Outside the high school program / Similarities 43 exercises (including 21 corrected)

a
-3-2-101234567-4-3-2-1123uv 1. Transformation and complex E.3933 Consider the plane provided with a reference frame O ; −→ i ; −→ j . Consider the point M of the plane of affix z . For each ques-tion, the point M is associated with a point M whose affix z is defined as a function of z . Determine the nature and characteristics of each of these ap-plications : a z = z − 1 + 2 · i b z = 2 · z + 3 − 2 · i c z = 1 2 − 3 2 · i · z + 1 + i E.3934 In the plane provided with the reference frame O ; −→ u ; −→ v . Consider the three plane similitudes : h is the homothety with center A of affix 1+i and ratio 2 ; t is the translation whose vector has affix − 1+2 · i ; r is the rotation of angle − 2 ı 3 and center B of affix 1 − 3 6 +i · − 1 2 − 3 3 . 1 Give the expression of the complex applications associ-ated with the similarities h , t and r . 2 Determine the complex expressions of the transforma-tions : a r ◦ h b h ◦ r c t ◦ r d r ◦ t 3 Consider the rotation r of center C of affix 2 and angle ı 3 . a Determine the complex expression of the transforma-tion r ◦ r . b Give the characteristics of the transformation r ◦ r . E.3941 In the complex referred to a direct orthonormal reference frame O ; −→ u ; −→ v , consider the direct similitude f of complex writing: z ↦−→ 3 2 · (1 − i) · z + 4 − 2 · i Indicate whether the following proposition is true or false and give a demonstration of the chosen answer: f = r ◦ h where h is the homothety of ratio 3 · 2 2 and of center the point Ω of affix − 2 − 2 · i and where r is the rotation of center Ω and angle − ı 4 . E.3935 Consider the plane O ; −→ u ; −→ v ; let A and A be the two points of the plane with affixes 1 2 − i and − 3+2 · i respectively. 1 Place the points A and A in the above reference frame. 2 a Determine the center of the homothety h of ratio 2 transforming the point A into A . b Let M be a point of the plane and z its affix. Let M be the image of M by h and z its affix. Show that we have the relation: z = 2 · z − 4 + 4 · i c Let B and C be the two points with affixes 1 2 − 2 · i and 9 2 − i respectively. Determine the affixes of the points B and C images of the points B and C by the homo-thety h . Place all these points on the graph. 3 Consider the point D of affix 3 . Let h be the homothety of center D with affix 3 and ratio − 3 2 . a Give the complex expression of the similarity h . b Determine the affixes of the images of A , B , C . c Draw the image of triangle ABC by homothety h . E.4028 In the complex plane provided with an orthonormal reference frame O ; −→ u ; −→ v . We denote by m a real number. Consider the transformation T m of the plane into itself which, to any point M of affix z associates the point M of affix z defined by: z = ( m + i) · z + m − 1 − i 1 Can we choose m such that T m is a translation? 2 Determine the real m such that T m is a rotation. Then specify the center and angle of this rotation. 2. Direct similarity E.3970 The complex plane is equipped with a direct orthonormal reference frame O ; −→ u ; −→ v of unit graphics 1 cm , we consider the points A , B , C , M , N and P with respective affixes : a = 1 + i ; b = − 1 + 2 · i ß ; c = 2 + 3 · i m = 7 − 5 · i ; n = 5 − i ; p = 9 + i https://chingmath.fr chapExoCorrec/3933 sacados/3933 chapExoCorrec/3934 sacados/3934 sacados/3941 Extrait de Liban Juin 2008 chapExoCorrec/3935 sacados/3935 -3-2-101234567-4-3-2-1123uv sacados/4028 chapExoCorrec/3970 sacados/3970
-8-6-4-202468-6-4-2246uvM0M1M2M3M4M5M6 1 a Place the points A , B , C , M , N and P on the coor-dinate plane. b Calculate the lengths of the sides of triangles ABC and NMP . c Deduce that these two triangles are similar. In the rest of the exercise, we will highlight the direct simi-larity that transforms triangle ABC into triangle MNP . 2 Let s be the direct similarity that transforms point A into N and point B into P . a Show that a complex notation of similarity s is : z = − 6 5 − 8 5 · i · z + 23 5 + 9 5 · i b Determine the ratio, the value of the angle rounded to the nearest degree, and the center of similarity s . c Verify that the similarity s transforms point C into M . E.3971 The complex plane is referred to a A ; −→ u ; −→ v direct orthonormal coordinate system. The graphical unit is 1 cm . Note i the complex number of modulus 1 and argument ı 2 . Consider the points B , C and H of affixes : b = 5 · i ; c = 10 ; h = 2 + 4 · i Construct a figure to be completed as the questions progress. 1 Study the position of point H : a Demonstrate that the point H belongs to the line ( BC ) . b Calculate h h − c , and deduce that : −−→ HC ; −−→ HA = − ı 2 [2 · ı ] 2 Study of a first similarity: a Calculate the ratios : BH AH ; BA AC ; AH CH b Show that there is a direct similarity S 1 that trans-forms the triangle CHA into the triangle AHB . c Determine the complex writing of this similarity S 1 as well as its characteristic elements. E.3972 The plane is equipped with a di-rect orthonormal reference frame O ; −→ u ; −→ v of units 1 cm . 1 Let the points C and D have respective affixes c =3 and d =1 − 3 · i , and S 1 the similarity that associates the point M of the plane with the point M 1 image of M by the symmetry axis symmetry O ; −→ u of the real numbers. a Place the points C and D then their respective images C 1 and D 1 by S 1 . The figure will be completed as the exercise progresses. b Give the complex expression of S 1 . 2 Let S 2 be the direct similarity defined by: the point C 1 and its image C with affix : c =1+4 · i ; the point D 1 and its image D d’affix : d = − 2+2 · i . a Show that the complex expression of S 2 est : z = i · z + 1 + i b Determine the characteristic elements of this similar-ity. 3 So S the similarity defined by: S = S 2 ◦ S 1 . Determine the complex expression of S . E.3936 The complex plane is referenced to a direct orthonormal coordinate system O ; −→ u ; −→ v . We consider the transformation f of the plane which, at any point M with affix z , associates the point M with affix z defined by: z = 2 · e 3 · i · π 8 · z We define a sequence of points ( M n ) as follows : the point M 0 has affix z 0 =i and for any natural number n : M n +1 = f M n . We denote z n the affix of the point M n . The points M 0 , M 1 , M 2 and M 3 are shown in the figure below : 1 Determine the nature and characteristic elements of the transformation f . 2 We note g the transformation f ◦ f ◦ f ◦ f . a Determine the nature and characteristic elements of the transformation g . b Deduce that for any natural number n : OM n +4 = 4 · OM n ; −−−→ OM n ; −−−−−→ OM n +4 = − ı 2 [2 · ı ] c Complete the figure by constructing the points M 4 , M 5 https://chingmath.fr chapExoCorrec/3971 sacados/3971 Extrait d'Asie Juin 2010 chapExoCorrec/3972 sacados/3972 Extrait d'Antilles Guyane Juin 2010 chapExoCorrec/3936 sacados/3936 -8-6-4-202468-6-4-2246uvM0M1M2M3M4M5M6
et M 6 . 3 Prove that for any natural number: z n = 2 n · e i · π 2 + 3 · n · π 8 4 Let there be two natural numbers n and p such that p n . a Express in terms of n and p a measure of : −−−→ OM p ; −−−→ OM n . b Prove that points O , M p and M n are aligned if, and only if, n − p is a multiple of 8 . E.4002 The complex plane is referred to a direct orthonormal frame O ; −→ u ; −→ v . Consider the application f of the plane which, to any point M d’affixe z , associates the point of affix z and g that which, to any point M d of affix z , associates the point of affix z  défini by: z = 1 + i · 3 2 · z ; z  = e i · π 5 · z 1 Precise the nature and characteristic elements of the f and g applications. 2 Consider the points A 0 and B 0 d affixes respectively: a 0 = 2 · e − 2 · i · π 3 ; b 0 = 4 · e − i · π 5 Let A n et B n les be a sequence of points defined by the recurrence relations : A n +1 = f A n ; B n +1 = g B n We note a n et b n les respective affixes of A n et B n . a What is the nature of each of the triangles OA n A n +1 ? b Determine the nature of the polygon A 0 A 1 A 2 A 3 A 4 A 5 . 3 a Montrer que les points B n sont situés sur un cercle dont on préciseira le center et le rayon. b Indicate a measure of the angle −−→ OB n ; −−−−→ OB n +2 . c Deduce the nature of the polygon B 0 B 2 B 4 B 6 B 8 . 4 Express a n and b n in terms of n . E.4116 The complex plane is equipped with an orthonormal basis O ; −→ u ; −→ v . Let A and C be the respective affix points : a = 3 + 5 · i ; c = 1 + 4 · i Let f be the transformation of the plane onto itself which, at any point M with affix z , associates the point M with affix z defined by: z = (2 − 2 · i) · z + 1 1 Let M be the affix point z = x + i · y , where we assume that x and y are relative integers. Let M be the image of M by f . Show that the vectors −−−→ CM and −→ CA are orthogonal if and only if x +3 · y =2 2 Consider the equation ( E ): x +3 · y =2 , where x and y are relative integers. a Check that the pair ( − 4 ; 2) is a solution of ( E ) . b Solve the equation ( E ) . c Deduce the set of points M whose coordinates are in-tegers belonging to the interval − 5 ; 5 and such that the vectors −−−→ CM and −→ CA are orthogonal. Place these points on the figure. 3. Indirect similarity E.4003 The complex plane is referred to a direct orthonormal reference frame O ; −→ u ; −→ v . Let A , B and C be the points with respective affixes : z A = 2 + i ; z B = 5 + 2 · i ; z C = i s 1 designates ( AB ) axis symmetry. 1 Demonstrate that s 1 transforms any point M of affix z into a point M such that : z = 4 5 + 3 5 · i · z + − 1 5 + 3 5 · i 2 Deduce the affix of C , image of C by symmetry of axis ( AB ) . 3 Demonstrate that the set of points M such that z is pure imaginary is the straight line ( D ) of equation 4 x +3 y =1 4 Verify that the point C belongs to ( D ) . E.3969 Let O ; −→ u ; −→ v be a direct or-thonormal datum of the complex plane (graphic unit : 4 cm ) We denote by A the point with affix : z A =1 . Consider the transformation T of the plane which, to any point M of affix z , associates the point of affix − z +2 . 1 Determine the respective images by the T transformation of the point A and the point Ω of affix 1+i · 3 . 2 Deduce the nature and characteristic elements of the T transformation. 3 Determine the image by the transformation T of the cir-cle C of center O and radius 1 . https://chingmath.fr chapExoCorrec/4002 sacados/4002 Extrait Antilles-Guyane Septembre 2008 sacados/4116 Extrait d'Amerique du Nord Juin 2007 chapExoCorrec/4003 sacados/4003 chapExoCorrec/3969 sacados/3969
-2-1234I-2-12JOABMNP E.4029 In the complex plane provided with a direct orthonormal frame of reference O ; −→ u ; −→ v , consider the two rectangles OABC and DEFG where the points A , B , C , D , E , F , G have respective affixes. z A = − 2 ; z B = − 2 + i ; z C = i ; z D = 1 z E = 1 + 3 · i ; z F = 5 2 + 3 · i ; z G = 5 2 Consider the indirect similarity s of complex writing: z = − 2 3 · i · z + 5 3 · i 1 Determine the image of rectangle DEFG by similarity s . 2 Consider the similarity g = s ◦ s . Determine the image of rectangle OABC by similarity g . 3 In this question, any trace of research, however incom-plete, or initiative, however unsuccessful, will be taken into account in the assessment. Does the similarity g have fixed points? What can we conclude for g ? E.4033 In the plane provided with the direct orthonormal frame O ; −→ u ; −→ v , consider points A of affix 3 · i and B of affix 6 . 1 Show that there is one direct similitude and only one that transforms A into O and O into B . Specify its character-istic elements. 2 Show that there is one and only one indirect similarity that transforms A into O and O into B . E.4113 The complex plane is provided with a direct orthonormal reference frame O ; −→ u ; −→ v . 1 Consider the similarity s admitting the complex writing: z = − 6 5 − 8 5 · i · z + 23 5 + 9 5 · i Determine the set of invariant points of the similitude s . 2 Consider the similarity ff admitting the complex writing: z = i · z + 1 + i Determine the set of invariant points of the similitude ff . E.4115 The complex plane is referenced to O ; −→ u ; −→ v orthonormal direct. Consider the points A and B of affixes 2 and 1 − i , respectively. Determine the complex writing of the axial symmetry of axis ( AB ) . 4. Annales - Direct similarities E.3949 The complex plane is equipped with a coordinate system O ; −→ OI ; −→ OJ orthonormal direct. We consider the points A and B with respective affixes : z A = 2 ; z B = 3 2 + i We consider the points M , N and P such that the triangles AMB , BNO and OPA are isosceles right triangles in the forward direction, as shown in the figure below : We denote s 1 the direct similarity with center A that trans-forms M into B . We denote s 2 the direct similarity with center O that trans-forms B into N . We consider the transformation : r = s 2 ◦ s 1 The aim of the exercise is to demonstrate in two dif-ferent ways that the lines ( OM ) and ( PN ) are perpen-dicular. 1 Using the transformations : a Give the angle and ratio of s 1 et of s 2 . b Determine the image of point M then that of point I by the transformation r . c Justify that r is a rotation of angle ı 2 whose center will be specified. d What is the image of point O by r ? e Deduce that the straight lines ( OM ) and ( PN ) sont perpendicular. 2 Using complex numbers: a Give the complex writings of s 1 et s 2 . The results from question 1 a will be used. b Determine the affixes z M and z N des points M et N . c Give, without justification, the affix z p of the point P puis demonstrate that the straight lines ( OM ) et ( PN ) sont perpendicular. https://chingmath.fr sacados/4029 Extrait de Metropole Septembre 2010 chapExoCorrec/4033 sacados/4033 Extrait Liban Mai 2006 sacados/4113 sacados/4115 chapExoCorrec/3949 sacados/3949 -2-1234I-2-12JOABMNP
ABCD E.3171 The complex plane is provided with a direct orthonormal reference frame O ; −→ u ; −→ v . We’ll take 5 cm as the graphic unit. Let f be the transformation which, to any point M of affix z , associates the point M of affix z defined by: z = 1 2 + 1 2 i z + 1 1 Justify that f is a direct similitude whose center Ω (of affix ! ) , ratio k and angle „ should be specified. 2 Let A 0 be the point O and, for any natural number n , let: A n +1 = f ( A n ) . a Determine the affixes of the points A 1 , A 2 , A 3 then place the points A 0 , A 1 , A 2 and A 3 . b For any natural number n , we pose u n =Ω A n . Jus-tify that the sequence ( u n ) is a geometric sequence and then establish that, for any natural number n : u n = 2 · 1 2 n c From what rank n 0 do all points A n belong to the disk with center Ω and radius 0.1 ? 3 a What is the nature of the triangle Ω A 0 A 1 ? Deduce, for any natural number n , the nature of the triangle Ω A n A n +1 . b For any natural number n , note ‘ n the length of the broken line A 0 A 1 A 2 : : : A n − 1 A n . This gives : ‘ n = A 0 A 1 + A 1 A 2 + · · · + A n +1 A n . Express ‘ n in terms of n . What is the limit of the sequence ( ‘ n ) ? E.3147 The plane is provided with a di-rect orthonormal coordinate system O ; −→ u ; −→ v (unit 1 cm ) . We’ll build a figure and complete it as we go along. 1 Let A be the point of affix 3, and r the rotation of center O and angle ı 3 . Let B , C , D , E and F be the respective images of the points A , B , C , D and E by the rotation r . Show that B has affix 3 2 + 3 3 2 i 2 Associate with each of the points C , D , E and F one of the affixes of the following set : − 3 ; − 3 2 + 3 √ 3 2 i ; 3 2 − 3 3 2 i ; − 3 2 − 3 3 2 i 3 a Determine r ( F ) . b What is the nature of the polygon ABCDEF ? 4 Let s be the direct similitude of center A , ratio 1 2 and angle ı 3 . Let s be the direct similitude of center E trans-forming F into C . a Determine the angle and ratio of s ? Deduce the angle and ratio of s ◦ s . b What is the image of point D by s ◦ s ? c Determine the complex writing of s . 5 Let A be the symmetrical of A with respect to C . a Without using complex numbers, determine s ( A ) and then the image of A by s ◦ s . b Calculate the affix of the point A . Then find the result of a using the complex writing s ◦ s . E.3203 The figure given in Appendix 2 will be completed as the questions are answered, and it will be returned with the copy. ABCD is a square such that −−→ AB ; −−→ AD =+ ı 2 . Let I be the center of the square ABCD . Let J be the midpoint of the segment [ CD ] . Let s be the direct similarity that transforms A into I and B into J . The aim of the exercise is to study certain properties of sim-ilarity s . In part A , we will use geometric reasoning ; in part B , we will use complex numbers. Part A 1 Determine the ratio and angle of similarity s . 2 We denote the center of this similitude by Ω . Γ 1 is the circle of diameter [ AI ] , Γ 2 is the circle of diameter [ BJ ] . Show that Ω is one of the points of intersection of Γ 1 and Γ 2 . Place Ω on the figure. 3 Give the image by s of the line ( BC ) . Deduce the point image by s of the point C , then the point K image by s of the point I . 4 We pose h = s ◦ s (composed of s with itself) . a Give the nature of the transformation h (specify its characteristic elements) . b Find the image of point A by h . Deduce that the points A , Ω and K are aligned. Part B The complex plan is referenced to a coordinate system A ; −→ u ; −→ v direct orthonormal basis, chosen so that the points A , B , C and D have respective affixes 0 , 2 , 2 + 2i and 2i . 1 Show that the complex form of the similarity is z = 1 2 i z + 1 + i . 2 Calculate the affix of the point Ω . 3 Calculate the affix of point E such that s ( E )=1 . Place point E on the figure. https://chingmath.fr sacados/3171 sacados/3147 sacados/3203 ABCD
ABCDEOIJH E.3940 Part A We assume the following result is known : An application f of the plane equipped with a direct orthonor-mal reference frame in itself is a direct similarity if, and only if, f admits a complex expression of the form : z = a · z + b where a ∈ C ∗ and b ∈ C . Course demonstration : we are in the complex plane. Prove that if A , B , A and B are four points such that A is distinct from B and A is distinct from B , then there exists a unique direct similarity transforming A into A and B into B . Part B In the complex plane equipped with a direct orthonormal co-ordinate system O ; −→ u ; −→ v , we consider the points A , B , C , D with respective affixes : z A = − 3 − i ; z B = 1 − i · 3 z C = 3 + i ; z D = − 1 + i · 3 1 a Give the modulus and argument of each of the four complex numbers z A , z B , z C and z D . b Construct the points A , B , C and D (we will use 2 cm as the graphic unit) . c Determine the midpoint of segment [ AC ] and that of segment [ BD ] . Calculate the quotient z B z A . Deduce the nature of the quadrilateral ABCD . 2 Consider the direct similarity g whose complex notation is : z = e − i · π 2 · z + 2 . a Determine the characteristic elements of g . b Construct the respective images E , F and J using a ruler and compass, g points A , C and O . c What can be observed about these points E , F and J ? Demonstrate this. E.3183 In the figure given below, consider the squares OABC and OCDE such that : −→ OA ; −−→ OC = −−→ OC ; −−→ OE = ı 2 We denote by I the middle of segment [ CD ] , by J the mid-dle of segment [ OC ] and by H the point of intersection of segments [ AD ] and [ IE ] 1 Justify the existence of a direct similarity s transforming A into I and D into E . 2 Determine the ratio of this similarity s . Assume that the angle of the similarity s is equal to ı 2 . 3 Give, without justification, the image of B by s . 4 Determine and place the image of C by s . 5 Let Ω be the center of the similarity s : a Show that Ω belongs to the circle of diameter [ AI ] and to that of diameter [ DE ] . b Show that Ω cannot be the point H . c Construct Ω . 6 Consider the direct orthonormal frame O ; −→ OA ; −−→ OC a Determine the complex writing of the similarity s . b Deduce the affix of the center Ω of s . https://chingmath.fr sacados/3940 Pondichery Avril 2008 sacados/3183 ABCDEOIJH
-2-12345678I-6-5-4-3-2-1234JOABC E.4031 Consider a direct square ABCD (i.e., a square ABCD such that −−→ AB ; −−→ AD = ı 2 [2 ı ] ) with center I . Let J , K and L be the respective midpoints of segments [ AB ] , [ CD ] and [ DA ] . Γ 1 denotes the circle with diameter [ AI ] and Γ 2 denotes the circle with diameter [ BK ] . Part A 1 Determine the ratio and angle of direct similarity s such that : s ( A ) = I ; s ( B ) = K 2 Show that circles Γ 1 and Γ 2 intersect at two distinct points : the point J and the center Ω of the direct simi-larity s . 3 a Determine the images by s the lines ( AC ) and ( BC ) . Deduce the image of point C by s . b Let E be the image of s under I . Prove that E is the midpoint of the segment [ ID ] . 4 In this question, any evidence of research, even if incom-plete, or initiative, will be taken into account in the as-sessment. Demonstrate that points A , Ω , and E are aligned. (The transformation t = s ◦ s may be considered) Part B Now, we consider that the side of the square measures 10 units and we place ourselves in the direct orthonormal refer-ence frame A ; 1 10 · −−→ AB ; 1 10 · −−→ AD . 1 Give the affixes of points A , B , C and D . 2 Demonstrate that the direct similarity s has the follow-ing complex notation : z = i 2 · z + 5 + 5 · i 3 Calculate the affix ! of the center Ω of s . 4 Calculate the affix z E of point E and find the alignment of points A , Ω and E . 5 Demonstrate that the lines ( AE ) , ( CL ) and ( DJ ) are concurrent at point Ω . E.4127 The complex plane is equipped with an orthonormal direct basis O ; −→ u ; −→ v . We consider the indirect similarity f of complex notation : z = 1 + i · 3 · z where z denotes the conjugate of z . Let the points A and B have respective affixes : z A = 6 + i · 2 ; z B = − 2 + i · 6 We note A and B the respective images of points A and B by f . A figure provided in the APPENDIX to the subject will be completed and submitted with the copy. The various constructions shall be made using a ruler and compass, and the construction lines shall be clearly visible. 1 a Write the affixes of points A and B in exponential form. b Show that triangle OAB is a right isosceles triangle. c Determine the nature of the triangle OA B . d Show that the affix z A de A vérifie the equality: z A = 2 · z A From this deduce the construction of A et B . 2 Note r the rotation of center O and angle measure ı 3 et s la orthogonal symmetry of axis O ; −→ u . We pose : g = r ◦ s . a Determine the complex writing of the transformation g . b Show that the points O and A are invariant by g . c Determine the nature of the transformation g . 3 a Montrer que l’on peut écrire f = h ◦ g , où h est une homothétie de center et de rapport à déterminer. b On the figure shown in APPENDIX , a point C is placed. Construct the image C of C by the transfor-mation f . 5. Annales - Indirect similarities E.3159 The plane is related to the or-thonormal frame O ; −→ u ; −→ v (graphic unit 4 cm ) Part I 1 Place the points I , J , H A , B , C , D with respective af-fixes : z I = 1 ; z J = i ; z H = 1 + i ; z A = 2 z B = 3 2 + i ; z C = 2i ; z D = − 1 https://chingmath.fr chapExoCorrec/4031 sacados/4031 Amerique du Sud Novembre 2009 5 points chapExoCorrec/4127 sacados/4127 -2-12345678I-6-5-4-3-2-1234JOABC chapExoCorrec/3159 sacados/3159
2 Let E be the symmetrical of B with respect to H . The perpendicular to the line ( AE ) passing through C and the parallel to the line ( OC ) passing through D intersect at F . Place E and F and check that the point F has affix z F = − 1+ 1 2 · i . 3 Show that the triangles OAB and OCF are isometric. Part II Consider the transformation f of the plane, of complex writ-ing : z = − i · z + 2i 1 Determine the images of points O , A , B by f . 2 a Show that f is a similitude. Is it an isometry? b Determine the set of points invariant by f . c Is the f transformation an axial symmetry? 3 Let t be the vector translation −→ IJ . Give the complex writing of t and that of its reciprocal t − 1 . 4 We pose : s = f ◦ t − 1 a Show that the complex writing of s is : z = − i · z + 1 + i b Show that I and J are invariant by s . Deduce the nature of s . c Deduce that f is the compound of a translation and an axial symmetry to be specified. E.3938 The plane P is referred to a direct orthonormal reference frame O ; −→ u ; −→ v . We will make a figure that we will complete with the various elements involved in the exercise. 1 Consider the points A of affix 1 and B of affix 1 . We call S the reflection (axial symmetry) of axis ( AB ) . Show that the image M by S of a point M of affix z has affix : z = − i · z + 1 + i 2 Note H the homothety with center A and ratio − 2 . Give the complex writing of H . 3 Note f the compound H ◦ S . a Show that f is a similitude. b Determine the complex writing of f . 4 We call M the image of a point M by f . a Demonstrate that the set of points M of the plane such that −−→ AM = −−−→ AM is the straight line ( AB ) . b Demonstrate that the set of points M of the plane such that −−→ AM = −−→ AM is the perpendicular at A to the line ( AB ) . E.3942 The complex plane is referenced to the origin O ; −→ u ; −→ v orthonormal direct the unit of mea-surement is 2 cm . Consider the points A , B , C , D , and E with respective coordinates : a = 2 ; b = 2 + 3 · i ; c = 3 · i d = − 5 2 ; e = − 5 2 1 Place these five points on a graph that will be completed as the exercise progresses. 2 Two rectangles are considered similar if, and only if, the ratio of their length to their width is the same for both rectangles. Show that OABC and ABDE are two rectangles and that they are similar. 3 Study of a direct similarity transforming OABC into ABDE . a Determine the complex notation of the direct similar-ity s that transforms O into A and A into B . b Demonstrate that similarity s transforms OABC into ABDE . c What is the angle of similarity s ? d Let Ω be the center of this similarity. Using the com-pound s ◦ s , demonstrate that point Ω belongs to lines ( OB ) and ( AD ) . Deduce the position of point Ω . 4 Study of an indirect similarity transforming OABC into BAED . a Show that the complex notation of the indirect similar-ity which transforms O into B and leaves A invariant is : z = 3 2 · i · z + 2 + 3 · i where z denotes the conjugate of the complex number z . b Show that s transforms OABC into BAED . c In this question, any evidence of research, even if in-complete, or unsuccessful attempts, will be taken into account in the assessment. Demonstrate that s is the composite of the reflection of axis ( OA ) followed by a direct similarity, the char-acteristic elements of which will be specified. https://chingmath.fr sacados/3938 Amerique du Sud Novembre 2007 sacados/3942 Centres etrangers Juin 2008
E.3945 The plan is referenced to the orthonormal coordinate system O ; −→ u ; −→ v . Graphic unit : 4 cm . Part I 1 Place points I , J , H , A , B , C , D of respective affixes : z I = 1 ; z J = i ; z H = 1 + i z A = 2 ; z B = 3 2 + i ; z C = 2 · i z D = − 1 2 Let E be the image of point B under the symmetry with center H . The perpendicular to line ( AE ) pass-ing through C and the parallel to the line ( OC ) passing through D intersect at F . Place E and F and verify that the point F has affix : z F = − 1 + 1 2 · i . 3 Show that triangles OAB and OCF are isometric. Part II Consider the transformation f of the plane, with complex writing: z = − i · z + 2 · i 1 Determine the images of the points O , A , B par f . 2 a Monstrate that f is a similitude. Is it an isometry? b Determine the set of points invariant by f . c Is the f transformation an axial symmetry? 3 So t the vector translation −→ IJ . Give the complex writing of t and that of its reciprocal t − 1 . 4 On poses s = f ◦ t − 1 : a Montrer that the complex writing of s est : z = − i · z + 1 + i b Show that I and J are invariant by s . Deduce the nature of s . c Deduce that f is the compound of a translation and an axial symmetry to be specified. E.4032 The plane P is referred to a O ; −→ u ; −→ v orthonormal direct reference frame. A figure will be made and completed with the various ele-ments involved in the exercise. 1 Consider the points A of affix 1 and B of affix i . We call S the reflection (axial symmetry) of axis ( AB ) . Show that the image M by S of a point m of affix z has affix : z = − i · z + 1 + i 2 Note H the homothety with center A and ratio − 2 . Give the complex writing of H . 3 Note f the compound H ◦ S . a Show that f is a similitude. b Determine the complex writing of f . 4 The image of a point M by f is called M  . a Demonstrate that the set of points M of the plane such that : −−−→ AM  = − 2 · −−→ AM is the straight line ( AB ) . b Show that the set of points M of the plane such that : −−−→ AM  = 2 · −−→ AM is the perpendicular in A to the line ( AB ) . 6. Annales - Point sequences E.3176 The plane is provided with a direct orthonormal reference frame O ; −→ u ; −→ v (graphic unit 4 cm ) . Let Ω be the point of affix 2. We call r the rotation of center Ω and angle ı 4 and h the homothety of center Ω and ratio 2 2 . 1 We pose ff = h ◦ r . a What is the nature of the transformation ff ? Specify its characteristic elements. b Show that the complex writing of ff is : z ↦−→ 1 + i 2 z + 1 − i c Let M be any point in the plane of affix z . We denote by M its image by ff and note z the affix of M . Show that : z − z = i · 2 − z 2 a Course question Prerequisites : geometric definitions of the modulus of a complex number and an argument of a non-zero complex number. Algebraic properties of modules and arguments. Show that : if A is a given point of affix a , then the image of the point P of affix p by the rotation of cen-ter A and angle ı 2 is the point Q of affix q such that : q − a =i( p − a ) . b Deduce from the previous questions the nature of the triangle Ω MM , for M distinct from Ω . 3 Let A 0 be the point with affix 2 + i . Consider the se-quence A n of points in the plane defined by: for any natural number n : A n +1 = ff A n . a Show that, for any natural number n , the affix a n of A n is given by: a n = 2 2 e i ( n +2) π 4 + 2 b Determine the affix of A 5 . 4 Determine the smallest integer n 0 such that we have : for n n 0 , the point A n is in the disk of center Ω and radius 0.01 . https://chingmath.fr sacados/3945 Nouvelle-Caledonie Novembre 2005 chapExoCorrec/4032 sacados/4032 Amerique du Sud Novembre 2007 5 points sacados/3176
E.3946 The complex plane is equipped with a direct orthonormal reference frame O ; −→ u ; −→ v . We will take 5 cm as the graphic unit. Let f be the transformation which, for any point M with affix z , associates the point M with affix z defined by: z = 1 2 + 1 2 · i · z + 1 1 Justify that f is a direct similarity whose center Ω (with affix ! ) , the ratio k and the angle „ . 2 We denote A 0 as the point O and, for any natural num-ber n , we set : A n +1 = f A n a Determine the affixes of points A 1 , A 2 , A 3 then place the points A 0 , A 1 , A 2 and A 3 . b For any natural number n , we set u n =Ω A n . Justify that the sequence ( u n ) is a geometric sequence, then establish that, for any natural number n : u n = 2 · 1 2 n c From which rank n 0 , do all points A n belong to the disk with center Ω and radius 0.1 ? 3 a What is the nature of the triangle Ω A 0 A 1 ? Deduce, for any natural number n , the nature of the triangle Ω A n A n +1 . b For any natural number n , we denote ‘ n the length of the broken line A 0 A 1 A 2 : : : A n − 1 A n . We thus have : ‘ n = A 0 A 1 + A 1 A 2 + · · · + A n − 1 A n . Express ‘ n in terms of n . What is the limit of the sequence ( ‘ n ) ? E.3947 The plane is equipped with a di-rect orthonormal reference frame O ; −→ u ; −→ v (graphical unit : 4 cm ) . Let Ω be the affix point 2 . We call r the rotation with center Ω and angle ı 4 and h the homothety with center Ω and ratio 2 2 . 1 We set : ff = h ◦ r . a What is the nature of the transformation ff ? Specify its characteristic elements. b Show that the complex notation of 4 ff is : z ↦−→ 1 + i 2 · z + 1 − i . c Let M be any point on the affix plane z . We denote by M its image by ff and we note z the affix of M . Show that : z − z = i · (2 − z ) 2 a Course question: Prerequisites : geometric definitions of the modulus of a complex number and the argument of a nonzero com-plex number. Algebraic properties of moduli and ar-guments. Prove that : if A is a given point with affix a , then the image of the point P with affix p by the rotation with center A and angle ı 2 is the point Q with affix q such that : q − a = i · ( p − a ) . b Deduce from the previous questions the nature of the triangle Ω MM , for M distinct from Ω . 3 Let A 0 be the point with coordinates 2 + i . Consider the sequence ( A n ) of points in the plane defined by: for any natural number n : A n +1 = ff ( A n ) . a Show that, for any natural number n , the suffix a n of A n is given by: a n = 2 2 2 · e i · ( n +2) π 2 + 2 b Determine the affix of A 5 . 4 Determine the smallest integer n 0 such that : for n n 0 , the point A n is in the disk with center Ω and radius 0.01 . 7. Annales - Arithmetic E.3943 The plane is referred to a O ; −→ u ; −→ v orthonormé direct reference frame. Let A and B be the points with respective affixes : z A = 1 − i ; z B = 7 + 7 2 · i 1 We consider the straight line ( d ) d of equation : 4 · x + 3 · y =1 . Demonstrate that the set of points of ( d ) dwhose coordi-nates are integers is the set of points M k (3 · k +1 ; − 4 · k − 1) when k describes the set of relative integers. 2 Determine the angle and ratio of the direct similitude of center A which transforms B into M − 1 ( − 2 ; 3) . 3 So s the transformation of the plane which, to any point M of affix z , associates the point M d’affix : z = 2 3 · i · z + 1 3 − 5 3 · i Determine the image of A by s , then give the nature and characteristic elements of s . 4 We denote B 1 the image of B by s and for any natural number n not equal to zero, B n +1 the image of B n by s : a Determine the length AB n +1 based on AB n . b From which integer n does the point B n , belong to the disk with center A and radius 10 − 2 ? c Determine the set of integers n for which A , B 1 and B n are aligned. https://chingmath.fr sacados/3946 Pondichery Avril 2006 sacados/3947 Amerique du Nord Mai 2006 sacados/3943 Metropoles Juin 2008
E.3944 1 The complex plane is mapped to an orthonormal coordi-nate system O ; −→ u ; −→ v . Let A , B and C be the respec-tive affix points : z A = 2 + i ; z B = 5 + 2 · i , z C = i s 1 denotes the axis symmetry ( AB ) . a Prove that s 1 transforms any point M with affix z into a point M with affix z such that : z = 4 5 + 3 5 · i · z + − 1 5 + 3 5 · i b Deduce the affix of C , image of C by the axis symme-try ( AB ) . c Demonstrate that the set of points M such that z is purely imaginary is the line ( D ) d’équation : 4 · x + 3 · y = 1 d Verify that the point C belongs to D . 2 a Prove that the lines ( D ) and ( AB ) intersect at a point Ω whose affix ! will be specified. b We denote by s 2 the symmetry of axis ( D ) and by f the transformation defined by f = s 2 ◦ s 1 . Justify that f is a direct similarity and specify its ratio. c Determine the images of points C and Ω by the trans-formation f . d Justify that f is a rotation whose center will be given. 3 In this question, candidates are asked to write down the steps of their approach on their answer sheet, even if it does not lead to a solution. a Determine the pairs of relative integers ( x ; y ) solutions to the equation : 4 · x + 3 · y = 1 . b Determine the points of ( D ) with integer coordinates whose distance to the point O is less than 9 . E.4071 ABC is an equilateral triangle such that : −−→ AB ; −→ AC = ı 3 + 2 · k · ı where k ∈ Z Let t be a fixed real number and let the points M , N and P , distinct from each other, be defined by: −−→ AM = t · −−→ AB ; −−→ BN = t · −−→ BC ; −−→ CP = t · −→ CA The purpose of the exercise is to demonstrate the existence of a single direct similarity ff that transforms points A , B and C into M , N and P , respectively, and to specify its charac-teristic elements. The plane is equipped with an orthonormal reference frame O ; −→ u ; −→ v direct. We note a , b , c , m , n and p , the respective affixes of points A , B , C , M , N , and P : 1 It should be noted that any similarity retains the barycen-ter. a Express m , n and p in terms of a , b , c and t . b Deduce that the two triangles ABC and MNP have the same center of gravity. We will denote G this center of gravity. c We assume that ff exists. Determine the image of G by ff . 2 Consider the rotation r with center G and angle 2 · ı 3 . a Verify that M is the barycenter of the system of points : ( A ; 1 − t ) ; ( B ; t ) and, from this, deduce that : r ( M )= N . We also assume that : r ( N )= P ; r ( P )= M . b Let ff 1 , the direct similarity of center G with ratio GM GA and angle −→ GA ; −−→ GM . Show that it transforms the points A , B and C into M , N and P , respectively. c Conclude on the existence and uniqueness of ff . https://chingmath.fr sacados/3944 La Reunion Juin 2008 sacados/4071 Guyane-Antilles Septembre 2007 5 points
E.3215 For each question, only one of the four proposed answers is correct. The candidate will in-dicate on the copy the number of the question and the letter corresponding to the chosen answer. Each correct answer earns 1 point. Each wrong answer deducts 0.5 point. An absence of response is counted as 0 points. If the total is negative, the mark is reduced to zero. No justification is required. 1 In the set of integers, consider the equation : x 2 − x + 4 ≡ 0 ( mod. 6) a All solutions are even integers. b There are no solutions. c Solutions verify x ≡ 2 ( mod. 6) d Solutions verify x ≡ 2 ( mod. 6) or x ≡ 5 ( mod. 6) . 2 We propose to solve the equation ( E ) : 24 x + 34 y = 2 , où x and y are relative integers. a The solutions of ( E ) are all of the form : ( x ; y ) =(34 k − 7 ; 5 − 24 k ) where k ∈ Z b The equation ( E ) has no solution. c The solutions of ( E ) are all of the form : ( x ; y ) =(17 k − 7 ; 5 − 12 k ) where k ∈ Z d The solutions of ( E ) are all of the form : ( x ; y ) =( − 7 k ; 5 k ) where k ∈ Z 3 Consider the two integers n =1789 and p =1789 2005 . Then we have : a n ≡ 4 ( mod. 17) and p ≡ 0 ( mod. 17) . b p is a prime number. c p ≡ 4 ( mod. 17) . d p ≡ 1 ( mod. 17) . 4 Consider the complex plane with respect to an orthonor-mal coordinate system, the points A and B with respec-tive affixes a and b . The triangle MAB is a right isosceles triangle with hypotenuse [ AB ] if, and only if, the point M with affix z is such that : a z = b − i a 1 − i b a − z = i( b − z ) c z − a = e i π 4 ( b − a ) d b − z = ı 2 ( a − z ) 5 Consider two distinct points A and B in the oriented plane ; let I be the midpoint of the segment [ AB ] . Let f be the direct similarity with center A , with ratio 2 and angle 2 ı 3 ; or g the direct similarity of center A , ratio 1 2 and angle ı 3 ; or h central symmetry with center I a h ◦ g ◦ f transforms A into B and is a rotation. b h ◦ g ◦ f is the reflection with the axis being the per-pendicular bisector of segment [ AB ] . c h ◦ g ◦ f is not a similarity. d h ◦ g ◦ f is the translation of vector −−→ AB . E.3218 For each of the six statements, say whether it is true or false, justifying the choice made. 1 The PGCD of 2004 and 4002 is 6 . 2 If p and q are two non-zero natural numbers, 2 pq − 1 is divisible by 2 p − 1 and 2 q − 1 . 3 For any n ∈ N ∗ , 2 n − 1 is never divisible by 9 . 4 The set of integer pairs solving the equation : 24 x + 35 y = 9 is the set of pairs : ( − 144+70 k ; 99 − 24 k ) where k ∈ Z 5 Let A and B be two distinct points of the plane ; if we denote f the homothety of center A and ratio 3 and g the homothety of center B and ratio 1 3 then g ◦ f is the translation of vector −−→ AB . 6 Let s be the similitude of complex writing z =i z +(1 − i) , the set of invariant points of s is a straight line. 8. Similarities E.3937 We will demonstrate without proof the following two properties : Property 1: Any indirect similarity that transforms a point M with affix z into a point M affix point z admits a complex expression of the form : z = a · z + b where a ∈ C ∗ and b ∈ C . Property 2: Let C be an affix point c . For any point D , distinct from C , with affix d and for any point E , distinct from C , with affix e , we have : −−→ CD ; −−→ CE = arg e − c d − c [2 · ı ] Question: Show that an indirect similarity transforms an oriented angle into its opposite. E.3939 The following result is assumed to be known : An application f of the plane provided with a direct orthonor- mal reference frame in itself is a direct similitude if, and only if, f admits a complex writing of the form : z = a · z + b where a ∈ C ∗ and b ∈ C . Course demonstration : we place ourselves in the complex plane. Show that if A , B , A and B are four points such that A is distinct from B and A is distinct from B , then there is a single direct similitude transforming A into A and B into B . E.3948 Course question: Prerequisites: Geometric definitions of the modulus of a complex number and the argument of a nonzero complex number. Algebraic properties of moduli and arguments. Prove that : if A is a given point with affix a , then the image of the point P with affix p by the rotation with center A and angle ı 2 is the point Q with affix q such that : q − a =i · ( p − a ) . https://chingmath.fr chapExoCorrec/3215 sacados/3215 France Septembre 2005 5 points chapExoCorrec/3218 sacados/3218 chapExoCorrec/3937 sacados/3937 Extrait d'Antilles-Guyane Juin 2010 chapExoCorrec/3939 sacados/3939 Extrait Pondichery Avril 2008 chapExoCorrec/3948 sacados/3948 Extrait d'Amerique du Nord Mai 2006
9. Unclassified financial years E.4315 Consider the equation : ( E ) : 3 · x − 2 · y = 1 1 a Show that the pair ( − 1 ; − 2) is a solution to ( E ) . b Determine all pairs ( x ; y ) of relative integers satisfying the equation ( E ) . 2 Let d and d be the lines with respective equations : y = 2 · x + 4 ; 3 · x − 2 · y = 1 a Verify that for any relative integer k , the point A k with coordinates ( k − 3 ; 2 · k − 2) belongs to the line d . We will assume that these are the only points of d with integer coordinates. b Show that the only points of with integer coordinates are the points B k with coordinates : ( 2 · k − 1 ; 3 · k − 2 ) where k ∈ Z . 3 a Are there two relative integers k and k such that : A k = B k ? b Determine the relative integers k and k such that the segment A k B k is parallel to the x-axis. c Find the integer q such that : −−−−→ A 3 q B 2 q =4 · −→ u 4 Let Ω be any point on the plane whose affix is denoted by ! . Let H be the midpoint of the segment A 6 B 4 . We denote by f the direct similarity with center Ω , ratio 1 2 and angle − ı 2 . a Give the complex notation for the similarity f . b Determine the point’s affix Ω so that the image of the point H is the origin O of the coordinate system. E.4653 Let be a direct equilateral triangle ABC and let be D a point on segment [ BC ] . The parallel to the line ( AC ) led by D intersects the line ( AB ) at E and the parallel to the line ( AB ) led by D intersects the line ( AC ) at F . Let the point G be the center of gravity of the triangle ABC and the points H and A , images of G and A by symmetry of axis ( BC ) . We define the points I and J respective centers of gravity of the triangles BDE and CDF . We define the direct similitudes S 1 , of center C , of ratio 3 , of angle ı 6 and S 2 , center B , ratio 1 3 , angle ı 6 and their composite : f = S 2 ◦ S 1 1 Determine the images of J and H by f . 2 Determine the nature and characteristic elements of f . 3 Deduce the nature of the triangle HIJ . https://chingmath.fr sacados/4315 sacados/4653