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

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1 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 ws included, the clss verge rose to 78. Wht ws the seventh student s score? 90 (B) (E) 98. The sides of right tringle re ll integers. Two of them re odd numbers tht differ by 50. Wht is the smllest possible vlue for the third side? 48 (B) (E) 7. Sen hs drwer contining dozen blck socks, dozen brown socks, dozen blue socks, nd dozen green socks, ll loose nd not properly pired. Sen tkes socks out of the drwer t rndom without replcement, in the drk. How mny socks must Sen tke out to ensure tht he hs pirs of socks of ech of the four colors mong those he hs chosen? 9 (B) 7 5 (E) 8 4. Let ( ) ( ) = + +, nd consider tringle with sides,, +. Wht is the re of the tringle? 4 (B) For which positive integers n is n k= n k= k k n integer? odd n only (B) even n only n = + 6k, integer k 0 n = + k, integer k 0 (E) n = only 6. A sphere of rdius r nd cube of edge length s hve the sme totl surfce re. Wht is s in terms of r? s = r (B) 4 π s = r π s = r π s = r (E) s = r π

2 Level I MAML Olympid 00 Pge of 6 7. A circle is inscribed in 60 sector of circle of rdius s shown in the digrm to the right. Wht is the rdius of the smll circle? (B) Figure for #7 8. Two loded dice ech hve the property tht or 4 is three times s likely to pper s,, 5, or 6 on ech roll. Wht is the probbility tht 7 will be the totl sum when the two dice re rolled? 8 (B) (E) Tringle ABC hs right ngle t C nd n ngle of 5 t A. Let BC =. Point D is the foot of the perpendiculr from C to the opposite side, nd point E is the foot of the perpendiculr from D to AC. In terms of, wht is the length EC? (B) 4 (E) The terms,, form n rithmetic sequence whose sum is 8. The terms +,, +, in tht order, form geometric sequence. Find the sum of ll possible vlues for. (B) (E). Consider cube of side. The centers of ech pir of fces of the cube shring common edge re connected to form regulr octhedron ( regulr polyhedron with 8 equilterl tringulr fces). Wht is the volume of this octhedron? (B) (E) 0

3 Level I MAML Olympid 00 Pge of 6. Wht is the sum of the slopes of the lines tngent to both circles: + y = nd ( 6) y 4 + =? + 5 (B) (E) None of these. Points S nd R re endpoints of dimeter of the circle with rdius, s shown in the figure to the right. PR is tngent to the circle t R, nd PS is secnt line intersecting the circle t Q nd S. If PR =, wht is the re of QRS? 4 9 (B) Figure for #` 4. In the Vigornii lphbet there re 8 consonnts nd 4 vowels. How mny different rrngements of 5 letters cn be mde if ectly two vowels must be used nd no repetition of letters is llowed? 6 (B) 0 400,600 (E) 40,0 5. The polr coordinte eqution sin( ) cos( ) distnce between two distinct points on this curve? r = θ θ defines curve in the plne. Wht is the lrgest (B) (E) 6. Find the solution set: 4+ 4 < nd. 4 0< < (B) < < > (E) 0< 7. Three circles of rdius r in the sme plne re eternlly tngent in pirs. Consider the tringle whose vertices re the centers of the circles. Wht percentge of the re of this tringle is not contined in ny of the circles (round to the nerest whole number)? 9 (B) 0 (E)

4 Level I MAML Olympid 00 Pge 4 of Find the sum of , where i = i i i i i i i (B) i 6i 8i (E) 7i π 9. Let R be the rottion by rdins counterclockwise bout the origin, let T be the trnsltion by the vector,0, nd let R be the rottion by 7 rdins counterclockwise bout the origin. Let P be 6π the point (,0). Wht re the coordintes of R( T( R( P )))? (E), (B) 0. Which of the following formuls define functions lwys equl to I. + tn ( ) II. sin cot + cos(4 ) csc ( ) IV. csc ( ) III. ( ) None (B) III only I nd III only II nd III only (E) I, II, III nd IV cos ( )?. Let >. Wht is the re of the region of the plne defined by ( y, ): 0, y 0, + y 0? { } 8 + (B) (E) the re is infinite. How mny pirs of deciml digits re there with the property tht if N is n integer ending in those two digits, then N ends in the sme two digits (repitition of the digits in pir is llowed)? (B) 4 (E) 5

5 Level I MAML Olympid 00 Pge 5 of 6. Let f( ) =, f( ) =, f( ) =, f4( ) =, f5( ) =. For 0,, which n in {,,,4,5 } is f( f4( )) = f( fn( ))? (B) 4 (E) 5 π 4. If tn =, then cot =? 4 + (B) Set A hs number of elements strictly between the number of elements in set B nd twice the number of elements in set B. Set B hs 6 more subsets thn set C (s usul, we count the empty set s subset of every set). Wht is the lrgest number of subsets A cn hve? 8 (B) (E) If b log 8 = 7 log 9 b nd log = log 4, wht is b? 79 4 (B) (E) 7. How mny different polynomils of the form fctors of the polynomil? b + c+ d with rtionl coefficients re (B) 5 7 (E) 5 8. Wht is the reminder when is divided by 4? (B) (E)

6 Level I MAML Olympid 00 Pge 6 of 6 9. Which of the following equls n sin + cos( k)? k= sin n + (B) sin n cos n ( ) ( ) sin ( n+ ) 0. A continer hs the shpe of right circulr cone with rdius R units nd height H units. It is originlly in n inverted position (verte down), nd prtilly filled with wter. If the depth of the wter is H, wht will the upper rdius of the wter be when the cone is returned to n upright position? H R (B) H RH (E) H R H R H 6 R H H

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?

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? 008 009 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