Outside the high school program / Barycentres 94 exercises (including 70 corrected)

a
3AB1 −AG12·−−ABABABABABABABABABABABABABAB2·−AG3·−−AB−0ABABABABABABABABABABABABAB3·−GA2·−−GB−0ABABABABABABABABABABABABAB5·−GA−3·−−GB−0ABABABABABABABABABABABABAB−2·−GA2·−−GB−0ABABABABABABABABABABABABAB−GA−4·−−GB−0ABABABABABABABABABABABABAB ACDEFMNPQRSGHJK AB AB 1. First approach E.2464 Archimedes (fl250 BC) was one of the first to give an explanation of the principle of moments and that of the barycenter. He writes in the treatise ˇ Sur le center de gravité de surface plane : ˇAny heavy body has a well-defined center of gravity at which any weight of the body can be considered as concentréı The diagram below shows a bar with two separate weights at-tached to its ends. The property of moments states that for the balance to be in equilibrium at point O if the moments m A × OA and m B × OB are equal. Determine the location of point O on this bar (this bar has been split into twenty separate parts) . E.2448 In each case, place the point G , if it exists, on the graduated lines below verifying the requested relationship : E.2449 Let A and B be two points in the plane. Let ¸ and ˛ be any two real numbers such that ¸ + ˛ =0 1 Consider the point G verifying −→ AG = ˛ ¸ + ˛ ·−−→ AB : a Show that the point G verifies the relationship : ¸ · −→ GA + ˛ · −−→ GB = −→ 0 b Show that the point G verifies the relation: −−→ BG = ¸ ¸ + ˛ · −−→ BA 2 Suppose now that the point G is defined by the relation: ¸ · −→ GA + ˛ · −−→ GB = −→ 0 Demonstrate that for any point M of the plane, we have the relationship below : ¸ · −−→ MA + ˛ · −−→ MB = ¸ + ˛ · −−→ MG E.2446 Below is a representation of a mobile. In order to return it to its original appearance, correctly replace the points A , C , D , E , F respectively on segments [ GH ] , [ MN ] , [ JK ] , [ QP ] , [ RS ] , so that each of the branches of this mobile is horizontal. 2. Representation with Thales E.2438 All constructions in this exercise must be carried out using compass and ruler only non-graduated : 1 Divide the segment [ AB ] shown below into four equal parts : 2 Share segment [ AB ] represented below in five equal parts : https://chingmath.fr chapExoCorrec/2464 sacados/2464 3AB1 chapExoCorrec/2448 sacados/2448 −AG12·−−ABABABABABABABABABABABABABAB2·−AG3·−−AB−0ABABABABABABABABABABABABAB3·−GA2·−−GB−0ABABABABABABABABABABABABAB5·−GA−3·−−GB−0ABABABABABABABABABABABABAB−2·−GA2·−−GB−0ABABABABABABABABABABABABAB−GA−4·−−GB−0ABABABABABABABABABABABABAB chapExoCorrec/2449 sacados/2449 chapExoCorrec/2446 sacados/2446 ACDEFMNPQRSGHJK chapExoCorrec/2438 sacados/2438 AB AB
ABC GABCABC E.2447 The figure below is composed of a seg-ment [ AC ] divided into five equal parts and a segment [ AB ] : 1 Place the point G barycenter of the weighted system : ( A ; 2) ; ( B ; 3) 2 Place the point H barycenter of the following system : ( A ; − 2) ; ( B ; 5) Use Thales’ theorem 3. Barycenter of two points E.2459 1 In each case, give the expression of the vector −→ AG as a function of the vector −−→ AB of the weighted sets below : a ( A ; 2) ; ( B ; 1) b ( A ; − 3) ; ( B ; 4) c A ; 3 4 ; B ; − 1 2 2 In each case, give the expression of the vector −−→ BG as a function of the vector −−→ BA of the weighted sets below : a ( A ; 1) ; ( B ; 2) b ( A ; 5) ; ( B ; − 1) c A ; 2 ; B ; 3 2 E.2462 Let A and B be two points in the plane. Let K be the symmetric of A with respect to B . Among the weighted systems below, indicate the system ad-mitting K for barycenter : a ( A ; 1) ; ( B ; − 1) b ( A ; − 1) ; ( B ; 1) c ( A ; − 2) ; ( B ; 1) d ( A ; 1) ; ( B ; − 2) E.2476 According to the course, if G is the barycenter of the system ( A ; ¸ ) ; ( B ; ˛ ) , we have the fol-lowing vector relationship : −→ AG = ˛ ¸ + ˛ · −−→ AB Use this formula to answer the following questions : 1 Let Z be the barycenter of the following weighted sys-tem : ( M ; m ) ; ( F ; ‹ ) Give the expression for the vector −−→ MZ as a function of −−→ MF . 2 Let H be the barycenter of the following weighted sys-tem : ( N ; b ) ; ( Q ; s ) Express −−→ HQ in terms of −−→ NQ . E.2675 Let A , B and C be three aligned points, verifying the following vector relation: −→ AC = 2 3 · −−→ AB 1 a Show that these three points verify the following re-lationship : −→ CA + 2 · −−→ CB = −→ 0 b Determine the coefficients of the system below so that C is the barycenter of this system : ( A ; ¸ ) ; ( B ; ˛ ) 2 Show that A is barycenter of the weighted system ( B ; 2) ; ( C ; − 3) 3 Determine the weighting coefficients of points A and C so that this system admits B as its barycenter. E.2671 The figure opposite shows two triangles ABC and A B C admitting the same point G as isobarycen-tre. Establish the following relationship : −−→ AA + −−→ BB + −−→ CC = −→ 0 E.4243 Let A and B be two distinct points in the plane. Determine the set of points M verifying the equality: −−→ MA · −−→ MB = 0 4. Suites and barycenters - Annales https://chingmath.fr chapExoCorrec/2447 sacados/2447 ABC chapExoCorrec/2459 sacados/2459 chapExoCorrec/2462 sacados/2462 chapExoCorrec/2476 sacados/2476 sacados/2675 chapExoCorrec/2671 sacados/2671 GABCABC sacados/4243 Extrait Liban Juin 2005
BCDEFGHIJKLMNOPA ABCDEFGHIJM E.3231 Part A Given two distinct points A 0 and B 0 of a straight line, we define the points : A 1 middle of segment [ A 0 B 0 ] ; B 1 barycentre de ( A 0 ; 1) ; ( B 0 ; 2) . Then, for any natural number n , A n +1 middle of segment [ A n B n ] and B n +1 barycentre of ( A n ; 1) ; ( B n ; 2) . 1 Place points A 1 , B 1 , A 2 and B 2 for A 0 B 0 =12 cm . What conjecture can be made about the points A n and B n when n becomes very large? 2 We equip the line ( A 0 B 0 ) with the coordinate system A 0 ; −→ i with −→ i = 1 12 · −−−→ A 0 B 0 . Let u n and v n be the re-spective x-coordinates of the points A n and B n . Prove that for any strictly positive natural number n , we have : u n +1 = u n + v n 2 ; v n +1 = u n + 2 v n 3 Part B Consider the sequences ( u n ) and ( v n ) defined by: u 0 = 0 v 0 = 12 ; u n +1 = u n + v n 2 v n +1 = u n + 2 v n 3 for all n ∈ N 1 Prove that the sequence ( w n ) defined by w n = v n − u n is a convergent geometric sequence and that all its terms are positive ; 2 Show that the sequence ( u n ) is increasing and then, that the sequence ( v n ) is decreasing. 3 Deduce from the previous two questions that the se-quences ( u n ) and ( v n ) are convergent and have the same limit. 4 Consider the sequence ( t n ) defined by: t n =2 u n +3 v n . Show that it is constant. Part C From the results obtained in Parts A and B , specify the limit position of points A n and B n when n tends towards plus infinity. 5. Reduction and barycenter of two points E.2893 1 We define the vector −→ u defined by the relation: −→ u = −→ AL + 2 · −→ AF a Draw a representative of the vector −→ u . b Determine the barycenter of the system : ( L ; 1) ; ( F ; 2) . c Reduce the writing of the vector −→ u to the form : −→ u = k · −→ AZ where k ∈ R and Z is one of the 15 points of the seg-ment. 2 For each of the following questions, give a simplification of the expression by introducing a well-chosen weighted system and its barycenter ; then plot a representative of this expression : a −−→ AB + −→ AF b −−→ AJ + 2 · −→ AL E.2861 Consider the segment [ AJ ] shown below subdivided into 9 equal parts and M a point of the plane : 1 Algebraically, justify the following equality: −−→ MC + 4 · −−→ MH = 5 −−→ MG 2 Complete the following equalities: a − 2 · −−→ MB + 3 · −−→ MD = ¸ · −−−→ M : : : b − 2 · −−→ MC − 5 · −−→ MJ = ˛ · −−−→ M : : : 6. Homogeneity rule E.2461 Justify that the two weighted systems below admit the same point as barycenter : A ; 1 ; B ; 2 ; A ; 3 ; B ; 6 E.2463 Let A and B be two points in the plane. 1 Give a weighted system from the points A and B whose barycenter verifies the following relationship : −→ AG = 2 3 · −−→ AB 2 Give a weighted system from the points A and B whose https://chingmath.fr chapExoCorrec/3231 sacados/3231 chapExoCorrec/2893 sacados/2893 BCDEFGHIJKLMNOPA sacados/2861 ABCDEFGHIJM chapExoCorrec/2461 sacados/2461 chapExoCorrec/2463 sacados/2463
KJLMN ABC barycenter verifies the following relationship : −−→ BG = 1 4 · −−→ AB E.2460 Below is a graduated line on which five points are placed. For each of the questions below, find the values of ¸ and ˛ so that the weighted set is the desired barycenter : 1 The weighted system L ; ¸ ; M ; ˛ has the point N as its barycenter. 2 The weighted system L ; ¸ ; J ; ˛ has as barycen-ter the point K . 3 The weighted system N ; ¸ ; M ; ˛ has as barycenter the point L . 7. Barycenter of three points E.2897 In each case, determine the values of the coefficients ¸ , ˛ , ‚ so that point G is the barycenter of the system : G ( A ; ¸ ) ; ( B ; ˛ ) ; ( C ; ‚ ) The points A , B , C satisfy the following relations : 1 3 · −→ GA − 2 · −−→ AB + −−→ GC = −→ 0 2 M represents any point on the plane : 5 · −−→ MG = 2 · −−→ MA − −−→ BC + 3 · −−→ MC E.2899 Consider in the plane three points A , B , C not aligned. Show that there is no point M verifying the relation: 3 · −−→ MA − 4 · −−→ MB + −−→ MC = −−→ AB E.4027 Consider the triangle ABC below. Each of its sides has been divided into 12 equal parts : We wish to determine the location of the point G verifying the vector relation: 4 · −→ GA + 6 · −−→ GB + −−→ GC = −→ 0 1 Consider the point M verifying the vector relationship : 2 · −−→ MA + 3 · −−→ MB = −→ 0 a Using the Chasles relationship, show that the point M verifies the relationship : −−→ BM = 2 5 · −−→ BA b Place the point M on the figure. c Using the Chasles relation and the definition of the point G , establish the following vector relation: 10 · −−→ GM + −−→ GC = −→ 0 d Deduce that the point G belongs to the straight line ( MC ) . 2 Consider the point N barycenter of the partial system : ( B ; 6) ; ( C ; 1) Place the point N on the figure. 3 Deduce the position of point G . E.4044 In the oriented plane, ABCD is a direct square ( −−→ AB ; −−→ AD )= ı 2 . Let I be its center and J the middle of [ AI ] . Determine the value of the real number m so that the point C is the barycenter of the following weighted system : ( A ; m ) ; ( B ; 1) ; ( D ; 1) https://chingmath.fr chapExoCorrec/2460 sacados/2460 KJLMN chapExoCorrec/2897 sacados/2897 chapExoCorrec/2899 sacados/2899 chapExoCorrec/4027 sacados/4027 ABC chapExoCorrec/4044 sacados/4044
ABCIJ ABC E.4045 To any point M of the plane, we associate the point M such that : −−−→ MM = −−→ MA + −−→ MB + 2 · −−→ MC Show that this transformation of the plane is the homothety of ratio − 3 and center G , where G denotes the barycenter of the system : ( A ; 1) ; ( B ; 1) ; ( C ; 2) E.4047 Of the following three proposals, only one is correct. The task is to determine the correct answer and justify the choice thus made. Consider a triangle ABC and note I the point such that : 2 · −→ IB + −→ IC = −→ 0 The points G , I and A are aligned when G is the barycenter of the system : a ( A ; 1) ; ( C ; 2) b ( A ; 1) ; ( B ; 2) ; ( C ; 2) c ( A ; 1) ; ( B ; 2) ; ( C ; 1) E.4049 Let A , B and C be three non-aligned points. 1 a Let M be the point defined by the vector relation: −−→ AM = ¸ · −−→ AB où ¸ ∈ R Determine the values of ¸ for which : the point M belongs to the segment [ AB ] ; the point M belongs to the set ( AB ) \ [ AB ] b Determine the conditions on the two real numbers a and b so that the barycenter G of the weighted system : ( A ; a ) ; ( B ; b ) belongs to segment [ AB ] . 2 Consider the barycenter H of the weighted system : H ( A ; a ) ; ( B ; b ) ; ( C ; c ) Determine the conditions on a , b and c so that the point H belongs inside the triangle ABC . E.4070 For the following statement, only one of the proposed propositions is correct. Give the correct answer: Let ABCD be a direct square −−→ AB ; −−→ AD = ı 2 . Let I be the center of the square. The set of points M of the plane such that : −−→ MA + −−→ MC = AB is : 1 the perpendicular bisector of [ AC ] . 2 the circumscribed circle of the square ABCD . 3 the perpendicular bisector of [ AI ] . 4 the circle inscribed in the square ABCD . E.2966 In the plane, consider the ABC triangle shown below where its sides are split into three equal parts. 1 a Justify that the following two weighted systems ad-mit the same barycenter : ( A ; 2) ; ( B ; 1) ; ( B ; 1) ; ( C ; 2) ( I ; 1) ; ( J ; 1) b Justify that the middle of segment [ IJ ] is the center of gravity of the triangle. 2 Place the following points using only an unlined ruler ; construction lines must be present on the figure, but no justification is required : a The point G center of gravity of the triangle ABC b The point H midpoint of segment [ AC ] . 8. Expression reduction E.2880 Let A , B , C be three non-aligned points in the plane. Consider the following three weighted systems and their associated barycenters : G ( A ; 2) ; ( B ; 1) ; H ( C ; − 1) ; ( B ; 2) I ( A ; 2) ; ( B ; 3) ; ( C ; − 1) 1 Justify that these three systems do indeed admit a barycenter. 2 a Justify the following vector equality: 2 · −→ IA + −→ IB = 3 · −→ IG b Demonstrate that the three points G , H , I are aligned. 3 Below is given the representation of three points in the plane that are not aligned and where the segment [ AB ] has been divided into 15 equal parts. Determine the posi- tion of point I on this figure ; no justification is required. https://chingmath.fr sacados/4045 sacados/4047 chapExoCorrec/4049 sacados/4049 chapExoCorrec/4070 sacados/4070 chapExoCorrec/2966 sacados/2966 ABCIJ chapExoCorrec/2880 sacados/2880 ABC
ABC ABC ABCD E.2891 In the plane, consider the three points A , B , C shown below : The marks shown on the segments or straight lines represent a regular division of each. M represents any point on the plane : 1 a Place, without justification, the point G barycenter of the weighted system ( A ; 2) ; ( B ; 1) ; ( C ; 1) b Reduce the vector expression below where M is any point in the plane : 2 · −−→ MA + −−→ MB + −−→ MC 2 Determine, graphically, the set of points M of the plane such that the vectors −→ u and −→ v , defined below, are collinear: −→ u = 2 · −−→ MA + −−→ MB + −−→ MC −→ v = 2 · −−→ MA − −−→ MB + 2 · −−→ MC E.2898 In the plane, consider three points A , B , C not aligned. We note : I the barycentre of the system ( B ; 1) ; ( C ; − 2) J the barycentre of the system ( A ; − 1) ; ( B ; 3) G barycentre of ( A ; 1) ; ( B ; − 1) ; ( C ; − 4) 1 Place the points I and J in the drawing below. 2 a Justify the following equality: −→ IA − −→ IB − 4 · −→ IC = − 4 · −→ IG b Deduce the equality: 2 · −→ IJ =4 · −→ IG . c Place the point G on the figure. 3 Leaving your construction lines, draw a representative of the vector: −→ GA − −−→ GB − 4 −−→ GC 9. Associativity rule E.2477 Consider the parallelogram ABCD be-low : Each side and diagonal has been deliberately divided into 12 equal parts. 1 Place the point G , barycenter of the following weighted system : ( A ; 1) ; ( B ; − 1) ; ( C ; 1) 2 Place the barycenter H of the points A , D , C weighted by 0, 1, 2 respectively. 3 Place the barycenter J of the following weighted system : ( D ; − 3) ; ( C ; 4) ; ( A ; 2) E.2475 Let ABCD be a parallelogram and O the center of symmetry of this quadrilateral. 1 a Determine non-zero coefficients of the weighted sys-tem ( A ; ¸ ) ; ( B ; ˛ ) ; ( C ; ‚ ) ; ( D ; ‹ ) so that it admits the point O for barycenter. b Is this solution unique? Give another weighted system, not obtained by homogeneity, admitting O as barycen-ter. 2 Let’s note E the symmetric of point O relative to point C . Determine a weighting of points A , C , D so that E be-comes the barycenter of these points. https://chingmath.fr chapExoCorrec/2891 sacados/2891 ABC sacados/2898 ABC chapExoCorrec/2477 sacados/2477 ABCD chapExoCorrec/2475 sacados/2475
ABCDIJKOMNPQ ABCGIDEFHJK ABC E.2485 Consider the figure below where ABCD is a rectangle. I is the midpoint of [ AB ] ; J is the midpoint of [ DO ] and K is the symmetric of O relative to the line ( CB ) . 1 a Let Q be the midpoint of segment [ OB ] . Determine the coefficients of a weighted system ( A ; ¸ ) ; ( B ; ˛ ) ; ( C ; ‚ ) admit the point Q as barycenter. b Deduce the value of ‹ so that the point J is the barycen-ter of : ( A ; ¸ ) ; ( B ; ˛ ) ; ( C ; ‚ ) ; ( D ; ‹ ) 2 a Let M and N be the respective middles of the seg-ments [ AD ] and [ BC ] . Using vector calculus, show that : −−→ DK − 3 · −−→ CK − 3 · −−→ BK + −−→ AK = −→ 0 b Determine the weighting given to the vertex of the rect-angle so that K is the barycentre of the points A , B , C and D . 3 a The point P is such that : −−→ CP = 3 2 ·−−→ CB Justify that I is the middle of segment [ MP ] . b Deduce non-zero weights of the four vertices of the rectangle so that I is the barycenter of ABCD . E.2482 Let ABCD be a square. Construct each barycenter proposed below in two distinct ways : 1 G is barycenter of : ( A ; 1) ; ( B ; 2) ; ( C ; 2) ; ( D ; 1) . 2 H is the barycenter of : ( A ; 2) ; ( B ; − 1) ; ( C ; 3) ; ( D ; 1) . E.2516 Consider the triangles ABC and DEF shown below. The points present on the segments show an equal share of the segment. 1 Determine the coefficients of the weighted system below so that G is the barycenter of this system : ( A ; ¸ ) ; ( B ; ˛ ) ; ( C ; ‚ ) 2 What values must the weighting coefficients of the points D , E and F have for H to be the barycenter of the tri-angle DEF ? E.2517 Consider the triangle ABC right-angled B shown below : The [ AC ] segment has been split into 3 equal parts. We provide the points A , B , C with weights 2, 1, -2 respec-tively; we note G the barycenter of this weighted system. 1 Using the ungraduated ruler and compass, place the barycenter I of the ( A ; 2) systemonthefigure ; ( B ; 1) 2 a Establish the two vector relationships below : https://chingmath.fr chapExoCorrec/2485 sacados/2485 ABCDIJKOMNPQ chapExoCorrec/2482 sacados/2482 chapExoCorrec/2516 sacados/2516 ABCGIDEFHJK chapExoCorrec/2517 sacados/2517 ABC
ABCIJK A0B0C0D0E0F0G0H0A1B1C1D1E1F1G1H1A2B2C2D2E2F2G2H2A3B3C3D3E3F3G3H3A4B4C4D4E4F4G4H4A5B5C5D5E5F5G5H5A6B6C6D6E6F6G6H6 ABCEF ACBACBACB −→ IA = − 1 2 · −→ IB ; −→ IC = − 1 2 · −→ IG b Place the point G on the figure using the ungraduated ruler and compass. 3 Deduce that the straight lines ( AC ) and ( GB ) are paral-lel. Construction lines must remain present on the figure. E.2672 Consider a triangle ABC and the points I , J , K verifying the following vector relations : −→ IA + −→ IB = −→ 0 ; −→ CJ = 1 3 · −−→ CB ; −−→ AK = 2 · −→ AC We wish to show that the points I , J and K are aligned. 1 a Show that B is the barycenter of the weighted sys-tem of points ( C ; − 2) ; ( J ; 3) b Deduce the following relationship : − 2 · −→ IC + 3 −→ IJ = −→ IB 2 Establish the following relationship : 2 · −→ IC − −→ IK = −→ IA 3 Deduce that I is barycenter of the system : ( J ; 3) ; ( K ; − 1) E.525 In the plane, consider the grid below : 1 Determine the barycenter G of the system : ( A 1 ; 1) ; ( C 5 ; 1) ; ( F 5 ; 2) 2 Using only the points shown on the figure, determine the barycenter H of the system : ( A 6 ; 2) ; ( D 6 ; − 2) ; ( B 4 ; − 3) 3 Determine the value of ¸ , a real number, so that the weighted system ( B 4 ; 2) ; ( H 1 ; 1) ; ( E 4 ; ¸ ) admits the point B 1 as barycenter. 10. Barycenter and line characterization E.2473 Let ABC be a triangle. F a point of segment [ BC ] verifying the relation −−→ BF = 3 4 ·−−→ BC . E is a point of [ AB ] verifying −−→ BE = 1 3 ·−−→ BA . 1 Determine the coefficients of the following weighted sys-tem : ( A ; ¸ ) ; ( B ; ˛ ) ; ( C ; ‚ ) so that : E is either the partial barycenter of the system ( A ; ¸ ) ; ( B ; ˛ ) ; F is either the partial barycentre of the system ( B ; ˛ ) ; ( C ; ‚ ) 2 Let G be the barycenter of this system ; justify that the point G is on the intersection of the straight lines ( AF ) and ( CE ) . E.2494 Consider below the ABC triangle, each side of which has been divided into six equal parts. Let ( A ; 1) ; ( B ; 2) ; ( C ; 1) be a weighted system and G its barycenter. 1 Let D be the barycenter of ( A ; 1) ; ( B ; 2) : a Give a vector relation characterizing the position of the point D on the line ( AB ) . b Place the point D on the figure above. 2 Note F the barycenter of ( B ; 2) ; ( C ; 1) . Place the point F on the figure above. 3 a Justify that G is a point on the line ( FA ) . b Justify that G is a point on the line ( DC ) . 4 Let’s note E the partial barycenter of points A and C in the system under consideration a Justify that E is the point of intersection of the straight lines ( BG ) and ( AC ) . https://chingmath.fr chapExoCorrec/2672 sacados/2672 ABCIJK chapExoCorrec/525 sacados/525 A0B0C0D0E0F0G0H0A1B1C1D1E1F1G1H1A2B2C2D2E2F2G2H2A3B3C3D3E3F3G3H3A4B4C4D4E4F4G4H4A5B5C5D5E5F5G5H5A6B6C6D6E6F6G6H6 chapExoCorrec/2473 sacados/2473 ABCEF sacados/2494 ACBACBACB
ACBACBACB ACBG ABCD b Graphically place the point E on the triangle above ; justify the position of E on the segment [ AC ] . E.2465 Consider below the ABC triangle, each side of which has been divided into six equal parts. 1 a Place the point E the middle of [ AC ] . b Place the point F barycenter of the following weighted system : A ; 1 ; B ; 2 c Place the point G barycenter of the following weighted system : C ; 2 2 ; B ; 2 2 Draw the three straight lines ( BE ) , ( FC ) and ( AG ) . Note that all three straight lines intersect. 3 Consider the point H barycenter of the following weighted system of three points : ( A ; 1) ; ( B ; 2) ; ( C ; 1) a Justify that H is the barycenter of the following weighted system : ( F ; 3) ; ( C ; 1) b Justify that H is also the barycenter of the following system : ( B ; 1) ; ( E ; 1) c Justify that the point H belongs to the line ( GA ) . 4 Deduce that the straight lines ( BE ) , ( FC ) and ( AG ) are concurrent. E.2466 Consider the triangle ABC shown below whose sides have all been subdivided into 7 equal parts and a point G placed inside the triangle ABC . Determine the weight assigned to the points A , B , C so that G is the barycenter of the triangle ABC . Justify your ap-proach. E.2515 Consider a triangle ABC . E and F are points belonging respectively to the straight lines ( AC ) and ( AB ) verifying the following vector relations : −−→ CE = 1 3 · −→ CA ; −−→ BF = − 3 4 · −−→ AB On considère le système ( A ; ¸ ) ; ( B ; ˛ ) ; ( C ; ‚ ) : 1 Determine the value of the weighting coefficients of this system so that : E is the barycenter of the points A and C . F is the barycenter of points A and B . 2 We note G the center of gravity of this weighting system and D the point of intersection of the straight lines ( BC ) and ( AG ) . a Justify that the point D is the partial barycenter of the points B and C . b Determine a vector relation characterizing the position of D on the line ( BC ) . E.2484 In the plane, consider a triangle ABC . Let I and J be two points verifying the following vector rela-tionships : −→ BI = 2 3 · −−→ BC ; −→ CJ = 1 4 · −→ CA Let O be the point of intersection of the straight lines ( AI ) and ( BJ ) . Note K the point of intersection of ( CO ) with the straight line ( AB ) . Determine the value of ¸ verifying the following relationship : −−→ BA = ¸ · −−→ BK 11. Barycenter of several points E.2862 Consider the four points A , B , C , D shown below : https://chingmath.fr chapExoCorrec/2465 sacados/2465 ACBACBACB chapExoCorrec/2466 sacados/2466 ACBG chapExoCorrec/2515 sacados/2515 chapExoCorrec/2484 sacados/2484 sacados/2862 ABCD
ABCDMNOP ABCDG600g400g200g600g ABCDFGH20cm40cm IJKLGABCD Determine the position of the barycenter of the following weighted system : ( A ; 3) ; ( B ; − 5) ; ( C ; 2) ; ( D ; 1) E.2863 In the plane, consider the quadrilateral ABCD below : Weighting the points on this figure, we obtain the points M , N , O , P partial barycenter respectively of the following pairs ( A ; B ) , ( B ; C ) , ( C ; D ) , ( D ; A ) : 1 a Determine the position of the partial barycenter of the triangle BCD . b Determine the position of the partial barycenter of the triangle ABC . 2 Deduce the position of the point G barycenter of the quadrilateral ABCD . E.2518 Consider the plane provided with four points A , B , C and D . Equipped with a weight, we obtain the following weighted sys-tem ( A ; 1) ; ( B ; 2) ; ( C ; 1) ; ( D ; − 2) ; note G its barycen-ter. 1 Write a vector relation involving these five points in the plane. 2 Deduce that A is barycenter of the system ( B ; ˛ ) ; ( C ; ‚ ) ; ( D ; ‹ ) ; ( G ; " ) whose coefficients will be determined. E.2892 A chandelier has four bulbs distinct in weight and size ; its iron frame is rectangular in shape and 30 cm by 10 cm in size. A representation of this chandelier is given below : 1 Draw in the plane the rectangle ABCD to scale 1 = 2 . We want to find the center of gravity of this fig-ure ; to do this, we associate the weighted system ( A ; 3) withthisproblem ; ( B ; 2) ; ( C ; 1) ; ( D ; 3) and find the position of the point G : 2 Place the following points : a I partial barycentre of points A and B . b J the partial barycenter of points B and C . c K the partial barycenter of points C and D . d L the partial barycenter A and D . 3 Deduce the position of point G on the figure. E.2894 Consider a homogeneous metal plate of constant thickness, and perform the following cut. The quadrilaterals ABCD and CDEF are two squares with sides 20 cm and 40 cm respectively 1 Determine the respective areas of the two squares. 2 Deduce the position of the center of gravity of this plate. E.2900 In the plane, consider the quadrilateral ABCD ; each of its sides has been divided into twelve equal parts. The points I , J , K , L are part of this marking; the segments [ IJ ] and [ KL ] have been split in the same way. Determine two non-homogeneous weights of the system A ; B ; C ; D admitting point G as barycenter. 12. Barycenter and coordinates https://chingmath.fr sacados/2863 ABCDMNOP chapExoCorrec/2518 sacados/2518 chapExoCorrec/2892 sacados/2892 ABCDG600g400g200g600g sacados/2894 ABCDFGH20cm40cm chapExoCorrec/2900 sacados/2900 IJKLGABCD
-4-3-2-12345678I-4-3-2-1234JOMABC E.2474 The plane is given the reference frame O ; −→ i ; −→ j . Consider the points A , B , C of respective coor-dinates (2 ; 5) , (3 ; − 2) , et ( − 3 ; − 1) Consider the following weighted system : ( A ; 1) ; ( B ; 2) ; ( C ; 4) Determine the coordinates of the point G barycenter of this system. E.2901 Consider the plane provided with the orthonormal reference frame O ; I ; J shown below : The points A , B , C have the coordinates : (2 ; 2) , ( − 3 ; 1) , ( − 3 ; − 3) . Consider the weighted system below having for point G as barycenter : G ( A ; 2) ; ( B ; − 1) ; ( C ; 1) 1 a Determine the coordinates of point G . b Place the point G in the datum. 2 Let M be the point in the plane with coordinate (4 ; 2) . Place the point H verifying the relation: −−→ MH = 2 · −−→ MA − −−→ MB + −−→ MC 13. Barycenter and complex numbers E.4073 In the complex plane referred to the reference frame O ; −→ u ; −→ v orthonormal direct, the point A has affix i . To any point M of affix z with z =i , we associate the point M whose affix is defined by: z = − z 2 z − i We name G the isobarycentre of the points A , M and M and g the affix of G : 1 Check the equality: g = 1 3 · ( z − i) 2 Deduce that : if M is a point on the circle with center A of radius r , then G is a point on the circle with center O of radius 1 3 · r . 3 Demonstrate that : arg( g )= − −→ u ; −−→ AM E.4108 The complex plane is provided with a reference frame O ; −→ u ; −→ v orthonormal direct. The graph-ical unit will be 1 cm . We denote by A , B and C the points of respective affixes : z A = 3 + 2 · i ; z B = − 3 ; z C = − 1 − 6 · i Determine and construct the set Γ 1 of points M of the plane such that : −−→ MA − −−→ MB + −−→ MC = 1 2 · −−→ MA + −−→ MC 14. Space and isobarycenter E.4084 The space is given an orthonormal reference frame O ; −→ i ; −→ j ; −→ k . Consider the points : P 1 ; 2 ; 3 ; Q 4 ; 2 ; − 1 ; R − 2 ; 3 ; 0 In a tetrahedron, the segment joining a vertex to the center of gravity of the opposite face is called the median. 1 Show that the tetrahedron OPQR is not regular. 2 We name P the center of gravity of the triangle OQR . Calculate the coordinates of P , center of gravity of tri- angle OQR . 3 Verify that a Cartesian equation of the plane ( OQR ) is : 3 · x + 2 · y + 16 · z = 0 4 Consider the following property ( P ) : P : Dans a tetrahedron, each median is orthog-onal to the opposite face. Is the property ( P ) true in any tetrahedron? https://chingmath.fr chapExoCorrec/2474 sacados/2474 chapExoCorrec/2901 sacados/2901 -4-3-2-12345678I-4-3-2-1234JOMABC chapExoCorrec/4073 sacados/4073 Extrait d'Antilles-Guyane Juin 2008 sacados/4108 chapExoCorrec/4084 sacados/4084 Extrait Pondichey Avril 2011
ABCDIJKLMN ABCDIJKLMN ABCDIJKLMN ABCDGHI E.4041 Space is provided with an or-thonormal reference frame O ; −→ i ; −→ j ; −→ k . Consider the points : A 1 ; − 1 ; 4 ; B 7 ; − 1 ; − 2 ; C 1 ; 5 ; − 2 1 a Calculate the coordinates of the vectors : −−→ AB ; −→ AC −−→ BC b Show that the triangle ABC is equilateral. c Deduce that x + y + z − 4=0 is a Cartesian equation of the plane ( ABC ) . 2 Determine the coordinates of point G isobarycenter of points A , B , C . E.4244 Consider a ABCDEFGH cube with edge 1. 1 Express the vector more simply: −−→ AB + −−→ AD + −→ AE 2 Deduce that the scalar product −→ AG · −−→ BD is zero. 3 Similarly demonstrate that the scalar product −→ AG · −−→ BE is zero. 4 Show that the line ( AG ) is orthogonal to the plane ( BDE ) . 15. Space and barycentre E.2486 Consider the tetrahedron ABCD below where I , J , K , L , M , N are the respective middles of the sides [ BC ] , [ DC ] , [ AD ] , [ AC ] , [ AB ] , [ BD ] . Using the isobarycenter of the tetrahedron, show the follow-ing two questions : 1 Show that the straight lines ( JM ) , ( KI ) , ( LN ) are con-current. 2 Show that the quadrilateral JLMN is a parallelogram. For the next two questions, we’ll use the theorems of classical geometry: 3 Show that the quadrilateral JLMN is a parallelogram. 4 Deduce that the three straight lines connecting the mid-dles of opposite sides are concurrent. E.2519 Consider the tetrahedron ABCD below : Consider : The point G barycentre of the system : ( D ; 2) ; ( B ; 2) . The point H barycenter of the system : ( A ; 3) ; ( B ; − 2) ; ( C ; 3) . We name I the middle of segment [ GH ] . Justify that the point I belongs to the plane ( ADC ) . https://chingmath.fr chapExoCorrec/4041 sacados/4041 sacados/4244 Extrait de Polynesie Juin 200 chapExoCorrec/2486 sacados/2486 ABCDIJKLMN ABCDIJKLMN ABCDIJKLMN chapExoCorrec/2519 sacados/2519 ABCDGHI
DABCHEFG E.4068 Consider a cube ABCDEFGH of edge length 3 . We choose the orthonormal reference frame D ; −→ i ; −→ j ; −→ k such that : −→ i = 1 3 · −−→ DA ; −→ j = 1 3 · −−→ DC ; −→ k = 1 3 · −−→ DH 1 Give the coordinates of the points A , C and E . 2 Determine the coordinates of the point I barycenter of the system ( C ; 2) ; ( E ; 1) 3 Determine the coordinates of the vectors −→ AE and −→ DI . E.4246 In space, consider three points A , B and C . Let G be the barycenter of the system : ( A ; 1) ; ( B ; 1) ; ( C ; 2) Consider The transformation which, to any point M of space, associates the point M such that : −−−→ MM = −−→ MA + −−→ MB + 2 · −−→ MC Justify that this transformation is the homothety of center G and ratio 3 . 16. Space, barycenter and straight line E.4252 In space provided with an orthonor-mal reference frame O ; −→ i ; −→ j ; −→ k , we give the three points : A 1 ; 2 ; − 1 ; B − 3 ; − 2 ; 3 ; C 0 ; − 2 ; − 3 We call G the barycenter of the weighted system : ( A ; 1) ; ( B ; − 1) ; ( C ; 2) 1 Demonstrate that the point G has coordinates 2 ; 0 ; − 5 . 2 Demonstrate that the line ( CG ) is orthogonal to the plane ( P ) . 3 Determine a parametric representation of the line ( CG ) . 4 Determine the coordinates of point H , intersection of plane ( P ) with line ( CG ) . E.4060 In space referred to an orthonor-mal frame of reference O ; −→ i ; −→ j ; −→ k , we have the points : A 0 ; 0 ; 2 ; B 0 ; 4 ; 0 ; C 2 ; 0 ; 0 We denote by G the isobarycenter of the points A , B and C . Say whether the following proposition is true or false : The line ( AG ) admits as parametric representation : x = t y = 2 t z = 2 − 2 t where t ∈ R E.4026 Consider space provided with a reference frame O ; −→ i ; −→ j ; −→ k orthonormé . For any real k , consider the point M k whose coordinates are defined by: x = 2 − k y = 3 · k z = 2 + k 1 a Déterminer point coordinates M k dans the following three cases : k = 0 ; k = 1 ; \quad k = − 1 b Justify that points M 0 , M 1 et M − 1 sont aligned. 2 We consider the point A of coordinates − 1 ; 9 ; 4 : a Does the point A appartient belong to the set of points M k ? b Does point A belong to the line M 0 M 1 ? 3 a Show that, for any real number k , the vectors −−−−→ M 0 M k and −−−−→ M 0 M 1 are collinear. b Deduce the nature of the set of points M k when k de-scribes R . c Without justification, give the nature of the set of points M k when k describes each of the following three sets : −∞ ; 0 ; 0 ; 1 ; 1 ; + ∞ https://chingmath.fr chapExoCorrec/4068 sacados/4068 DABCHEFG sacados/4246 Inspir de Pondichery Avril 2009 sacados/4252 Extrait de Liban Juin 2011 chapExoCorrec/4060 sacados/4060 chapExoCorrec/4026 sacados/4026
OABCDEFG ABCDEFGH ABCDMNOP E.4042 Consider the cube OABCDEFG of edge length 1 shown below. The completed graph is not required to be returned with the copy. Let be The points P and Q such that : −−→ OP = 2 · −→ OA ; −−→ OQ = 4 · −−→ OC We call R the barycenter of the weighted points ( B ; − 1) and ( F ; 2) . Space is provided with the orthonormal reference frame O ; −→ OA ; −−→ OC ; −−→ OD . 1 a Show that the point R has coordinates 1 ; 1 ; 2 . b Demonstrate that the points P , Q and R are not aligned. c What is the nature of the triangle PQR ? 2 Consider the plane ( P ) of equation : 4 · x + 2 · y + z − 8 = 0 a Show that the points P , Q , R verify the equation. b Verify that the point D does not belong to the plane ( PQR ) . 17. Space, barycenter and weighting search E.2881 Consider the ABCDEFGH paral-lelepiped shown below : The following weighted system admits the point N as barycen-ter : ( A ; 1) ; ( B ; − 1) ; ( E ; − 1) ; ( D ; − 1) 1 a Give a representative, using the points on this figure, of the following sum : −−→ GE + −−→ GB + −−→ GD b Establish that point N is the midpoint of segment [ AG ] . 2 The space is given the reference frame A ; −−→ AB ; −−→ AD ; −→ AE . Using the coordinates of the points, establish that the point N is the midpoint of the segment [ AG ] . 3 Consider the system E ; F ; A ; H ; determine the weighting of this system so that its barycenter is the point N E.2870 Consider the tetrahedron ABCD shown below : The edges of this tetrahedron have been split into equal parts. The points M , N , O , P are respectively points of the edges [ BC ] , [ CD ] , [ AD ] , [ AB ] . 1 Determine the weighting, if any, of the vertices of the tetrahedron so that the points M , N , O , P are the par-tial barycenters of the ends of the edges to which they belong. 2 Let G be the barycenter of this tetrahedron : a Justify that the point G belongs to the line ( PN ) . b Deduce that the points M , N , O , P are coplanar. https://chingmath.fr chapExoCorrec/4042 sacados/4042 OABCDEFG chapExoCorrec/2881 sacados/2881 ABCDEFGH sacados/2870 ABCDMNOP
ABCDMNOP ABCD E.2872 Consider the tetrahedron ABCD shown below : The edges of this tetrahedron have been split into equal parts. The points M , N , O , P are respectively points of the edges [ BC ] , [ CD ] , [ AD ] , [ AB ] . 1 Determine the weighting, if any, of the vertices of the tetrahedron so that the points M , N , O , P are the par-tial barycenters of the ends of the edges to which they belong. 2 Let G be the barycenter of this tetrahedron : a Justify that the point G belongs to the line ( PN ) . b Deduce that the points M , N , O , P are coplanar. E.3114 Space is provided with an orthonor-mal reference frame O ; −→ i ; −→ j ; −→ k . Consider the three points A , B and C of coordinates : A − 1 ; 1 ; 3 ; B 2 ; 1 ; 0 ; C 4 ; − 1 ; 5 Can we write C as the barycenter of the points A and B ? E.3271 Consider the tetrahedron ABCD ; let I be the midpoint of the segment [ AB ] and J be the mid-point of [ CD ] . 1 a Let G 1 be the barycenter of the weighted point sys-tem : ( A ; 1) ; ( B ; 1) ; ( C ; − 1) ; ( D ; 1) Express −−→ IG 1 as a function of −−→ CD . Place I , J and G 1 on figure (see appendix sheet) . b Let G 2 be the barycenter of the weighted point system ( A ; 1) ; ( B ; 1) ; ( D ; 2) . Show that G 2 is the midpoint of segment [ ID ] . Place G 2 . c Prove that IG 1 DJ is a parallelogram. Deduce the position of G 2 relative to the points G 1 and J . 2 Let m be a real number. Let G m be the barycenter of the weighted point system : ( A ; 1) ; ( B ; 1) ; ( C ; m − 2) ; ( D ; m ) a Specify the set E of values of m for which the barycen-ter G m exists. In the following questions, we assume that the real number m belongs to the set E . b Prove that G m belongs to the plane ( ICD ) . c Show that the vector m −−−→ JG m is constant. d Deduce the set F of points G m when m describes the set E . E.4043 Space is provided with an or-thonormal reference frame O ; −→ i ; −→ j ; −→ k . Consider the points : A − 1 ; 1 ; 3 ; B 2 ; 1 ; 0 ; C 4 ; − 1 ; 5 Can we write the point C as the barycenter of the points A and B . 18. Space, barycenter and geometric locus E.4040 Space is referred to an orthonor-mal reference frame O ; −→ i ; −→ j ; −→ k . Consider the points A 3 ; 1 ; 3 and B − 6 ; 2 ; 1 . Determine the characteristics of the set of points M in space such that : 4 · −−→ MA − −−→ MB = 2 E.4048 Space is provided with a reference frame O ; −→ i ; −→ j ; −→ k . Consider the points : A 1 ; − 1 ; 3 ; B 0 ; 3 ; 1 ; C 6 ; − 7 ; − 1 D 2 ; 1 ; 3 ; E 4 ; − 6 ; 2 1 a Show that the barycenter of the system : ( A ; 2) ; ( B ; − 1) ; ( C ; 1) est le point E . b Deduce the set Γ of points M in space such that : 2 · −−→ MA − −−→ MB + −−→ MC = 2 · √ 21 https://chingmath.fr sacados/2872 ABCDMNOP sacados/3114 Extrait de Pondichery Avril 2010 sacados/3271 Antilles-Guyane Juin 2004 4 points ABCD chapExoCorrec/4043 sacados/4043 chapExoCorrec/4040 sacados/4040 chapExoCorrec/4048 sacados/4048
A0A1B0B1u024681012 EHGFADCB 2 Show that the points A , B and D define a plane. 3 Determine a parametric representation of the line ( EC ) . E.4059 Indicate whether the following proposition is true or false : Let B and C be two points in space. The set of points M of space such that : −−→ MB + −−→ MC = −−→ MB − −−→ MC is the sphere of diameter [ BC ] . 19. Old annals and sequels (before 2012) E.3432 Part A Consider the sequences of points A n and B n defined for all natural numbers n as follows : on an oriented axis O ; −→ u given in the appendix, the point A 0 has abscissa 0 and the point B 0 has abscissa 12. Point A n +1 is the barycenter of points ( A n ; 2) and ( B n ; 1) , the point B n +1 is the centroid of the weighted points ( A n ; 1) and ( B n ; 3) . 1 On the graph, place the points A 2 , B 2 . 2 We define the sequences a n and b n as the respective x-coordinates of the points A n and B n . Show that : a n +1 = 2 · a n + b n 3 We also assume that : b n +1 = a n +3 · b n 4 Part B 1 Consider the sequence u n defined, for any natural num-ber n , by: u n = b n − a n . a Show that the sequence u n is geometric. Explain why. b Give the expression of u n as a function of the natural number n . c Determine the limit of u n . Interpret this result geo-metrically. 2 a Prove that the sequence a n is increasing (you can use the sign of u n ) . b Study the variations of the sequence b n . 3 What can be said about the previous results regarding the convergence of the sequences a n and b n ? Part C 1 Consider the sequence v n defined, for any natural num-ber n , by: v n = 3 · a n + 4 · b n Show that the sequence v n is constant. 2 Determine the limit of the sequences a n and b n . 20. Old annals and spaces (before 2012) E.3265 Consider the cube aBCDEFGH shown opposite ; O 1 and O 2 are the centers of squares ABCD and EFGH , and I is the center of gravity of triangle EBD . Let m be a real number and G m the barycenter of the weighted point system : ( E ; 1) ; ( B ; 1 − m ) ; ( G ; 2 m − 1) ; ( D ; 1 − m ) Part A 1 Justify the existence of point G m . 2 Specify the position of point G 1 . 3 Verify that G 0 = A . Deduce that points A , I , and G are aligned. 4 Prove that −−−→ AG m = m −−→ AO 2 . Deduce the set of points G m when m runs through the set of real numbers. 5 a Verify that the points A , G m , E and O 1 are copla-nar. b Determine the value of m for which G m lies on the line ( EI ) . Part B In this question, the space is referenced to the orthonormal coordinate system A ; −−→ AB ; −−→ AD ; −→ AE . 1 Prove that the line ( AG ) is orthogonal to the plane ( EBD ) . Deduce a Cartesian equation for the plane ABD . 2 Determine the coordinates of the point G m . 3 For what values of m is the distance from G m to the plane ( EBD ) equal to 3 3 ? https://chingmath.fr chapExoCorrec/4059 sacados/4059 sacados/3432 Antilles-Guyane Juin 2006 5 points A0A1B0B1u024681012 sacados/3265 EHGFADCB
E.3269 In the affine plane, consider ABC a right-angled triangle in A , I the middle of the segment [ AB ] and J the center of gravity of ABC . For any real m , different from − 1 3 , note G m the barycenter of the weighted point system : S m = ( A ; 1) ; ( B ; m ) ; ( C ; 2 m ) For any point M of the plane, note : −→ V M = 3 −−→ MA − −−→ MB − 2 −−→ MC For each of the following six statements, say whether it is true ( V ) or false ( F ) . Each correct answer gives 0.5 point, each wrong or illegible an-swer deducts 0.25 point, no answer gives or deducts no points. Any negative total would be reduced to 0 . . Statement T or F G 1 is the midpoint of the segment [ CI ] G 1 is the barycenter of ( J ; 2) ; C ; 2 3 For any point M : −→ V M = −−→ AB +2 −→ AC For all ; , distinct from − 1 3 , −−−→ AG m is collinear with −−−→ AG − 1 IBG − 1 2 is a right triangle For any point P of ( AG − 1 ) , there exists a real number m such that P = G m E.4064 In the plane ( P ) , consider the tri-angle ABC isocèle at A , of height [ AH ] tel that AH = BC =4 . The unit of measurement is the centimetre. 1 Justifying the construction, place the point G , barycen-ter of the weighted point system : ( A ; 2) ; ( B ; 1) ; ( C ; 1) 2 We denote the point M any point of ( P ) . a Show that the vector −→ V =2 · −−→ MA −−−→ MB −−−→ MC est a vec-tor whose norm is 8 . b Determine and construct the set E 1 of points M du plane such that : 2 · −−→ MA + −−→ MB + −−→ MC = −→ V 3 Consider the weighted point system : ( A ; 2) ; ( B ; n ) ; ( C ; n ) where n est a fixed natural number. a Show that the barycenter G n of this system of weighted points exists. Place G 0 , G 1 , G 2 . b Show that the point G n belongs to the segment AH . c Calculate the distance AG n as a function of n and de-termine the limit of AG n when n tends towards + ∞ . Specify the limit position of G n when n tends towards + ∞ . d Let E n be the set of points M in the plane such that : 2 · −−→ MA + n · −−→ MB + n · −−→ MC = n · −→ V Show that E n is a circle that passes through the point A . Specify its center and radius, denoted by R n . e Construct E 2 . E.4069 The complex plane is referenced to a direct orthonormal coordinate system O ; −→ u ; −→ v (graphi-cal unit : 2 cm ) . Consider the points A , B , and C with respective affixes : z A = − 3 2 + i · 3 2 ; z B = z A ; z C = − 3 Part A 1 Write the complex numbers z 1 and z B in exponential form. 2 Place the points A , B and C . 3 Show that triangle ABC is equilateral. Part B Let f be the application which, for any point M in the plane of affix z , associates the point M with affix z = 1 3 · i · z 2 . We note O , A , B and C the points respectively associated by f with the points O , A , B and C . 1 a Determine the exponential form of the affixes of points A , B et C . b Place the points A , B and C . c Demonstrate the alignment of points O , A and B as well as that of points O , B and A . 2 Let G be the isobarycenter of points O , A , B and C . Let G be the point associated with G by f . a Determine the affixes of points G and G . b Is point G the isobarycenter of points O , A , B and C ? 3 Prove that if M belongs to the line ( AB ) then M be-longs to the parabola with equation : y = − 1 3 · x 2 + 3 4 (We are not asking you to draw this parabola) E.4241 Consider a tetrahedron ABCD . We denote I , J , K , L , M , N the respective middles of the edges [ AB ] , [ CD ] , [ BC ] , [ AD ] , [ AC ] and [ BD ] . We denote by G the isobarycenter of the points A , B , C and D . 1 Show that the straight lines ( IJ ) , ( KL ) and ( MN ) are concurrent at G . In the rest of the exercise, it is assumed that : AB = CD ; BC = AD ; AC = BD (The tetrahedron ABCD is said to be equifacial because its faces are isometric) . 1 a What is the nature of the quadrilateral IKJL ? Also specify the nature of the quadrilaterals IMJN and KNLM . b Deduce that ( IJ ) and ( KL ) are orthogonal. It will be admitted that, similarly, the straight lines ( IJ ) and ( MN ) are orthogonal and the straight lines ( KL ) and ( MN ) are orthogonal. https://chingmath.fr chapExoCorrec/3269 sacados/3269 chapExoCorrec/4064 sacados/4064 sacados/4069 Liban Juin 2009 5 points sacados/4241 Extrait de Pondichery Avril 2008
ABCDA E.3204 In space provided with an or-thonormal frame of reference O ; −→ i ; −→ j ; −→ j , we give the points : A 2 ; 1 ; 3 ; B − 3 ; − 1 ; 7 ; C 3 ; 2 ; 4 1 Show that the points A , B et C ne are not aligned. 2 So ( d ) the parametric representation line: x = − 7 + 2 t y = − 3 t z = 4 + t où t ∈ R a Show that the line ( d ) est orthogonal to the plane ( ABC ) . b Give a Cartesian equation of the plane ( ABC ) . 3 So H the point common to the line ( d ) et and the plane ( ABC ) . a Show that H is the centroid of ( A ; − 2) , ( B ; − 1) and ( C ; 2) . b Determine the nature of the set Γ 1 , points M in space such that : − 2 −−→ MA − −−→ MB + 2 −−→ MC · −−→ MB − −−→ MC = 0 Specify the characteristic elements. c Determine the nature of the set Γ 2 , points M in space such that : − 2 −−→ MA − −−→ MB + 2 −−→ MC = 29 Specify the characteristic elements. d Specify the nature and give the characteristic elements of the intersection of the sets Γ 1 and Γ 2 . e Does point S − 8 ; 1 ; 3 belong to the intersection of sets Γ 1 and Γ 2 . E.4085 Part I In this part, ABCD is a regular tetrahedron, i.e. a solid whose four faces are equilateral triangles. A is the center of gravity of the triangle BCD . In a tetrahedron, the segment joining a vertex to the center of gravity of the opposite face is called the median. Thus, the segment [ AA ] is a median of the tetrahedron ABCD . 1 We wish to prove the following property: ( P 1 ) : Dans a regular tetrahedron, each median is orthogonal to the opposite face. a Show that : −−→ AA · −−→ BD = 0 ; −−→ AA · −−→ BC = 0 . (We can use the middle I of the segment [ BD ] et the middle J du segment [ BC ] .) b Deduce that the median ( AA ) is orthogonal to the face BCD . A similar argument shows that the other medians of the regular tetrahedron ABCD are also orthogonal to their opposite faces. 2 G is the isobarycenter of points A , B , C , and D . We wish to prove the following property: ( P 2 ) : The medians of a regular tetrahedron are concurrent at G . Using the associativity of the centroid, show that G be-longs to the line ( AA ) , then conclude. Part II We equip the space with an orthonormal reference frame O ; −→ i ; −→ j ; −→ k . Consider the points : P 1 ; 2 ; 3 ; Q 4 ; 2 ; − 1 ; R − 2 ; 3 ; 0 1 Show that the tetrahedron OPQR is not regular. 2 Calculate the coordinates of P , the center of gravity of triangle OQR . 3 Verify that a Cartesian equation of the plane ( OQR ) is : 3 · x + 2 · y + 16 · z = 0 4 Is the property ( P 1 ) true in any tetrahedron? https://chingmath.fr chapExoCorrec/3204 sacados/3204 Liban mai 2006 5 points chapExoCorrec/4085 sacados/4085 Pondichey Avril 2011 5 points ABCDA
E.3207 For each of the following five state-ments, indicate whether it is true or false and provide evidence for your answer. An answer without evidence will not earn any points. In the space relative to a reference point O ; −→ i ; −→ j ; −→ k or-thonormal, we give the points : A 0 ; 0 ; 2 ; B 0 ; 4 ; 0 ; C 2 ; 0 ; 0 We denote by I the midpoint of the segment [ BC ] , by G the isobarycenter of the points A , B and C , and H as the orthog-onal projection of point O onto plane ( ABC ) . Proposition 1: ˇall points M in the space such that −−→ AM ·−−→ BC =0 is the plane ( AIO ) ı. Proposition 2: ˇthe set of points M in the space such that −−→ MB + −−→ MC = −−→ MB −−−→ MC is the sphere with diameter [ BC ] ı. Proposition 3: ˇThe volume of the tetrahedron OABC is equal to 4ı. Proposition 4: ˇthe plane ( ABC ) has the equation 2 x + y +2 z =4 and the point H has coordinates 8 9 ; 4 9 ; 8 9 ı. Proposition 5: ˇthe right ( AG ) admits for parametric rep-resentation : x = t y = 2 t z = 2 − 2 t where t ∈ R ı. E.3223 Three distinct points A , B , and C are given in the plane, which are not collinear. An urn U contains six cards that are indistinguishable to the touch, bearing the numbers − 2 , − 1 , 0 , 1 , 2 , and 3 . An urn V contains five cards that are indistinguishable to the touch ; four cards bear the number 1 and one card bears the number − 1 . A card is drawn at random from each of the urns. The draws are equiprobable. Let a be the number read on the card from U and b the number read on the card from V . 1 Justify that the weighted points ( A ; a ) , ( B ; b ) and ( C ; 4) have a barycenter. We denote it by G . 2 a Determine the probability of each of the following events : E 1 : ˇ G belongs to the line ( BC ) ‘’; E 2 : ˇ G belongs to the segment [ BC ] ‘’. b Show that the probability of event E 3 : ˇ G is located inside triangle ABC and does not belong to any of its sidesı is equal to 2 5 . Considerations of sign may be used. 3 Let n be a non-zero natural number. We repeat n times under the same conditions the test consisting of drawing a card from each of the urns U and V and then consid-ering the barycenter G of question 1 . We denote by X the random variable taking as values the number of realizations of the event E 3 . a Determine the integer n so that the expectation of the random variable X is equal to 4. b Determine the smallest integer n so that the probabil-ity of having at least one of the barycenters located inside the triangle ABC is greater than or equal to 0.999 . https://chingmath.fr chapExoCorrec/3207 sacados/3207 chapExoCorrec/3223 sacados/3223
EHGFADCB E.3224 Answers to this exercise are to be written on the attached sheet. Any ambiguous answer will be considered as a non-answer . For each of the five questions, one or more answers are correct. The candidate must write T (true) or F (false) in the corresponding box . No justification is required. For each question, 3 correct an-swers earn 1 point and 2 correct answers earn 1 2 point. Let ABCDEFGH be a cube of side 1 . Choose the orthonor-mal coordinate system A ; −−→ AB ; −−→ AD ; −→ AE . Let I and J be the midpoints of segments [ EF ] and [ FG ] , respectively. L is the centroid of ( A ; 1) ; ( B ; 3) . Let ( ı ) be the plane with equation : 4 x − 4 y +3 z − 3=0 1 The coordinates of L are: a 1 4 ; 0 ; 0 b 3 4 ; 0 ; 0 c 2 3 ; 0 ; 0 2 The plane ( ı ) is the plane : a ( GLE ) b ( LEJ ) c ( GFA ) 3 The plane parallel to the plane ( ı ) passing through I intersects the line ( FB ) at M with coordinates : a 1 ; 0 ; 1 4 b 1 ; 0 ; 1 5 c 1 ; 0 ; 1 3 4 a The lines ( EL ) and ( FB ) intersect at a point N which is the image of M under the symmetry with center B . b The lines ( EL ) and ( IM ) are parallel. c The lines ( EL ) and ( IM ) intersect. 5 The volume of the tetrahedron FIJM is : a 1 36 b 1 48 c 1 24 E.3233 The space E is referenced to an orthonormal basis O ; −→ i ; −→ j ; −→ k . Consider the points A , B and C with respective coordinates 1 ; 0 ; 2 , 1 ; 1 ; 4 and − 1 ; 1 ; 1 . 1 a Show that points A , B , and C are not collinear. b Let −→ n be the coordinate vector 3 ; 4 ; − 2 . Check that the vector −→ n is orthogonal to the vectors −−→ AB and −→ AC . Deduce a Cartesian equation of the plane ( ABC ) . 2 Let P 1 and P 2 be the planes with the respective equa-tions : 2 x + y + 2 z + 1 = 0 ; x − 2 y + 6 z = 0 a Show that the planes P 1 and P 2 intersect along a line D for which we will determine a system of parametric equations. b Are the line D and the plane ( ABC ) intersecting or parallel? 3 Let t be any positive real number. Consider the barycen-ter G of the points A , B and C assigned the respective coefficients 1 , 2 and t . a Justify the existence of point G for any positive real number t . Let I be the barycenter of points A and B assigned the respective coefficients 1 and 2 . Determine the co-ordinates of point I . Express the vector −→ IG in terms of the vector −→ IC . b Show that the set of points G when t describes the set of positive or zero real numbers is the segment [ IC ] excluding the point C . For what value of t does the midpoint J of the segment [ IC ] coincide with G ? https://chingmath.fr chapExoCorrec/3224 sacados/3224 EHGFADCB chapExoCorrec/3233 sacados/3233
E.3253 For each of the five questions, only one of the three statements is correct. Candidates should indicate the question number and the letter corresponding to their chosen answer on their answer sheet. No justification is required. A correct answer is worth 1 point ; an incorrect answer deducts 0.5 points ; no answer is worth 0 points. If the to-tal is negative, the score is reduced to zero. The space is reported to an orthonormal reference frame O ; −→ i ; −→ j ; −→ k . Consider the points A 3 ; 1 ; 3 and B − 6 ; 2 ; 1 . The plane P has the following Cartesian equation : x +2 y + 2 z =5 1 All points M in space such as : 4 · −−→ MA − −−→ MB = 2 is : a a plane in space b une sphère c l’ensemble vide 2 The coordinates of point H , orthogonally projected from point A onto the plane P are: a 11 3 ; 1 3 ; 1 3 b 8 3 ; 1 3 ; 7 3 c 7 3 ; − 1 3 ; 5 3 3 The sphere with center B and radius 1: a intersects the plane P following a circle b is tangent to the plane P c does not intersect the plane P 4 Consider the line D of the space passing through A and with direction vector −→ u 1 ; 2 ; − 1 and the right D of équations paramétriques x = 3 + 2 t y = 3 + t z = t where t ∈ R The lines D and D are: a coplanar and parallel b coplanar and secant c non coplanaires 5 The set of points M in space equidistant from points A and B is : a the line with parametric equations : x = − 3 2 − t y = 3 2 − 7 · t z = 2 + t where t ∈ R b the Cartesian equation plane : 9 x − y +2 z +11=0 . c the Cartesian equation plane : x +7 y − z − 7=0 . E.3241 Let ABCD be a tetrahedron such that ABC , ABD , ACD are three isosceles right triangles A with : AB = AC = AD = a . Let A 1 be the center of gravity of triangle BCD . 1 Show that line ( AA 1 ) is orthogonal to plane ( BCD ) . (For example, we can calculate −−→ AA 1 · −−→ CD and −−→ AA 1 · −−→ BC ) 2 Expressing the volume of the tetrahedron ABCD in two different ways, calculate the length of the segment [ AA 1 ] . 3 Let G be the isobarycenter of the tetrahedron ABCD and I the midpoint of [ BC ] . a Show that G belongs to the segment [ AA 1 ] and deter-mine the length AG . b Determine the set of points M in space such that : −−→ MA + −−→ MB + −−→ MC + −−→ MD = 2 · −−→ MB + −−→ MC 4 Let H be the image of A under the symmetry with center G . a Prove that : 4 · −→ GA + −→ AC + −−→ AD = −−→ BA . b Prove the equality: HC 2 − HD 2 = −−→ DC ·−−→ BA . c Deduce that : HC = HD . Reminder : the volume of a pyramid with height h and associated base area b is : V = 1 3 · b · h https://chingmath.fr chapExoCorrec/3253 sacados/3253 chapExoCorrec/3241 sacados/3241
EHGFADCB E.3194 ABCDEFGH is the edge cube 1 shown on the attached sheet, which must be completed and returned with the copy. The space is referenced to the or-thonormal coordinate system A ; −−→ AB ; −−→ AD ; −→ AE . Part A. A triangle and its center of gravity 1 Prove that triangle BDE is equilateral. 2 Let I be the center of gravity of triangle BDE . a Calculate the coordinates of I . b Prove that −→ AI = 1 3 −→ AG . What can we deduce for points A , I , G ? 3 Prove that I is the orthogonal projection of A onto the plane ( BDE ) . Part B. A particular line For any real number k , we define two points M k and N k , as well as a plane P k as follows : M k is the point on the line ( AG ) such that −−−→ AM k = k · −→ AG ; P k is the plane passing through M k and parallel to the plane ( BDE ) ; N k is the point of intersection of the plane P k and the line ( BC ) . 1 Identify P 1 3 , M 1 3 and N 1 3 using previously defined points. Calculate the distance M 1 3 N 1 3 . 2 Calculation of N k coordinates. a Calculate the coordinates of M k in the frame A ; −−→ AB ; −−→ AD ; −→ AE . b Determine an equation of the plane P k in this reference frame. c Deduce that the point N k has coordinates 1 ; 3 · k − 1 ; 0 . 3 For what values of k is the line ( M k N k ) orthogonal to both the lines ( AG ) and ( BC ) ? 4 For what values of k is the distance M k N k minimal? 5 Draw on the figure given in the appendix the section of the cube by the plane P 1 2 . Draw the line M 1 2 N 1 2 in the same figure. E.3250 For each of the eight statements (in quotation marks) below, indicate whether it is true or false. Candidates should indicate the question number and write ˇtrueı or ˇfalseı . A correct answer is worth 0.5 points, an incorrect answer deducts 0.25 points, and no answer neither adds nor deducts points. Any negative total will be reduced to zero. 1 ˇ If a is any real number and f is a function defined and strictly decreasing on a ; + ∞ , then lim x ↦→ + ∞ f ( x )= −∞ ı 2 Let f and g be two functions defined on 0 ; + ∞ , g not canceling each other out : ˇ If lim x ↦→ + ∞ f ( x )= −∞ and if lim x ↦→ + ∞ g ( x )=+ ∞ then lim x ↦→ + ∞ f ( x ) g ( x ) = − 1 ı 3 ˇ If f is a function defined on 0 ; + ∞ such that 0 f ( x ) √ x on [0 ; + ∞ [ then lim x ↦→ + ∞ f ( x ) x =0 ı 4 Consider a coordinate system O ; −→ i ; −→ j of the plan. ˇ If f is a function defined on R ∗ then the line of x =0 is an asymptote to the curve representing f in the coordinate system O ; −→ i ; −→ j ı 5 ˇ The function f defined on R by f ( x )=( x 2 +3 x +1)e x is a solution on R of the differential equation : y − y =(2 x +3)e x ı 6 Let A , B , C be three points in the plane. We call I the barycenter of the points A and B assigned respectively the coefficients 3 and − 2 . ˇ If G is the centroid of points A , B and C are assigned the coefficients 3 , − 2 and 1 , then G is the midpoint of the segment [ CI ] ı 7 Let A , B , C be three points in the plane and G the cen-troid of A , B and C are assigned the coefficients 3 , − 2 and 1 , respectively. ˇ The set of points M in the plane such that : 3 · −−→ MA − 2 · −−→ MB + −−→ MC =1 is the circle with center G and radius 1 ı. 8 Let A and B be two distinct points on the plane. Let M be any point on the plane. ˇ The scalar product −−→ MA · −−→ MB is zero if, and only if, M = A or M = B ı. 21. Unclassified financial years https://chingmath.fr chapExoCorrec/3194 sacados/3194 EHGFADCB chapExoCorrec/3250 sacados/3250 Liban juin 2005 4 points
ABCDEFGH ABCDEFGHIJM 10cm6cm8cmABCKG E.3135 Consider in space a cube of side 3 cm , denoted ABCDEFGH and shown below : Let I be the barycenter of the weighted points ( E ; 2) and ( F ; 1) , J that of ( F ; 1) and ( B ; 2) and finally K that of ( G ; 2) and ( C ; 1) . We want to determine the set of points M equidistant from I , J and K . We note Δ this set 1 Place the points I , J and K on the figure above. 2 Let Ω be the point of Δ located in the plane ( IJK ) . What does this point represent for the IJK triangle? For the remainder of the exercise, we now place ourselves in the following orthonormal frame of reference : A ; 1 3 −−→ AD ; 1 3 −−→ AB ; 1 3 −→ AE 3 Give the coordinates of the points I , J and K . 4 Let P 2 ; 0 ; 0 and Q 1 ; 3 ; 3 be two points to be placed on the figure. Show that the straight line ( PQ ) is orthog-onal to the plane ( IJK ) . 5 Let M be a point in space with coordinates ( x ; y ; z ) . a Demonstrate that M belongs to Δ if, and only if, the triplet ( x ; y ; z ) is a solution of a system of two linear equations that we will write down. What is the nature of Δ ? b Verify that P and Q belong to Δ . Draw Δ on the figure. 6 a Determine a normal vector to the plane ( IJK ) and deduce a Cartesian equation of this plane. b Then determine the exact coordinates of Ω E.3808 Let A , B be two distinct fixed points of a circle C of center I and M be any point on this circle C . The point D is defined by: −→ IA + −→ IB + −−→ IM = −→ ID 1 Prove that the scalar products −−→ AD · −−→ BM and −−→ BD · −−→ AM are zero. Deduce to which particular straight lines of the triangle ABM the point D belongs and then specify the nature of the point D for the triangle AMB . 2 Let G be the isobarycenter of the points A , B , M . Ex-press −→ ID as a function of −→ IG . E.4313 The figure opposite shows a cube ABCDEFGH of edge 1 . We denote by I and J the respective middles of the edges [ BC ] and [ CD ] . Let M be any point on segment [ CE ] . Throughout the exercise, we place ourselves in the orthonor-mal reference frame A ; −−→ AB ; −−→ AD ; −→ AE . 1 a Give, without justification, the coordinates of the points C , E , I and J . b Justify the existence of a real t belonging to the inter-val 0 ; 1 , such that the coordinates of point M are 1 − t ; 1 − t ; t . 2 a Demonstrate that the points C and E belong to the mediator plane of the segment [ IJ ] . b Deduce that the triangle MIJ is an isosceles triangle at M . c Express IM 2 in terms of t . E.1665 In the diagram, consider the configura-tion shown below : 1 Write the drawing instructions for this configuration, be-ginning with the sentence : ˇDraw triangle ABC such that : AC =10 cm ; BC =8 cm ; AB =6 cm ‘’ 2 The following constructions must be drawn using a ruler and a compass : a Reproduce this figure to scale. b Draw the circle with center K and passing through point G . What do you notice? https://chingmath.fr sacados/3135 France Septembre 2006 6 points ABCDEFGH chapExoCorrec/3808 sacados/3808 Extrait Antilles-Guyane Septembre 2003 sacados/4313 ABCDEFGHIJM chapExoCorrec/1665 sacados/1665 10cm6cm8cmABCKG