CONCURS DE ADMITERE (facultate) 18 iulie 2004

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1 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, UNIVERSITATEA DIN BUCURETI Fultte de Mtemt Iformt CONCURS DE ADMITERE fultte 8 ule I ALGEBR Fe polomul f X X X X R[ X ] Determ umerele rele,, stfel îât f, r restul împrr polomulu f l X este egl u Petru umerele,, determte l putul, gs rdle polomulu f y Fe G, y R, y Art : y D A, B G, tu AB G G împreu u îmulre mtrelor este grup el II III IV y y, Fu C este u omorfsm de f : G, f y l grupul multpltv, C l grupul G, ude C C \ { ANALIZ MATEMATIC Fe fu f : R R, f S se lulee f s se rte f este strt restore S se lulee f s se rte f este fue ove f f lm f : ; R, f S se lulee Fe fu [ ] f S se gses umerele rele A, B, C stfel îât s vem A B C f petru ore [ ;] S se lulee f d GEOMETRIE I TRIGONOMETRIE Se osder erul de eue y y drept de eue y α, α R S se determe vlorle prmetrulu α stfel îât drept s fe eteror erulu Îtr-u tetredru ABCD se osder AC BD S se rte seue determt de terse tetredrulu u u pl prlel u muhle AC BD este u dreptugh INFORMATIC Se osder tloul de umere rele A,,, Se er urmtorele : U lgortm petru ordore restore elemetelor tloulu A

2 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, U lgortm performt preferl lr petru elmre dulurlor d tloul ordot Presupuâd,,, sut oordotele persoe flte pe o drept, s se determe o oordot u proprette duere tuturor elor persoe pe po s se relee u efort totl mm efortul petru o perso este dst de l po s l l po Eemplu : Petru,,,6,9,6, A :,,,,6,6,9 ;,,,6,9 ; Prer : Petru fere lgortm ve sre dee solue, preum omplette s î tmp î fue de Nu pot f foloste tlour suplmetre Dtele de trre A se osder tte, r petru dtele de ere se v pre lo lor Petru uul d prmele dou pute se v d solu su form ue proedur / fu Psl, C su C, r î rest fd sufet pseudood NOT : Tmp de luru ore Se er trtte suete l legere SOLUIE : I Folosd Teorem lu Beout, odle d eu se redu l sstemul f f petru re vem suesv : f f 8 8 Polomul fd u oefe rel vem î

3 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, oture : Deoree polomul re oefe rel, tu vâd o rd mgr pe re rd ojugt este pe - Folosm Shem lu Horer vem : Polomul out este X X re re rdle : ± ± De rdle lu f sut :,,, Fe A, B G Atu :, y,, v R, y, v stfel îât y v yv v y A, B Avem : AB Notâd y v y v yv yv v y vem AB,, R d oper u umere rele yv v y yv y v v yv y y v v y v y v y v fd produsul dou umere rele eule Atu AB G L putul preedet s- demostrt G este îhs l îmulre mtrelor Verfm omele grupulu : - Îmulre mtrelor este sotv, pr urmre îmulre mtrelor d G este sotv - Elemetul eutru l îmulre mtrelor este I re se gsete î G petru, y y - Tote mtrele d G sut versle, vâd determtul y y y, m treue verft vers se fl y y y v î G Îtr-devr : d A, B AB I reult y v

4 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, yv y v v y yv de ude vy y v v y y y re sut umere rele, r y v, ee e sgur y y v A G v y v,, A, B Deoree y v yv v y vy v y AB, r BA se oserv y v yv v y vy AB BA, d - Fe A B G d omuttvtte îmulr umerelor rele reult omuttvtte îmulr pe G Î ose G, este grup el Verfm îtâ d f este morfsm de grupur Fe y, v C f y v f yv v y yv v y y v f y f v y v yv y v Avem : morfsm de grupur Verfm jetvtte lu f : fe y f v f f e jetv y y v v, de f este stfel îât y, v C y v y v y Fe A G y C y w y y A G, reult C f w A y De w D w um evdet se oe surjetvtte lu f De f este o jee um er morfsm de grupur reult f este u omorfsm de grupur II Fd sum dtre o fue poloml u epoel, mele dervle pe R, reult f este dervl pe R f ': R R, f ' l Deoree fu det fu poloml î suprutr sut strt restore, reult f este strt restore, sum de dou fu strt restore Fd sum dtre o fue poloml u epoel, mele deft dervle pe R, reult f este de dou or dervl pe R

5 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, f '': R R, f '' l Deoree >, R, r l > reult f '' >, R, de ude reult f este strt ove de ove f k f f f k k lm lm lm k k k k k k lm lm lm lm Am folost lmt fudmetl : k lm, k N,, R, > A B C A B C f, [ ; ], [ ; ] A B C, [ ; ] A A A B B C C, [ ; ] A B A B B C A A C, [ ; ] A B C A C B A B A B A A A A 6 A C A C A C d f d d d 9 l l l l l l l III Drept y α este eteror erulu y, d ; r, d um d dst de l etrul erulu l drept erulu de etru α > α < α > α > su α < α ; ; este m mre deât, d > α su 5

6 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, Fe terse tetredrulu u plul prlel u AC BD ptrulterul MNPQ u AB BC P CD M, N, Q AD Prlelsmul plulu u AC teorem D o drept este prlel u u pl, tu ore lt pl re oe drept dt tersete plul dt, o fe dup o drept prlel u drept dt sgur petru plele ABC respetv ACD relle : MN AC PQ AC, re d trtvtte rele de prlelsm dreptelor î spu mpl MN PQ Alog, dr petru plele BCD respetv ABD prlelsmul plulu de tersee u BD sgur Pr urmre ptrulterul MNPQ vâd lturle opuse NP MQ prlele dou âte dou este u prlelogrm Deoree dreptele AC BD sut perpedulre, r MN AC MQ BD reult dreptele MN MQ sut perpedulre, ee e demostre prlelogrmul MNPQ este dreptugh IV Idee : prurgem vetorul de l p l od testm po dou elemete oseutve D este sut î orde desrestore, ele se omut D l sfârtul prurger vetorulu u s- produs o omutre tu vetorul este ordot, ltfel se re prurgere Complette : î ul el m defvorl, î re vetorul l este ordot desrestor, umrul de pereh de testt este -, r umrul de prurger este tot - oper ee e produe o omplette de Pseudood : M Cât tmp M eeut M Petru k,- D k > k tu eeut M P k k k k P Vetorul ordot se gsete î ee loe vetorul l Psl : proedure ord; eg m:; whle m do eg m:; for k: to - do f [k]>[k] the eg m:; C P N D Q B M A 6

7 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, ed; C ed; p:[k]; [k]:[k]; [k]:p; ed; vod ord { m; whle m { m; for k;k<-;k f [k]>[k] { m; p[k]; [k][k]; [k]p; Idee : prurgem vetorul de l p l od testm po dou elemete oseutve D este sut dete, l dole se elm Complette : î ul el m defvorl, î re vetorul ordot este u tote elemetele dete, umrul de pereh de testt este -, ee e produe o omplette de oper Pseudood : k Cât tmp k< eeut D k k tu eeut Petru k,- eeut - ltfel kk Vetorul fr dulur se gsete î ee loe vetorul l Psl : proedure elm; eg k:; whle k< do eg f [k][k] the eg for :k to - do []:[]; :-; ed else k:k; ed; ed; C 7

8 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, vod elm { k; whle k< { f [k][k] { for k;<-; [][]; -; else kk; Idee : prurgem vetorul de l p l od lulm efortul totl petru duere tuturor persoelor î po uret Reem mmul -l fm Complette : î ul el m defvorl, î re efortul mm se fl î ultm poe, se produe o omplette de oper Pseudood : SmMIt Petru, eeut s; for j, eeut sss - j d s<sm tu eeut Sms Sre Coordot se gsete î Psl : proedure oordot; eg Xm:; Sm:MIt; for : to do eg s:; for j: to do s:s s[]-[j]; f s<sm the eg Sm:s; Xm:; ed; ed; wrtel' ',[Xm]; ed; C vod oordot { Xm; Sm ; for ;<; 8

9 Uverstte d Buuret Fultte de Mtemt Admtere î fultte 8 ule Solue, redtre Ctedr de Mtemt-formt Leulu Teolog Greo-Ctol, Setor, Buuret, { s; for j;j<;j sss[]-[j]; f s<sm { Sms; Xm; out<<"coordot este "<<[Xm]<<edl; Ctedr de Mtemt-Iformt Leulu Teolog Greo-Ctol, Setor, Buuret, 9

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