Thirty-fourth Annual Columbus State Invitational Mathematics Tournament. Instructions

Transcription

1 Thirty-fourth Annul Columbus Stte Invittionl Mthemtics Tournment Sponsored by Columbus Stte University Deprtment of Mthemtics Februry, 008 ************************* The Mthemtics Deprtment t Columbus Stte University welcomes you to our cmpus nd to this yer s tournment. We wish you success on this test nd in your future studies. Instructions This is 90-minute, 50-problem, multiple choice exm. There re five possible responses to ech question. You should select the one best nswer for ech problem. In some instnces this my be the closest pproximtion rther thn n exct nswer. You my mrk on the test booklet nd on the pper provided to you. If you need more pper or n extr pencil, let one of the monitors know. When you re sure of n nswer circle the choice you hve mde on the test booklet. Crefully trnsfer your nswers to the score sheet. Completely drken the blnk corresponding to the letter of your response to ech question. Mrk your nswer boldly with No. pencil. If you must chnge n nswer, completely erse the previous choice nd then record the new nswer. Incomplete ersures nd multiple mrks for ny question will be scored s n incorrect response. The exmintion will be scored on the bsis of + for ech correct nswer, for ech incorrect selection, nd 0 for ech omitted item. Ech student will be given n initil score of +00. Pre-selected problems will be used s tie-brekers for individul wrds. These problems, designted with n sterisk (*), in order of considertion re:, 4,, 4, 6, 40, 4, nd 4. Throughout the exm, AB will denote the line segment from point A to point B nd AB will denote the length of AB. Pre-drwn geometric figures re not necessrily drwn to scle. The mesure of the ngle ABC is denoted by m ABC. Review nd check your score sheet crefully. Your student identifiction number nd your school number must be encoded correctly on your score sheet. When you complete your test, bring your pencil, scrtch pper nd nswer sheet to the test monitor. Leve the room fter you hve hnded in your nswer sheet. Plese leve quietly so s not to disturb the other contestnts. Do not congregte outside the doors by the testing re. You my keep your copy of the test. Your sponsor will hve copy of solutions to the test problems. Do not open your test until instructed to do so!

2 ) If x+ y = 0 nd x 0, wht is the vlue of ( y) x ( x y) A) B) C) y D) x E) ) If point ( x, y ), not on the xes, stisfies both of the inequlities y< x nd y< x, in which qudrnts could the point be locted A) I, II B) II, III C) I, III D) III, IV E) I, IV ) Wht re the lst two digits of A) 75 B) 65 C) 5 D) 5 E) 5 4) Little Billy s score on this yer Columbus Stte Invittion Mthemtics Tournment test is 40. (See the rule on the first pge.) His score would be if he hd been given + points for ech correct nswer, - point for ech incorrect nswer, 0 for ech left blnk, nd n initil score of 0. How mny did he leve blnk A) 0 B) 5 C) 5 D) 6 E) 0 5) It tkes 40 minutes for Moe to pint room lone nd it tkes 60 minutes for Joe to pint the sme room lone. If they decide to pint the room together, how long will it tke them to pint one-third of the room A) 0 min B) 8 min C) 0 min D) min E) 4 min 6) A tnk contins 0 gllons of 90% slt solution. How mny gllons of 50% slt solution must be dded to the tnk to yield 70% slt solution A) 6 gl B) 7 gl C) 8.5 gl D) 9 gl E) 0 gl 7) Dick nd Jne hve totl of $480. Dick tkes one-third of Jne s money. Then Dick tkes one-fourth of the money tht Jne hs left. Jne nd Dick now hve equl mounts of money. How much money did Jne hve originlly A) $480 B) $50 C) $0 D) $00 E) $80 8) Exctly /4of the boys nd exctly 4/5 of the girls in clss pssed n lgebr test. If n equl number of boys nd girls pssed the test, wht frction of the entire clss pssed the test A) /5 B) 4/7 C) 6/ D) 8/5 E) 4/

3 9) Compny A will rent cr for $5 per dy plus $0.0 per mile, wheres compny B chrges $0 per dy plus $0. per mile. John needs cr for 5 dys. For wht rnge of milege will John sve money by renting cr from compny B A) More thn 000 miles B) No more thn 00 miles C) More thn 0 miles D) More thn 50 miles E) Less thn 50 miles 0) Two verticl poles, 6ft nd 4ft high, re erected feet prt on level ground. Stright wires re ttched from the top of ech pole to the bottom of the other pole. Wht is the verticl distnce from the ground to the intersection of the wires A) 5.6 ft B) 7ft C) 8ft D) 8.6ft E) 9.6ft ) Wht is the coefficient of 6 5 x y in the expnsion of (x ) y A) B)07756 C) D) E) 555 ) A three-digit number equls 9 times the sum of its digits. If the digits re reversed, the resulting number is greter thn the given number by 97. The tens digit exceeds the units digits by. Find the number. A) 84 B) 85 C) 86 D) 487 E) 588 ) A certin rectngle hs n re of 0 squre inches. Incresing its width by 4 inches nd decresing its length by inches increses its re by 4 squre inches. Find the perimeter of the originl rectngle. A) 44 in B) 46 in C) 5 in D) 68 in E) 86 in 4) How mny ones re there if 008 (bse 0) is written in bse A) 5 B) 6 C) 7 D) 8 E) 9 5) Find the sum of the digits of the only rel solution to the eqution 7 + = 0. x x 7 A) B) C) D) 4 E) 5

4 6) A :40 scle model of prism is shown on the right. If the volume of the scled model is5 cm, wht is the volume of the ctul prism A) 00 D) 0. cm B) m E) 8000 cm C) cm m 7) Find the sum of ll rel nd complex solutions of the eqution x = 7. A) 0 B) C) 6 D) i E) i 8) Solve the eqution 6 x 6 x =. A) / B) C) D) E) / x x 8 9) Find the rel solution of the eqution =. A) / B) log C) 0.7 D) 0.8 E) log 0) If cos x = sin x, then wht is the vlue of sin x cos x A) / B) C) D) /5 E) 4/5 + bn ) * Define sequence by b = nd bn + = for integers n. Wht is the vlue bn ofb 008 A) B) C) D) E) ) A gme is plyed by selecting four-digit number from 0000 to 9999, nd the winning prize is $5000. Suppose you py $ to ply the gme. Wht is the expected vlue of your gin or loss in dollrs A) 500 B) C) 0. D) 0.4 E) 0.5 4

5 ) Suppose there re five problems on test with 5 possible responses for ech problem. If you guess ech nswer, wht is the probbility tht you get more thn correct nswers A) B) C) D) 0.05 E) 0.5 4) * How mny zeros re there t the end of 008! A) 0 B) 450 C) 500 D) 008 E) 550 5) The three circles in the figure centered t A, B, nd C re tngent to one nother nd hve rdii 7, nd 6, respectively. Find the re of tringle ABC. A) 54 B) 64 C) 74 D) 84 E) 94 A B C 6) How mny distinct permuttions re there of the letters in the word MATHEMATICS, using ll the letters A) B) C) D) 640 E) ) A regulr six-pointed str is inscribed in circle. Let be the edge length. Express the re of the str in terms of. A) B) C) D) 6 E) 6 8) If > 0, wht is the vlue of log log A) 0 B) C) D) log E)

6 9) For x nd x 0, wht is the formul for cot(sin x) A) x x B) x C) x D) x E) x x 4 0) Let Y = (04 ) + 6(04 ) + 4(04) +. How mny positive integer fctors does Y hve A) 6 B) 4 C) 5 D) 6 E) 5 ) * In the tringle DEF, the points A, B, nd Cre locted on the sides of the tringle (see figure), such tht DC = EC, EB= BF, nd FA = AD. Find the rtio (Are DEF) /(Are ABC). A) / B) 4/ C) D) E) 7/ ) Given circle of rdius, wht is the probbility tht the length of the chord between ny two rndomly chosen points on the circle will be greter thn or equl to A) / B) / C) /4 D) /4 E) /6 ) Five distinct integers re picked t rndom from {0,,,, 4, 5, 6, 7, 8, 9}. Wht is the probbility tht, mong those selected, the third smllest is A) 5 8 B) 5 45 C) 5 84 D) 68 E) 8 4) * How mny distinct permuttions re there of letters selected from the letters in the word MATHEMATICS A) 8 B) 6 C) 99 D) 65 E) 990 6

7 5) Three mutully tngent circles hve rdii,, nd inches, respectively. Find the re of the white region between the circles. A) in B) in C) 0.4 in D) 0.46 in E) 0.75 in 6) * In tringle ABC lines CE nd AD re drwn so tht AE EB = 4 rtio CP PE. CD 4 DB = nd. Let P be the intersection point of CE nd AD. Find the A C P E D B A) B) 4 C) 5 D) 6 E) 7 7) Suppose x is complex number stisfying the eqution nturl number k. Wht is the vlue of x k k + x π x + = cos for some x k A) 0 B) C) D) k E) k 8) If sin x+ sin y = nd cos x+ cos y =, wht is the vlue of cos( x y) A) B) /4 C) / D) / E) / 9) The distnce to three corners of squre ABCD from point X in its interior is XA = 0, XB = 0, nd XC = 0. Find the distnce XD. A) 8 B) 5 C) 0 D) 0 E) ) * Every nturl number which is not power of cn be written s sum of two or more consecutive integers. There is only one representtion of 008 in the form 008 = + ( + ) + + bwhen > 0. Find + b. A) 6 B) 5 C) 5 D) 44 E) 45 7

8 4) * If the sum of two solutions of the eqution other two solutions. + = 0 is 0, find the 4 x x x x A) ± i B) ± i C) ± D) ± E) ± i 4) * Let f ( x ) be polynomil of degree 007 stisfying f( k) = for k =,,,008. k + Wht is the vlue of f ( ) A) 004 B) 490 C) 0 D) E) 4) If bb is the representtion of 008 in bse c nd b>, 0, wht is b+ c A) 0 B) C) D) E) 4 44) For rel number > 0, how mny rel solutions stisfy the eqution + + x = x A) 0 B) C) D) E) 4 45) Find the sum of the digits of the gretest integer n with the property tht n divides n A) B) 7 C) 8 D) 9 E) 46) A inch dimeter sphericl scoop of ice crem rests on cone tht hs inch dimeter opening nd inch height. Find the dimeter of the intersection of the sphere nd the cone. A) in B).5 in C) in D) in E) in 47) Find the sum if 0< <. A) + B) C) ( ) D) E) 8

9 48) Find the vlue of in the prbol line to the prbol. x = + y so tht the line y x = + is tngent A) /4 B) /8 C) 7/64 D) /7 E) 9/6 49) Two crs re trveling t the sme constnt speed towrd ech other on stright rod. A plne flying 50 miles per hour psses over the second cr two hours fter pssing over the first cr. The plne continues to fly in the sme direction nd is 400 miles from the crs when they pss ech other. Find the speed of the crs. A) 50 mph B) 55 mph C) 60 mph D) 65 mph E) 70 mph 50) Evlute the nonterminting continued frction A) B) C) D) + 5 E) 5 9