ASUPRA UNOR PROBLEME PROPUSE LA CONCURSURI INTERJUDEŢENE

Size: px
Start display at page:

Download "ASUPRA UNOR PROBLEME PROPUSE LA CONCURSURI INTERJUDEŢENE"

Transcription

1 DIDTI MTHEMTI, Vol. 33(15), pp SUPR UNOR PROLEME PROPUSE L ONURSURI INTERJUDEŢENE Dniel Văăreţu strt. This pper will present the solutions of two prolems whih were proposed to Grigore Moisil nd Mrin Ţrină inter-ounties Romnin mthemtil ompetitions. In Romni, these ompetitions re onsidered mongst the most diffiult ones, eing good opportunity for students to trin themselves nd to self-ssess their potentil in view of their prtiiption in the finl stge of the Ntionl Mthemtis Olympid. s n evidene for the lst sttement stys the ft tht the prolems proposed in these ompetitions re nnully pulished in the Mthemtis Review/ Gzet Mtemtiă. We lso hve to mention the ft tht the students results in these ompetitions re speil riteri for their dmission to the Fulty of Mthemtis nd Informtis of eş-olyi University from luj-npo. MS Key words. Mthemtil ompetitions, geometry of tringle 1 Fie un triunghi suţitunghi orere şi O entrul erului său irumsris. Se onsideră dreptele, respetiv perpendiulre în O pe dreptele O, O respetiv O, unde,,, şi,. Notăm u H, H respetiv H ortoentrele triunghiurilor, respetiv. Să se demonstreze ă segmentele H, H şi H pot fi lturile unui triunghi. Dniel Văăreţu onursul Interjudeţen de Mtemtiă şi Informtiă Grigore Moisil ediţi XXVI-, 11, Târgu Mureş Soluţi 1. m(ô) = m(ô) = 9 m( ) m( ) = m( ) nlog m( ) = m(ĉ). Dei este ntiprlelă l. nlog este ntiprlelă l şi este ntiprlelă l. Ne intereseză de fpt ii ă triunghiurile, şi sunt semene u triunghiul (u eleşi măsuri de unghiuri). vem: tg O + tg O + tg O OH = tg + tg + tg OH = tg O + tg O + tg O tg + tg + tg

2 114 D. Văăreţu OH = tg O + tg O + tg O. tg + tg + tg H O H H dunăm memru u memru şi oţinem: O + OH + OH = tg O + tg O + tg O tg + tg + tg tg ( O + O ) + tg ( O + O ) + tg ( O + O ) + tg + tg + tg O + OH + OH = OH + tg O + tg O + tg O tg + tg + tg unde m nott u H ortoentrul triunghiului şi u,, mijloele segmentelor [ ], [ ] şi [ ] re sunt de fpt şi mijloele lturilor,,.

3 3 supr unor proleme propuse l onursuri interjudeţene 115 Dr triunghiul este triunghiul medin, dei re unghiurile de măsuri m(â), m( ), m(ĉ). Dei tg O + tg O + tg O = tg + tg + tg pentru ă este vetorul OH, unde H este ortoentrul triunghiului re oinide u O. m oţinut dei: OH = OH + OH + OH din relţi lui Sylvester O + O + O = OH + OH + OH H + H + H = diă segmentele H, H şi H pot form un triunghi. Soluţi. H = R os, unde R este rz erului irumsris triunghiului re este semene u triunghiul (invers semene diă este ntiprlelă l ). Din semănre lor vem: R R = R h unde R = O este rz erului irumsris triunghiului şi înălţime orespunzătore lui în triunghiul, ir h este înălţime din triunghiului şi nlog: R = R H = R os h h H = R h os, H = R h os. H + H > H R os + R h h os os > R h + os > os. h h h Dr h = sin, h = sin, h = sin. Rezultă: os os sin + os sin > os sin os R sin sin + os R sin sin > os R sin sin sin os + sin os > sin os

4 116 D. Văăreţu 4 sin + sin > sin sin( + ) os( ) > sin os os( ) > os os( ) os > sin + sin > ( π ) ( sin sin π ) < os os > devărt. nlog se demonstreză H + H > H şi H + H > H. Fie O şi I entrul erului irumsris respetiv entrul erului însris triunghiului orere. Se onsideră dreptele, respetiv perpendiulre în O pe dreptele O, O respetiv O, unde,,, şi,. Fie H I şi O I ortoentrul respetiv entrul erului irumsris triunghiului I I I, unde I, I şi I sunt entrele erurilor însrise în triunghiurile, şi şi I entrul erului însris în triunghiul, unde, şi sunt mijloele lturilor, şi. Să se demonstreze ă: IH I = O I I. Dniel Văăreţu onursul Interjudeţen de Mtemtiă şi Informtiă Mrin Ţrină, ediţi XII-, 1, Turd Soluţie.

5 5 supr unor proleme propuse l onursuri interjudeţene 117 O Vetorul de poziţie l entrului erului însris în triunghiul este: r I = r + r + r = sin r + sin r + sin r. + + sin + sin + sin vem: m(â) = m(â) = m(â), m( ) = m( ) = m( ), m(ĉ) = m(ĉ) = m(ĉ), dei în zul triunghiurilor, şi vem: sin O I + sin O I + sin O I O I I = sin + sin + sin O I I = O I I = sin O I + sin O I + sin O I sin + sin + sin sin O I + sin O I + sin O I sin + sin + sin

6 118 D. Văăreţu 6 (m lut origine vetorilor puntul O I.) dunăm memru u memru şi vem din relţi lui Sylvester: = sin ( O I + O I I + = O I I + O I H I = O I I + O I I + O I I O I ) + sin ( O I + O I ) + sin ( O I + O I ) sin + sin + sin sin O I + sin O I + sin O I = O I I + O I I sin + sin + sin (pentru ă m(â ) = m(â), m( ) = m( ), m(ĉ ) = m(ĉ) şi,, sunt mijloele segmentelor, şi O I H I O I I = O I I IH I = O I I. ILIOGRFIE [1] onursul de Mtemtiă şi Informtiă Grigore Moisil, Ediţi XXVI-, Tg. Mureş, 1-3, prilie 11, prezentre de Eugeni Du şi Dorel I. Du, Gzet Mtemtiă seri, nr.5/11, pg [] onursul interjudeţen de Mtemtiă şi Informtiă Mrin Ţrină, Ediţi XII, Turd, 11-1 mi 1, prezentre de Dorel I. Du şi Gheorghe Loonţ, Gzet Mtemtiă seri, nr. 11/1, pg Fulty of Mthemtis nd omputer Siene eş-olyi University Str. Kogălnienu, no luj-npo, Romni e-mil: Primit l redţie: Noiemrie 15

Notes on Spherical Triangles

Notes on Spherical Triangles Notes on Spheril Tringles In order to undertke lultions on the elestil sphere, whether for the purposes of stronomy, nvigtion or designing sundils, some understnding of spheril tringles is essentil. The

More information

Section 6.1 Law of Sines. Notes. Oblique Triangles - triangles that have no right angles. A c. A is acute. A is obtuse

Section 6.1 Law of Sines. Notes. Oblique Triangles - triangles that have no right angles. A c. A is acute. A is obtuse Setion 6.1 Lw of Sines Notes. Olique Tringles - tringles tht hve no right ngles h is ute h is otuse Lw of Sines - If is tringle with sides,, nd, then sin = sin = sin or sin = sin = sin The miguous se (SS)

More information

Titlul lucrării propuse pentru participarea la concursul pe tema securității informatice

Titlul lucrării propuse pentru participarea la concursul pe tema securității informatice Titlul lucrării propuse pentru participarea la concursul pe tema securității informatice "Îmbunătăţirea proceselor şi activităţilor educaţionale în cadrul programelor de licenţă şi masterat în domeniul

More information

Reflexia şi refracţia luminii. Aplicaţii. Valerica Baban

Reflexia şi refracţia luminii. Aplicaţii. Valerica Baban Reflexia şi refracţia luminii. Aplicaţii. Sumar 1. Indicele de refracţie al unui mediu 2. Reflexia şi refracţia luminii. Legi. 3. Reflexia totală 4. Oglinda plană 5. Reflexia şi refracţia luminii în natură

More information

Fubini for continuous functions over intervals

Fubini for continuous functions over intervals Fuini for ontinuous funtions over intervls We first prove the following theorem for ontinuous funtions. Theorem. Let f(x) e ontinuous on ompt intervl =[, [,. Then [, [, [ [ f(x, y)(x, y) = f(x, y)y x =

More information

Defining the Rational Numbers

Defining the Rational Numbers MATH10 College Mthemtis - Slide Set 2 1. Rtionl Numers 1. Define the rtionl numers. 2. Redue rtionl numers.. Convert etween mixed numers nd improper frtions. 4. Express rtionl numers s deimls.. Express

More information

Olimpiad«Estonia, 2003

Olimpiad«Estonia, 2003 Problema s«pt«m nii 128 a) Dintr-o tabl«p«trat«(2n + 1) (2n + 1) se ndep«rteaz«p«tr«telul din centru. Pentru ce valori ale lui n se poate pava suprafata r«mas«cu dale L precum cele din figura de mai jos?

More information

Great Is the Love/Hay Gran Amor. Jaime Cortez. Unison Keyboard

Great Is the Love/Hay Gran Amor. Jaime Cortez. Unison Keyboard 887 OCP sheet music Great Is the Love/Hay Gran Amor aime Cortez Unison Keyboard The material that you have requested is copyrighted. Copyright la requires you to obtain a license from the copyright holder

More information

SAMPLE KYRIE. Dm (Em) Dm (Bm) (Bm) (G) (Em) (Bm) (D) Chri ste. ri e e. son. ri e e lé. Gm7 F (G) Gm7. (Bm) (Em7) (D) (Em7) (D) son. Chri ste.

SAMPLE KYRIE. Dm (Em) Dm (Bm) (Bm) (G) (Em) (Bm) (D) Chri ste. ri e e. son. ri e e lé. Gm7 F (G) Gm7. (Bm) (Em7) (D) (Em7) (D) son. Chri ste. KYRIE Capo 3: () m () m () m () m () m () () B e e (7) m7 lé () m () m lé son. Ky r e e () son. Chr ste SMPLE Text: raduale Romanum, 1974. Musc: Chant Mass; raduale Romanum, 1974; gutar acc. 1995, OCP.

More information

Auditul financiar la IMM-uri: de la limitare la oportunitate

Auditul financiar la IMM-uri: de la limitare la oportunitate Auditul financiar la IMM-uri: de la limitare la oportunitate 3 noiembrie 2017 Clemente Kiss KPMG in Romania Agenda Ce este un audit la un IMM? Comparatie: audit/revizuire/compilare Diferente: audit/revizuire/compilare

More information

(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ

(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ Tringles nd Trigonometry Prepred y: S diyy Hendrikson Nme: Dte: Suppose we were sked to solve the following tringles: Notie tht eh tringle hs missing informtion, whih inludes side lengths nd ngles. When

More information

Subiecte Clasa a VI-a

Subiecte Clasa a VI-a (40 de intrebari) Puteti folosi spatiile goale ca ciorna. Nu este de ajuns sa alegeti raspunsul corect pe brosura de subiecte, ele trebuie completate pe foaia de raspuns in dreptul numarului intrebarii

More information

Versionare - GIT ALIN ZAMFIROIU

Versionare - GIT ALIN ZAMFIROIU Versionare - GIT ALIN ZAMFIROIU Controlul versiunilor - necesitate Caracterul colaborativ al proiectelor; Backup pentru codul scris Istoricul modificarilor Terminologie și concepte VCS Version Control

More information

A NOVEL ACTIVE INDUCTOR WITH VOLTAGE CONTROLLED QUALITY FACTOR AND SELF-RESONANT FREQUENCY

A NOVEL ACTIVE INDUCTOR WITH VOLTAGE CONTROLLED QUALITY FACTOR AND SELF-RESONANT FREQUENCY BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LX (LXIV), Fasc. 4, 2014 Secţia ELECTROTEHNICĂ. ENERGETICĂ. ELECTRONICĂ A NOVEL ACTIVE INDUCTOR

More information

2. Setări configurare acces la o cameră web conectată într-un router ZTE H218N sau H298N

2. Setări configurare acces la o cameră web conectată într-un router ZTE H218N sau H298N Pentru a putea vizualiza imaginile unei camere web IP conectată într-un router ZTE H218N sau H298N, este necesară activarea serviciului Dinamic DNS oferit de RCS&RDS, precum și efectuarea unor setări pe

More information

Sylvain Guillaumet Composer, Interpreter, Teacher

Sylvain Guillaumet Composer, Interpreter, Teacher Sylvain Guilumet Composer, Interpreter, Teacher rance, Châteauroux About the artist Musician, composer and author, he multipes the musical experiences, both in the interpretation of the writing Today,

More information

Metrici LPR interfatare cu Barix Barionet 50 -

Metrici LPR interfatare cu Barix Barionet 50 - Metrici LPR interfatare cu Barix Barionet 50 - Barionet 50 este un lan controller produs de Barix, care poate fi folosit in combinatie cu Metrici LPR, pentru a deschide bariera atunci cand un numar de

More information

Double Integrals over Rectangles

Double Integrals over Rectangles Jim Lmbers MAT 8 Spring Semester 9- Leture Notes These notes orrespond to Setion. in Stewrt nd Setion 5. in Mrsden nd Tromb. Double Integrls over etngles In single-vrible lulus, the definite integrl of

More information

Procesarea Imaginilor

Procesarea Imaginilor Procesarea Imaginilor Curs 11 Extragerea informańiei 3D prin stereoviziune Principiile Stereoviziunii Pentru observarea lumii reale avem nevoie de informańie 3D Într-o imagine avem doar două dimensiuni

More information

D în această ordine a.î. AB 4 cm, AC 10 cm, BD 15cm

D în această ordine a.î. AB 4 cm, AC 10 cm, BD 15cm Preparatory Problems 1Se dau punctele coliniare A, B, C, D în această ordine aî AB 4 cm, AC cm, BD 15cm a) calculați lungimile segmentelor BC, CD, AD b) determinați distanța dintre mijloacele segmentelor

More information

Spherical Geometry. This is an article from my home page:

Spherical Geometry. This is an article from my home page: Spheril Geometry This is n rtile from my home pge: www.olewitthnsen.dk Ole Witt-Hnsen nov. 6 Contents. Geometry on sphere.... Spheril tringles...3. Polr tringles...4 3. The right-ngle spheril tringle...6

More information

SOLVING TRIANGLES USING THE SINE AND COSINE RULES

SOLVING TRIANGLES USING THE SINE AND COSINE RULES Mthemtics Revision Guides - Solving Generl Tringles - Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES

More information

Lecture 22: Digital Transmission Fundamentals

Lecture 22: Digital Transmission Fundamentals EE 400: Communiation Networks (0) Ref: A. Leon Garia and I. Widjaja, Communiation Networks, 2 nd Ed. MGraw Hill, 2006 Latest update of this leture was on 30 200 Leture 22: Digital Transmission Fundamentals

More information

EN teava vopsita cu capete canelate tip VICTAULIC

EN teava vopsita cu capete canelate tip VICTAULIC ArcelorMittal Tubular Products Iasi SA EN 10217-1 teava vopsita cu capete canelate tip VICTAULIC Page 1 ( 4 ) 1. Scop Documentul specifica cerintele tehnice de livrare pentru tevi EN 10217-1 cu capete

More information

PROCEDURA PRIVIND DECONTURILE. 2. Domeniu de aplicare Procedura se aplică în cadrul Universităţii Tehnice Cluj-Napoca

PROCEDURA PRIVIND DECONTURILE. 2. Domeniu de aplicare Procedura se aplică în cadrul Universităţii Tehnice Cluj-Napoca PROCEDURA PRIVIND DECONTURILE 1. Scpul: Descrie structura si mdul de elabrare si prezentare a prcedurii privind dcumentele care trebuie intcmite si cursul acestra, atunci cind persana efectueaza un decnt.

More information

SAMPLE. End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

SAMPLE. End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below. End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte 2 Complete the missing numers in the sequenes elow. 50 25 00 75 8 30 3 28 2 9 Put irle round two of the shpes elow whih hve 3 shded. 3

More information

The First TST for the JBMO Satu Mare, April 6, 2018

The First TST for the JBMO Satu Mare, April 6, 2018 The First TST for the JBMO Satu Mare, April 6, 08 Problem. Prove that the equation x +y +z = x+y +z + has no rational solutions. Solution. The equation can be written equivalently (x ) + (y ) + (z ) =

More information

MONOCHRONICLE STRAIGHT

MONOCHRONICLE STRAIGHT UPDATED 09-2010 HYDROCARBON Hydrocrbon is poncho-style cowl in bulky-weight yrn, worked in the round. It ws designed to be s prcticl s it is stylish, with shping tht covers the neck nd shoulders nd the

More information

NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES

NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES At Mth Univ Comenine Vol LXXXV, (06, pp 07 07 NEW OSTROWSKI-TYPE INEQUALITIES AND THEIR APPLICATIONS IN TWO COORDINATES G FARID Abstrt In this pper, new Ostrowski-type inequlities in two oordintes re estblished

More information

Updating the Nomographical Diagrams for Dimensioning the Concrete Slabs

Updating the Nomographical Diagrams for Dimensioning the Concrete Slabs Acta Technica Napocensis: Civil Engineering & Architecture Vol. 57, No. 1 (2014) Journal homepage: http://constructii.utcluj.ro/actacivileng Updating the Nomographical Diagrams for Dimensioning the Concrete

More information

REVIEW QUESTIONS TOPIC 5 TRIGONOMETRY I FLUENCY

REVIEW QUESTIONS TOPIC 5 TRIGONOMETRY I FLUENCY TOPIC 5 TRIGONOMETRY I REVIEW QUESTIONS FLUENCY The most urte mesure for the length of the third side in the tringle elow is: A 4.83 m B 23.3 m C 3.94 m D 2330 mm E 4826 mm 2 Wht is the vlue of x in this

More information

PACHETE DE PROMOVARE

PACHETE DE PROMOVARE PACHETE DE PROMOVARE Școala de Vară Neurodiab are drept scop creșterea informării despre neuropatie diabetică și picior diabetic în rândul tinerilor medici care sunt direct implicați în îngrijirea și tratamentul

More information

ITEC2620 Introduction to Data Structures

ITEC2620 Introduction to Data Structures /5/20 ITEC220 Introdution to Dt Strutures Leture 0 Gme Trees Two-Plyer Gmes Rules for gme define the sttespe Nodes re gme sttes Links re possile moves Build serh tree y rute fore Exmple I Exmple II A Our

More information

OPTIMIZING TOOLS DIAMETERS AND TOOL PATH STYLE TO IMPROVE TIME MACHINING

OPTIMIZING TOOLS DIAMETERS AND TOOL PATH STYLE TO IMPROVE TIME MACHINING BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LXI (LXV), Fasc. 4, 2015 Secţia CONSTRUCŢII DE MAŞINI OPTIMIZING TOOLS DIAMETERS AND TOOL PATH

More information

Ghid identificare versiune AWP, instalare AWP şi verificare importare certificat în Store-ul de Windows

Ghid identificare versiune AWP, instalare AWP şi verificare importare certificat în Store-ul de Windows Ghid identificare versiune AWP, instalare AWP 4.5.4 şi verificare importare certificat în Store-ul de Windows Data: 28.11.14 Versiune: V1.1 Nume fişiser: Ghid identificare versiune AWP, instalare AWP 4-5-4

More information

Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Leture 16 Double integrls Dn Nihols nihols@mth.umss.edu MATH 233, Spring 218 University of Msshusetts Mrh 27, 218 (2) iemnn sums for funtions of one vrible Let f(x) on [, b]. We n estimte the re under

More information

Triangles and parallelograms of equal area in an ellipse

Triangles and parallelograms of equal area in an ellipse 1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper

More information

INFORMAȚII DESPRE PRODUS. FLEXIMARK Stainless steel FCC. Informații Included in FLEXIMARK sample bag (article no. M )

INFORMAȚII DESPRE PRODUS. FLEXIMARK Stainless steel FCC. Informații Included in FLEXIMARK sample bag (article no. M ) FLEXIMARK FCC din oțel inoxidabil este un sistem de marcare personalizată în relief pentru cabluri și componente, pentru medii dure, fiind rezistent la acizi și la coroziune. Informații Included in FLEXIMARK

More information

Capete terminale şi adaptoare pentru cabluri de medie tensiune. Fabricaţie Südkabel Germania

Capete terminale şi adaptoare pentru cabluri de medie tensiune. Fabricaţie Südkabel Germania CAPETE TERMINALE ŞI ADAPTOARE PENTRU CABLURI DE MEDIE TENSIUNE Capete terminale şi adaptoare pentru cabluri de medie tensiune. Fabricaţie Südkabel Germania Terminale de interior pentru cabluri monopolare

More information

ANTICOLLISION ALGORITHM FOR V2V AUTONOMUOS AGRICULTURAL MACHINES ALGORITM ANTICOLIZIUNE PENTRU MASINI AGRICOLE AUTONOME TIP V2V (VEHICLE-TO-VEHICLE)

ANTICOLLISION ALGORITHM FOR V2V AUTONOMUOS AGRICULTURAL MACHINES ALGORITM ANTICOLIZIUNE PENTRU MASINI AGRICOLE AUTONOME TIP V2V (VEHICLE-TO-VEHICLE) ANTICOLLISION ALGORITHM FOR VV AUTONOMUOS AGRICULTURAL MACHINES ALGORITM ANTICOLIZIUNE PENTRU MASINI AGRICOLE AUTONOME TIP VV (VEHICLE-TO-VEHICLE) 457 Florin MARIAŞIU*, T. EAC* *The Technical University

More information

Patterns and Relationships

Patterns and Relationships Series Techer Ptterns nd Reltionships opyright 009 3P Lerning. All rights reserved. First edition printed 009 in Austrli. A ctlogue record for this ook is ville from 3P Lerning Ltd. ISBN 978-1-91860-3-4

More information

Przesłuchania do chóru "Muzyka zespołu Queen symfonicznie" TENOR

Przesłuchania do chóru Muzyka zespołu Queen symfonicznie TENOR Sorano Alto Tenor Bass "Somebody to Love" Przesłuchania do chóru "Muzyka zesołu Queen symonicznie" b b TENOR Can a ny bo dy ind me some bo dy to love "Bohemian Rhasody" 6 bb n b r n Is this the real lie

More information

CHAMPIONS LEAGUE 2017 SPONSOR:

CHAMPIONS LEAGUE 2017 SPONSOR: NOUA STRUCTURĂ a Ch League Pe viitor numai fosta divizie A va purta numele Champions League. Fosta divizie B va purta numele Challenger League iar fosta divizie C se va numi Promotional League. CHAMPIONS

More information

Determine currents I 1 to I 3 in the circuit of Fig. P2.14. Solution: For the loop containing the 18-V source, I 1 = 0.

Determine currents I 1 to I 3 in the circuit of Fig. P2.14. Solution: For the loop containing the 18-V source, I 1 = 0. Prolem.14 Determine currents 1 to 3 in the circuit of Fig. P.14. 1 A 18 V Ω 3 A 1 8 Ω 1 Ω 7 Ω 4 Ω 3 Figure P.14: Circuit for Prolem.14. For the loop contining the 18-V source, Hence, 1 = 1.5 A. KCL t node

More information

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Person Edution Limited Edinurgh Gte Hrlow Essex M20 2JE Englnd nd ssoited ompnies throughout the world Visit us on the World Wide We t: www.personed.o.uk Person Edution Limited 2014 ll rights reserved.

More information

RADIO TRANSCEIVER CONSUMPTION MODELING FOR MULTI-HOP WIRELESS SENSOR NETWORKS

RADIO TRANSCEIVER CONSUMPTION MODELING FOR MULTI-HOP WIRELESS SENSOR NETWORKS U.P.B. Sci. Bull., Series, Vol., Iss., 200.. ISSN 1454-2331 RADIO TRANSCEIVER CONSUMPTION MODELING FOR MULTI-HOP WIRELESS SENSOR NETWORKS Dan TUDOSE 1, Laura GHEORGHE 2, Nicolae ȚĂPUȘ 3 Acest articol propune

More information

Theme: Don t get mad. Learn mod.

Theme: Don t get mad. Learn mod. FERURY When 1 is divided by 5, the reminder is. nother wy to sy this is opyright 015 The Ntionl ouncil of Techers of Mthemtics, Inc. www.nctm.org. ll rights reserved. This mteril my not be copied or distributed

More information

Evoluția pieței de capital din România. 09 iunie 2018

Evoluția pieței de capital din România. 09 iunie 2018 Evoluția pieței de capital din România 09 iunie 2018 Realizări recente Realizări recente IPO-uri realizate în 2017 și 2018 IPO în valoare de EUR 312.2 mn IPO pe Piața Principală, derulat în perioada 24

More information

Automatic Strategy Verification for Hex

Automatic Strategy Verification for Hex utomti Strtegy Verifition for Hex Ryn B. Hywrd, Broderik rneson, nd Philip Henderson Deprtment of Computing Siene, University of lert, Edmonton, Cnd {hywrd,roderi,ph}@s.ulert. strt. We present onise nd/or-tree

More information

Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is

More information

EVALUATION OF THE YARN QUALITY CHARACTERISTICS THROUGH SYNTHETIC INDICATORS

EVALUATION OF THE YARN QUALITY CHARACTERISTICS THROUGH SYNTHETIC INDICATORS BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LVI (LX), Fasc. 2, 2010 SecŃia TEXTILE. PIELĂRIE EVALUATION OF THE YARN QUALITY CHARACTERISTICS

More information

CAIETUL DE SARCINI Organizare evenimente. VS/2014/0442 Euro network supporting innovation for green jobs GREENET

CAIETUL DE SARCINI Organizare evenimente. VS/2014/0442 Euro network supporting innovation for green jobs GREENET CAIETUL DE SARCINI Organizare evenimente VS/2014/0442 Euro network supporting innovation for green jobs GREENET Str. Dem. I. Dobrescu, nr. 2-4, Sector 1, CAIET DE SARCINI Obiectul licitaţiei: Kick off,

More information

Samantha s Strategies page 1 of 2

Samantha s Strategies page 1 of 2 Unit 1 Module 2 Session 3 Smnth s Strtegies pge 1 of 2 Smnth hs been working with vriety of multiplition strtegies. 1 Write n expression to desribe eh of the sttements Smnth mde. To solve 18 20, I find

More information

ARE THE STATIC POWER CONVERTERS ENERGY EFFICIENT?

ARE THE STATIC POWER CONVERTERS ENERGY EFFICIENT? ARE THE STATIC POWER CONVERTERS ENERGY EFFICIENT? Ion POTÂRNICHE 1,, Cornelia POPESC, Mina GHEAMALINGA 1 Corresponding member of the Academy of Technical Sciences of Romania ICPE ACTEL S.A. Abstract: The

More information

Student Book SERIES. Patterns and Algebra. Name

Student Book SERIES. Patterns and Algebra. Name E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing

More information

Aspecte controversate în Procedura Insolvenţei şi posibile soluţii

Aspecte controversate în Procedura Insolvenţei şi posibile soluţii www.pwc.com/ro Aspecte controversate în Procedura Insolvenţei şi posibile soluţii 1 Perioada de observaţie - Vânzarea de stocuri aduse în garanţie, în cursul normal al activității - Tratamentul leasingului

More information

Evaluating territories of Go positions with capturing races

Evaluating territories of Go positions with capturing races Gmes of No Chne 4 MSRI Pulitions Volume 63, 2015 Evluting territories of Go positions with pturing res TEIGO NAKAMURA In nlysing pturing res, or semeis, we hve een fousing on the method to find whih plyer

More information

Areas of Composite Figures 8.4. ACTIVITY: Estimating Area. How can you find the area of. a composite figure?

Areas of Composite Figures 8.4. ACTIVITY: Estimating Area. How can you find the area of. a composite figure? .4 Areas of Composite Figures a composite figure? How can you find the area of 1 ACTIVITY: Estimating Area Work with a partner. a. Choose a state. On grid paper, draw a larger outline of the state. b.

More information

SAG MITTIGATION TECHNICS USING DSTATCOMS

SAG MITTIGATION TECHNICS USING DSTATCOMS Eng. Adrian-Alexandru Moldovan, PhD student Tehnical University of Cluj Napoca. REZUMAT. Căderile de tensiune sunt una dintre cele mai frecvente probleme care pot apărea pe o linie de producţie. Căderi

More information

Prima. Evadare. Ac9vity Report. The biggest MTB marathon from Eastern Europe. 7th edi9on

Prima. Evadare. Ac9vity Report. The biggest MTB marathon from Eastern Europe. 7th edi9on Prima Evadare Ac9vity Report 2015 The biggest MTB marathon from Eastern Europe 7th edi9on Prima Evadare in numbers Par%cipants subscribed 3.228, 2.733 started the race and 2.400 finished the race 40 Photographers

More information

SOFTWARE CONTROL USED FOR AC MOTORS

SOFTWARE CONTROL USED FOR AC MOTORS BULETINUL INSTITUTULUI POLITEHNIC DIN IŞI Publicat de Universitatea Tehnică Gheorghe sachi din Iaşi Tomul LVI (LX), Fasc. 3, 2010 Secţia ELECTROTEHNICĂ, ENERGETICĂ, ELECTRONICĂ SOFTWRE CONTROL USED FOR

More information

Example. Check that the Jacobian of the transformation to spherical coordinates is

Example. Check that the Jacobian of the transformation to spherical coordinates is lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.

More information

Rem Ahsap is one of the prominent companies of the market with integrated plants in Turkey, Algeria and Romania and sales to 26 countries worldwide.

Rem Ahsap is one of the prominent companies of the market with integrated plants in Turkey, Algeria and Romania and sales to 26 countries worldwide. Ȋncepându-şi activitatea ȋn 2004, Rem Ahsap este una dintre companiile principale ale sectorului fabricǎrii de uşi având o viziune inovativǎ şi extinsǎ, deschisǎ la tot ce ȋnseamnǎ dezvoltare. Trei uzine

More information

Hamida Aziz. SOS: Save our Sam!

Hamida Aziz. SOS: Save our Sam! Hamida Aziz SOS: Save our Sam! English Readers a1 a2 palier 1 Hamida Aziz SOS: Save our Sam! Before Reading 1. Look at the cover and find where the boy is. Tick the right answer: 2. Look at the title and

More information

USING SERIAL INDUSTRIAL ROBOTS IN CNC MILLING PROCESESS

USING SERIAL INDUSTRIAL ROBOTS IN CNC MILLING PROCESESS BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LXI (LXV), Fasc. 3, 2015 Secţia CONSTRUCŢII DE MAŞINI USING SERIAL INDUSTRIAL ROBOTS IN CNC MILLING

More information

Question Paper Wednesday 13 Thursday 14 January 2010

Question Paper Wednesday 13 Thursday 14 January 2010 KEY SKILLS INFORMATION AND COMMUNICATION TECHNOLOGY Level 3 ArtComp [KSI31] Question Pper Wenesy 13 Thursy 14 Jnury 2010 Do NOT open this Question Pper until you re tol to y the invigiltor THERE ARE THREE

More information

ACTA TECHNICA NAPOCENSIS

ACTA TECHNICA NAPOCENSIS 143 TECHNICAL UNIVERSITY OF CLUJ-NAPOCA ACTA TECHNICA NAPOCENSIS Series: Applied Mathematics, Mechanics, and Engineering Vol. 59, Issue I, March, 2016 AUTOMATED EQUIPMENT FOR STAMPED SHEET METAL PARTS

More information

Numele şi prenumele elevului... Data susţinerii testului...

Numele şi prenumele elevului... Data susţinerii testului... TEST DE EVALUARE INIȚIALĂ Limba engleză, Clasa a VII-a Numele şi prenumele elevului... Data susţinerii testului... Pentru rezolvarea corectă a tuturor exerciţiilor se acordă 90 de puncte. Din oficiu se

More information

The driving force for your business.

The driving force for your business. Performanţă garantată The driving force for your business. Aveţi încredere în cea mai extinsă reţea de transport pentru livrarea mărfurilor în regim de grupaj. Din România către Spania în doar 5 zile!

More information

Class D Power Amplifiers

Class D Power Amplifiers Class D Power Amplifiers A Class D amplifier is a switching amplifier based on pulse-width modulation (PWM) techniques Purpose: high efficiency, 80% - 95%. The reduction of the power dissipated by the

More information

Patterns and Algebra

Patterns and Algebra Student Book Series D Mthletis Instnt Workooks Copyright Series D Contents Topi Ptterns nd funtions identifying nd reting ptterns skip ounting ompleting nd desriing ptterns numer ptterns in tles growing

More information

PL 221 S-DIAS Pixel LED Module

PL 221 S-DIAS Pixel LED Module PL 221 S-DIAS Pixel LED Module Date of creation: 13.03.2018 Version date: 16.05.2018 Article number: 20-018-221-E Publisher: SIGMATEK GmbH & Co KG A-5112 Lamprechtshausen Tel.: +43/6274/4321 Fax: +43/6274/4321-18

More information

Mods euro truck simulator 2 harta romaniei by elyxir. Mods euro truck simulator 2 harta romaniei by elyxir.zip

Mods euro truck simulator 2 harta romaniei by elyxir. Mods euro truck simulator 2 harta romaniei by elyxir.zip Mods euro truck simulator 2 harta romaniei by elyxir Mods euro truck simulator 2 harta romaniei by elyxir.zip 26/07/2015 Download mods euro truck simulator 2 harta Harta Romaniei pentru Euro Truck Simulator

More information

Symphony No. 9 Last Movement Ludwig van Beethoven

Symphony No. 9 Last Movement Ludwig van Beethoven Bass Friedrich chiller König Friedrich Wilhelm III von Preußen ymhony No 9 Last Movement Ludwig van Beethoven? b4 Presto {h = 96} Temo I 7 Adagio cantabile Allegro {q = 88} Temo I Vivace 9 4 8 4 10 8 b

More information

SLOVAK UNIVERSITY OF TECHNOLOGY Faculty of Material Science and Technology in Trnava. ELECTRICAL ENGINEERING AND ELECTRONICS Laboratory exercises

SLOVAK UNIVERSITY OF TECHNOLOGY Faculty of Material Science and Technology in Trnava. ELECTRICAL ENGINEERING AND ELECTRONICS Laboratory exercises SLOVAK UNIVERSITY OF TECHNOLOGY Fulty of Mteril Siene nd Tehnology in Trnv ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory exerises Róbert Riedlmjer TRNAVA 00 ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory

More information

Regular languages can be expressed as regular expressions.

Regular languages can be expressed as regular expressions. Regulr lnguges cn e expressed s regulr expressions. A generl nondeterministic finite utomton (GNFA) is kind of NFA such tht: There is unique strt stte nd is unique ccept stte. Every pir of nodes re connected

More information

PARLAMENTUL EUROPEAN

PARLAMENTUL EUROPEAN PARLAMENTUL EUPEAN 2004 2009 Comisia pentru piața internă și protecția consumatorilor 2008/0051(CNS) 6.6.2008 PIECT DE AVIZ al Comisiei pentru piața internă și protecția consumatorilor destinat Comisiei

More information

8.1. The Sine Law. Investigate. Tools

8.1. The Sine Law. Investigate. Tools 8.1 Te Sine Lw Mimi 50 ermud Tringle ermud 1600 km Sn Jun 74 Puerto Rio Te ermud Tringle, in te nort tlnti Oen, is te lotion of severl unexplined plne nd sip disppernes. Vrious teories ve een suggested

More information

ISWC T Amor Real. original vocal version **** in Spanish **** with English translation. Op.2 VOCAL PART

ISWC T Amor Real. original vocal version **** in Spanish **** with English translation. Op.2 VOCAL PART SWC T-077481-2 Amo Re oigin voc vesion **** in Spanish **** with English tanslation Op.2 VOCAL PART oigin voc vesion * * * * in Spanish * * * * with English tanslation Voc Pat AMOR REAL Op.2 Lyics, slation,

More information

THIS LECTURE looks at bell ringing (the posh name is Tintinnalogia) which as. WE NORMALLY think of a bell as hanging mouth down. If we swing the bell,

THIS LECTURE looks at bell ringing (the posh name is Tintinnalogia) which as. WE NORMALLY think of a bell as hanging mouth down. If we swing the bell, 7 Bells THIS LECTURE looks t ell ringing (the posh nme is Tintinnlogi) whih s n orgnize tivity hs een roun for long time. Inee, n importnt ook y Stemn on the sujet ws pulishe in 1668 (two yers fter the

More information

The PWM switch model introduced by Vatché Vorpérian in 1986 describes a way to model a voltage-mode switching converter with the VM-PWM switch model.

The PWM switch model introduced by Vatché Vorpérian in 1986 describes a way to model a voltage-mode switching converter with the VM-PWM switch model. The PWM swith model introdued by Vthé Vorpérin in 1986 desribes wy to model voltge-mode swithing onverter with the VM-PWM swith model. The lrge-signl model is equivlent to d trnsformer whose turns rtio

More information

GLONASS PhaseRange biases in RTK processing

GLONASS PhaseRange biases in RTK processing ASS PhseRnge ises in RTK proessing Gle Zyrynov Ashteh Workshop on GSS Bises 202 Bern Switzerlnd Jnury 8-9 202 Sope Simplified oservtion models for Simplified oservtion models for ASS FDMA speifi: lok nd

More information

+ sin bsin. sin. tan

+ sin bsin. sin. tan 6. Spheril rignmetri Frmule Just s in plne gemetry, there re useful trignmetri frmule whih relte the sides nd vertex ngles f spheril tringles: Csine Frmul [6.1] s ss + sin sin s Sine Frmul [6.] sin sin

More information

APSI WIFI, LLC. Company S Monroe Plaza Way Suite A Sandy, UT 84070

APSI WIFI, LLC. Company S Monroe Plaza Way Suite A Sandy, UT 84070 APSI WIFI, LLC Address 9121 S Monroe Plaza Way Suite A Sandy, UT 84070 Publication number WO/2015/161133 Application number PCT/US2015/026259 Publication date 2015-10-22 Filing Date 2015-04-16 Publication

More information

Ziua Copilului. Fabuloase și ieftine! Preț special. Preț special. set creativ Label Waffles. Roabă

Ziua Copilului. Fabuloase și ieftine! Preț special. Preț special. set creativ Label Waffles. Roabă Cadouri de Ziua Copilului Fabuloase și ieftine! Preț special set creativ Label Waffles cu accesorii, 4 tuburi plastilină parfumată și diverse forme Preț special Roabă cu imprimeu Paw Patrol Ofertă valabilă

More information

10.4 AREAS AND LENGTHS IN POLAR COORDINATES

10.4 AREAS AND LENGTHS IN POLAR COORDINATES 65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the

More information

RESEARCH CONCERNING THE INFLUENCE OF ANGLE OF FILING FROM THE KNIFE BLADES VINDROVERS ON THE MECHANICAL WORK ON CUTTING

RESEARCH CONCERNING THE INFLUENCE OF ANGLE OF FILING FROM THE KNIFE BLADES VINDROVERS ON THE MECHANICAL WORK ON CUTTING BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LIX (LXIII), Fasc. 2, 13 SecŃia CONSTRUCłII DE MAŞINI RESEARCH CONCERNING THE INFLUENCE OF ANGLE

More information

Math Circles Finite Automata Question Sheet 3 (Solutions)

Math Circles Finite Automata Question Sheet 3 (Solutions) Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how

More information

A Novel Virtual Anchor Node-based Localization Algorithm for Wireless Sensor Networks

A Novel Virtual Anchor Node-based Localization Algorithm for Wireless Sensor Networks Novel Virtul nhor Noe-se Loliztion lgorithm for Wireless Sensor Networks Pengxi Liu Xinming Zhng Shung Tin Zhiwei Zho Peng Sun Deprtment of Computer Siene n Tehnology University of Siene n Tehnology of

More information

Annual Project meeting and Workshop 8: W8. Managing research data workshop

Annual Project meeting and Workshop 8: W8. Managing research data workshop Modernization of academic library services in Moldova, funded by Norwegian Cooperation Programme in Higher Education with Eurasia. Project number: CPEA-2015/10014 Annual Project meeting and Workshop 8:

More information

Praise Be to You/Song of Compassion. Assembly, SATB Choir, Descant, Organ, and Brass Quartet

Praise Be to You/Song of Compassion. Assembly, SATB Choir, Descant, Organ, and Brass Quartet Chorl Hymns Prise Prise Be You/ng Compssion Assly, SATB Choir, Descnt, Orgn, Brss Qurtet Bo Hurd INLANDIA, 11 10 11 10 11 10 en Sieli, 1865 1957 Arrnged y Crig Kgsury 5 Tpt I Tpt II Tr I Tr II Orgn INTRO

More information

VOLTAGE SAG IMPROVEMENT BY PARTICLE SWARM OPTIMIZATION OF FUZZY LOGIC RULE BASE

VOLTAGE SAG IMPROVEMENT BY PARTICLE SWARM OPTIMIZATION OF FUZZY LOGIC RULE BASE VOL., NO. 7, PRIL 206 ISSN 89-6608 RPN Journl of Engineering nd pplied Sienes 2006-206 sin Reserh Pulishing Network (RPN). ll rights reserved. VOLTGE SG IMPROVEMENT Y PRTILE SWRM OPTIMIZTION OF FUZZY LOGI

More information

Dear Family, Literature Look for these books in a library. Use, and. fl ashcards to compare the groups of objects.

Dear Family, Literature Look for these books in a library. Use, and. fl ashcards to compare the groups of objects. Chapter 7 Dear Famiy, My cass started Chapter 7 today. In this chapter, I wi compare and order numbers to show greater than or ess than. I wi aso use , and = to compare numbers. Love, Vocabuary Power

More information

Probability and Statistics P(A) Mathletics Instant Workbooks. Copyright

Probability and Statistics P(A) Mathletics Instant Workbooks. Copyright Proility nd Sttistis Student Book - Series K- P(A) Mthletis Instnt Workooks Copyright Student Book - Series K Contents Topis Topi - Review of simple proility Topi - Tree digrms Topi - Proility trees Topi

More information

ON THE TRANSIENTS OPTIMIZATION AND THE POWER FACTOR CORRECTION OF THE STATIC CONVERTERS

ON THE TRANSIENTS OPTIMIZATION AND THE POWER FACTOR CORRECTION OF THE STATIC CONVERTERS U.P.B. Sci. Bull., Series C, Vol. 70, No. 1, 2008 ISSN 1454-234x ON THE TRANSIENTS OPTIMIZATION AND THE POWER FACTOR CORRECTION OF THE STATIC CONVERTERS N. FULGA 1, M. O. POPESCU 2, Claudia POPESCU 3 Obiectivul

More information

(1) Non-linear system

(1) Non-linear system Liner vs. non-liner systems in impednce mesurements I INTRODUCTION Electrochemicl Impednce Spectroscopy (EIS) is n interesting tool devoted to the study of liner systems. However, electrochemicl systems

More information

WCCI FAIRIES

WCCI FAIRIES 1 1 st Prize JT «SuperProlem-10», 2016 Задача G12 xnxrx?x? Kx?x?x?x x?xpx?x? Px?x?x?x x?x?x?xk Rx?x?x?x h#2 2 solutions 1+6+5n AntiCire neutrl piees 6, e6, 4, g6, 5 - Nightrier (nn) 6,e6 - Equihopper (ne)

More information

Answers for Chapter 12 Masters

Answers for Chapter 12 Masters Answers for Chapter 2 Masters Scaffolding Answers Scaffolding for Getting Started Activity pp. 55 56 A. 20-sided die: one on the die, 20 numbers on the die, 2 0 Spinner A: one on the spinner, 0 numbers

More information

Planificarea anuala a activitatii de invatare clasa a 9-a (L3) Manualul: ENTERPRISE 2

Planificarea anuala a activitatii de invatare clasa a 9-a (L3) Manualul: ENTERPRISE 2 Planificarea anuala a activitatii de invatare clasa a 9-a (L3) Manualul: ENTERPRISE 2 Nr. crt. Continutul tematic al unitatii de invatare Competente specifice vizate Nr. de ore alocate 1. UNIT 1 PEOPLE

More information

Balancing Your Life. Ideas that might help you

Balancing Your Life. Ideas that might help you Blning Your Life Ides tht might help you Pul Hoskin Summer 2007 Let s e honest if one lists off the responsiilities nd hoies tht eh of us hve nd ssigns weekly hourly time tht eh needs to e fulfilled, then

More information