Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll. If the top of the ldder slips 4 ft., then the mount the foot of the ldder slips in feet is 9. 5 c. 5 d. 8 e. 4. The numer of circulr pipes with n inside dimeter of inch tht will crry the sme mount of wter s pipe with n inside dimeter of 6 inches is 9. 6 c. d. 6 e. 8. The vlue of y = CE in the digrm elow such tht ABC is similr to ADE is c. c c. d. c e. 4. Two trins, one 500 ft. long nd the other 400 ft. long, on prllel trcks, cn pss ech other in 9 sec. when moving in opposite directions. When moving in the sme direction, the fster trin cn pss the slower trin in 8 sec. The speed of the fster trin in ft. per sec. is 75. 50 c. 60 d. 90 e. 00 5. The verge mrk on n exm ws 60%, nd the verge mrks for those pssing nd filing the exm were 75% nd 55%, respectively. The percentge of students pssing the exm ws 50. 40 c. 5 d. 0 e. 5 6. The vlue of n such tht 0 0 0 0 0 0 0 0 + + + + + + + = 64 n is 6. 9 4 c. 5 d. 4 5 e. 5 7
7. The sum of two numers is 0 nd their product is 0. The sum of their reciprocls is 0. c. d. e. 4 8. At dnce prty group of oys nd girls exchnge dnces s follow: one oy dnces with 5 girls, second oy dnces with 6 girls, nd so on, nd the lst oy dnces with ll the girls. If nd g denote the numer of oys nd girls, respectively, then = g. = c. = g 4 d. = g 5 e. n.o.t. g 9. Two circles, ech of positive rdius r inches, re plced so they re tngent to ech other nd re on stright line s elow. A smller circle of positive rdius x inches is nestled etween them so tht it is tngent to the circles nd the line. The rtio of r to x is 5..5 c. 4 d. e..5 0. A thtu fills in minimum of minutes with the cold wter on full tp. It fills in minimum of 9 minutes with the hot wter on full tp. The thtu empties in minimum of 6 minutes with the drin plug removed. Assuming the thtu cn hold more thn 0 gllons, then the minimum mount of time in minutes it tkes the thtu to fill with oth hot nd cold wter on full tp nd the drin plug removed is 0. 45 c. 4 d. 6 e. 8. The minute nd hour hnd of clock point in the sme directions t time etween 8 m nd 9 m. The numer of minutes fter 8 m t this time rounded to deciml plces is 4.60. 44. c. 44.7 d. 4.67 e. 4.64
. A rectngle of perimeter 8 inches cn e divided into three identicl squres s elow. Its re in squre inches is 405. 750 c. 868 d. 768 e. 59. A cylindricl oil tnk, lying horizontlly, hs n interior length of 0 ft. nd n interior dimeter of 6 ft. If the surfce level of the oil it contins is 40 sq. ft., nd the tnk is t lest hlf-full, then the depth of the oil in feet is 4. 0 c. + d. + 5 e. + 5 4. A stone is dropped into well nd the report of the stone striking the ottom is herd 7.7 seconds lter. If the stone flls 6t ft. in t seconds nd the speed of sound is, 0 ft. per sec., then the depth of the well in feet is 784. 4 c. 568 d. 56.8 e. 64 5. In group of 00 tourists, 5 knew neither French nor Itlin, 65 knew French nd 77 knew Itlin. The sum of the digits of the numer of tourists tht knew Itlin ut not French is. c. 8 d. 6 e. 7 6. A smll school hs 00 students nd rooms A, B, nd C. After the st period, hlf the students in room A move to room B, 5 of the students in room B move to room C, nd of the students in room C move to room A, nd these re the only chnges. If the numer of students in ech room does not chnge, then the numer of students in room A is 50. 40 c. 0 d. 0 e. 0 7. The mximum vlue of k for which x 6x k + nd x 0x k + hve common zero is 0. c. 4 d. 6 e. n.o.t.
4 8. The length of the shortest elt in inches tht will go round two tngent circles of dimeters 6 inches nd 8 inches s indicted elow is + 6. + 7 c. + 5 d. + 4 e. 9. A ot hs speed of 5 mph in still wter. In current of 5 mph it trvels certin distnce downstrem nd returns. Assuming ll speeds re expressed in mph, the rtio of the verge speed of the round trip to the verge speed of the corresponding trip in still wter is 8 7. 8 9 c. 7 8 d. 4 5 e. cnnot e determined y nyone 0. The numer of positive integers tht leve reminder of 4, s defined y the Division Algorithm, when divided into 4049 is 8. 7 c. 6 d. 5 e.. Let n e positive integer. If the reminders of 77 nd re 7 nd 9, respectively, when divided y n, then the sum of the se 0 digits of n re 7. 6 c. 5 d. e.. Six people line up to use drink mchine tht ccepts only nickels nd dimes in which ech drink costs 5 cents. If of the 6 people only hve nickel ech, only hve dime ech, nd the drink mchine is initilly out of chnge, then the proility, in percent, the six people line up rndomly in such wy they ech cn purchse drink is 5. 8 c. 0 d. 5 e. 5. Two plyers re delt n crds. Assume the plyers cn e delt n unequl numer of crds nd ech plyer must receive t lest crd. The numer of different hnds possile is n. n c. n n! d. ( n ) e. n
5 4. If 0 0 0 0 (6 6 )(6 6 A B A B ) 8 8, + = nd AB, re rel numers, then B is 0. 0 c. 0 d. 40 e. cnnot e determined y nyone 5. The numer of rel ordered pirs ( xy, ) t which the system x= sin( t) nd t [0,4], intersects itself is y t t =, for 0. c. d. 4 e. 8 6. If + =, then + equls 0. - c. ( ) d. ( ) e. 7. A oy wishes to clim ldder tht hs 500 rungs (steps). He my go up the ldder only in positive integer multiples of 00 steps or down the ldder in only positive integer multiples of 98 steps, ut he must remin on the ldder, once on the ldder. Assuming the rungs re enumerted from ottom to top from to 500 nd he strts with oth feet on the ground, then the rung with the highest numer he cn step on is 00. 40 c. 4 d. 49 e. 498 8. The smllest positive rel numer p such tht x p implies x + x x differs from y less thn.0 is 6.4. 5.74 c..4 d. 5.7 e. 5.76 9. A mn orn etween 800 nd 850 ws x yers old in the yer x. The sum of the digits of the yer he ws orn is 5. c. 6 d. 8 e. 7
6 0. Given the right tringle AOC elow, if OBA is similr to CBO, OA =, nd OC = 6, then BD is A B O D C 6 5. 5 c. d. 5 e. 9 5
7 Tie Breker Nme School Code Work the tie reker prolem on this pge, ter off this sheet, nd then hnd it in long with your nswer form. Prtil credit will e given. Tie Breker: Prove tht if m is positive integer, then the eqution x y mx my m xy + = + + + hs exctly m solutions ( xy, ) in which xy, re positive integers. The qulity nd completeness of the solution will e tken into considertion.