 are 1, 2, 3, 4, 6, 8, and12. De(initel/ /esHHH



11/11

!o we ha&e 12, 18, 20, and 24.4 Ans.


State 2003 Mathcounts

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