Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?

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Maximilian Quinn
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1 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing red king nd blck queen? How mny regulr polygons cn tessellte plne? A cube of side length hs its side lengths incresed by fctor of. By how much hs the surfce re incresed? Wht is the volume of right circulr cylinder with rdius π nd height 009? points ech Stcey decides to dd the numbers from 1 to her fvorite number. If she gets sum of 7, nd relizes tht she forgot to dd one number, wht is Stcey s fvorite number? 7 How mny integer fctors does 009 hve? 8 Wht is the probbility tht I get heds when I flip 8 fir coins? 9 If + b nd + b 1, then wht is the vlue of + b? 10 Using chords to cut circle, wht is the mimum number of pieces tht cn be mde? points ech 11 An equilterl tringle is inscribed in circle tht is inscribed in nother equilterl tringle. If point is selected t rndom inside the lrger tringle wht is the probbility tht the point will lie inside the circle but outside the smller tringle? 1 Wht is the reminder when is divided by + 1? 1 Wht is the gretest common fctor of 97 nd 9? 1 How mny integer vlues of n eist such tht < n 009 < 10? 1 Wht is the totl distnce trvelled by bll tht rebounds to 7 of its drop height when dropped from 009 foot building?

2 Log1 Contest Round Alph Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing red king nd blck queen? How mny regulr polygons cn tessellte plne? A cube of side length hs its side lengths incresed by fctor of. By how much hs the surfce re incresed? Wht is the smllest positive vlue of in rdins such tht the epression ( ( tn )) sin cos equls one? points ech Stcey decides to dd the numbers from 1 to her fvorite number. If she gets sum of 7, nd relizes tht she forgot to dd one number, wht is Stcey s fvorite number? 7 How mny integer fctors does 009 hve? 8 Wht is the probbility tht I get heds when I flip 8 fir coins? 9 If + b nd + b 1, then wht is the vlue of + b? 10 sides of tringle re prt of n rithmetic sequence, if ll sides re integers between 7 nd 1, inclusive, then wht is the probbility tht the tringle is obtuse? points ech 11 An equilterl tringle is inscribed in circle tht is inscribed in nother equilterl tringle. If point is selected t rndom inside the lrger tringle wht is the probbility tht the point will lie inside the circle but outside the smller tringle? 1 Wht is the reminder when is divided by + 1? 1 Wht is the gretest common fctor of 97 nd 9? 1 How mny integer vlues of n eist such tht < n 009 < 10? 1 If the vector <,, b > is orthogonl to <,, > nd <,1, >, wht is + b?

3 Log1 Contest Round Mu Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing red king nd blck queen? How mny regulr polygons cn tessellte plne? e Evlute: e 1 lim 1 + Wht is the smllest positive vlue of in rdins such tht the epression ( ( tn )) sin cos equls one? points ech Stcey decides to dd the numbers from 1 to her fvorite number. If she gets sum of 7, nd relizes tht she forgot to dd one number, wht is Stcey s fvorite number? 7 How mny integer fctors does 009 hve? 8 Wht is the probbility tht I get heds when I flip 8 fir coins? 9 Wht is the probbility tht when I integrte the function f ( ) cos over n intervl of width π, tht I obtin positive vlue? 10 sides of tringle re prt of n rithmetic sequence, if ll sides re integers between 7 nd 1, inclusive, then wht is the probbility tht the tringle is obtuse? points ech 11 An equilterl tringle is inscribed in circle tht is inscribed in nother equilterl tringle. If point is selected t rndom inside the lrger tringle wht is the probbility tht the point will lie inside the circle but outside the smller tringle? 1 Wht is the reminder when is divided by + 1? 1 Wht is the gretest common fctor of 97 nd 9? 1 If y when 0 nd y y, then wht is y when? 1 If the vector <,, b > is orthogonl to <,, > nd <,1, >, wht is + b?

4 Log1 Contest Round Individul Answers Thet Answers Alph Answers Mu Answers e 80π π π π π π

5 Log1 Contest Round Individul Solutions Th Al Mu Solution The first five numbers of the Fiboncci sequence re 1, 1,,, nd. The sum of these numbers is 1. The probbility of drwing red king nd blck queen is the sum of the probbility of drwing red king nd then blck queen nd the probbility of drwing blck queen nd then red king: 8 +. Or, we cn simply tke the number of wys of 1 1 drwing red king () times the number of wys of drwing blck queen () nd dividing by the number of wys of choosing crds; C. In order for regulr polygon to tessellte plne one of its interior ngles must be fctor of 0. There re only regulr polygons tht hve this ttribute: tringle, qudrilterl, nd hegon. The surfce re of cube cn be epressed s e, where e is the side length of the cube. Thus: ( e ) e 8e 8( ) 19 surfcere This should look fmilir s n ltered form of the definition of e, 1 lim 1 +. Since ln() is continuous, we cn tke logs nd tke the limit outside. We cn then write the problem s: ln(1 + e ) ep lim,this cn be solved using L Hopitls rule to get e ep(1)e. The volume of right cylinder cn be determined by the epression: π r h. Plug in the vlues: π π π ( ) ( ) Since the mimum for the sine function in the epression is 1, we set the epression equl to 1 nd solve for. cos ( ( tn )) 1 ( tn )) sin () 1 sin cos π tn cos 0 tn ( 0) 0 π But since 0 is not positive, the net vlue for which tn 0 is π. This problem cn be epressed by the eqution: n ( n + 1) 7 + k, where n is Stcey s fvorite number nd k is the number she skipped. Thus: n ( n + 1) 7 + k n + n 89 + k Since n + n is close to n, we find the smllest perfect squre lrger thn 89. This hppens to be 900. Thus Stcey s fvorite number is 0.

6 7 7 7 Prime fctoriztion of 009: Thus the totl number of positive fctors is, yet to ccount for negtive fctors s well double tht vlue Binomil Probbility: ( ) b ( + b )( b + b ) + b + b 1 ( + b) + b + b ( + b ) + b 1 + b b + b ( + b )( + b b ) π cos d sin + π sin( + π ) sin( ) sin( ) cos( π ) + sin( π ) cos( ) sin( ) sin( ) This will be positive whenever sin() is negtive which is 1/ the time.. 10 The first chord cuts the circle into two pieces, the second (intersecting the first) dds more for totl of. The third (intersecting both previous chords but not t their intersection) dds, so 7 totl. The th dds more nd the th, more for totl of 1 pieces There eist 1 tringles tht hve these described ttributes (9-8-7, , , , , , , , , 1-1-1, , , 1-1-1, , nd Using the lw of cosines, for the lrgest ngle to be obtuse, we must hve + b < c. This only hppens in the tringles: , nd The center of the circle will lso be the centroid of both tringles. The distnce from the centroid to the fr verte is / the medin (nd ltitude in n equilterl tringle). If we left s equl the rdius of the circle, then s will be the height of the smll tringle nd s the height of the lrger tringle. Then, s will be the side length of the smller tringle nd s the side length of the lrger tringle. The desired probbility is then the (Are of the circle minus the re of the smller π 9 tringle) divided by the re of the lrger tringle.

7 1 1 1 The reminder cn be found by division (synthetic or otherwise) or by the reminder theorem which sttes the reminder when P() is divided by (-) is P(). So, using - 1, we get reminder of Since , nd number tht divides 97 nd 9 must divide 98, therefore the gcf of 97 nd 9 is the sme s 9 nd 98. Now 9 (98)+ so ll we need is the gcf of 98 nd. Continuing this equls the gcf of nd 9 which equls the gcf of 9 nd 18 which equls the gcf of 18 nd 79 which equls 79. This methods is due to Euler. 1 1 By rising ech number to the power n nd then compring the vlues, we see tht the inequlity only works for vlues of n between nd 10, inclusive. Thus 7. 1 This eqution cn be solved by seprting the nd y terms. yy dy y d ydy d ydy d 1 y + C y + C y, 0, C 9 y ( y ) The totl distnce is the sum of the distnce the bll trvels down nd the distnce the bll trvels up; both of which re infinite geometric sequences with common rtio of 7. The downwrd distnce hs first term of 009 while the upwrd distnce hs first term equl to the downwrd sequence s second term, ( 7)( ) Two vectors re orthogonl or perpendiculr if their dot product is 0. So we hve () + + b 0 ( ) + b 0 Solving for nd b gives b-, so +b-.

Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38

Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38 Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score