{"id":2840,"date":"2023-12-06T15:54:32","date_gmt":"2023-12-06T14:54:32","guid":{"rendered":"https:\/\/math.univ-lyon1.fr\/dream\/?p=2840"},"modified":"2025-12-05T11:58:50","modified_gmt":"2025-12-05T10:58:50","slug":"le-champ-de-colonnes","status":"publish","type":"post","link":"https:\/\/math.univ-lyon1.fr\/dream\/?p=2840","title":{"rendered":"Le champ de colonnes"},"content":{"rendered":"\r\n<h2>\u00c9nonc\u00e9 du probl\u00e8me<\/h2>\r\n<p>On se place aux bords d&rsquo;un champ constitu\u00e9 d&rsquo;une infinit\u00e9 de colonnes verticales, identiques, qui sont rep\u00e9r\u00e9es par des points sur la grille ci-contre<\/p>\r\n<p>Les colonnes ont toutes des coordonn\u00e9es enti\u00e8res strictement positives et on se place sur les bords correspondant aux demi-droites des axes des abscisses et des ordonn\u00e9es. Depuis ce bord on pointe un laser \u00e0 travers ce champ de colonnes.<\/p>\r\n<p><em>Trouver toutes les trajectoires o\u00f9 le laser ne rencontre aucune colonne. Comment les caract\u00e9riser ? Comment le prouver ?<\/em><\/p>\r\n<p><img loading=\"lazy\" decoding=\"async\" title=\"CHAMPCOLONNES\" src=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-300x282.png\" alt=\"CHAMPCOLONNES\" width=\"300\" height=\"282\" \/><\/p>\r\n<p><iframe loading=\"lazy\" src=\"https:\/\/drive.google.com\/file\/d\/1g7jpcgM7ftsV0azWdniU1oJ-eXMCpN3s\/preview\" width=\"640\" height=\"480\"><\/iframe><\/p>\r\n<p><a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=79\" target=\"_self\"> Retour Panier \u00e0 probl\u00e8mes <\/a><\/p>\r\n<h3>Connaissances math\u00e9matiques<\/h3>\r\n<p>\u2666\u00a0Caract\u00e9risation d&rsquo;une droite (point de vue g\u00e9om\u00e9trique, vectoriel ou affine)<br \/>\u2666\u00a0Equations de droite (g\u00e9n\u00e9rale ou r\u00e9duite)<br \/>\u2666\u00a0Vecteur directeur, vecteurs colin\u00e9aires<br \/>\u2666 Appartenance d&rsquo;un point \u00e0 une droite (par \u00e9quation, par colin\u00e9arit\u00e9. . .)<br \/>\u2666\u00a0Ensembles de nombres<br \/>\u2666\u00a0Propri\u00e9t\u00e9 des rationnels<br \/>\u2666\u00a0PGCD, PPCM de deux entiers<br \/>\u2666\u00a0Equations diophantiennes<\/p>\r\n<h3>Les raisonnements utilis\u00e9s<\/h3>\r\n<p>\u2666 Disjonction de cas<br \/>\u2666 Raisonnement par l&rsquo;absurde<br \/>\u2666 Raisonnement par condition n\u00e9cessaire et suffisante (analyse-synth\u00e8se)Caract\u00e9risation d&rsquo;une droite (point de vue g\u00e9om\u00e9trique, vectoriel ou affine) <a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/Le-champ-de-colonnes.pdf\" target=\"_blank\" rel=\"noopener\"> T\u00e9l\u00e9charcher le document complet <\/a><\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>Dans un champ infini de colonnes, quelles trajectoires d\u2019un laser depuis les axes \u00e9vitent toutes les colonnes\u202f? Comment les d\u00e9crire et le d\u00e9montrer\u202f?<\/p>\n","protected":false},"author":3,"featured_media":2841,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[107],"tags":[],"niveau":[114],"type_situation":[116],"theme_maths":[120,118],"probleme_associe":[],"type-expe":[],"class_list":{"0":"post-2840","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-situation-probleme","8":"niveau-lycee","9":"type_situation-situation-probleme","10":"theme_maths-espace-et-geometrie","11":"theme_maths-nombres-et-calculs","12":"czr-hentry"},"rttpg_featured_image_url":{"full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"landscape":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"portraits":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"thumbnail":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-150x150.png",150,150,true],"medium":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-300x282.png",300,282,true],"large":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"1536x1536":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"2048x2048":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"tc-grid-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"tc-grid":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-376x350.png",376,350,true],"tc-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-270x250.png",270,250,true],"slider-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-376x300.png",376,300,true],"slider":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-376x300.png",376,300,true],"tc-sq-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"tc-ws-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES.png",376,354,false],"tc-ws-small-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-376x297.png",376,297,true],"tc-slider-small":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/CHAMPCOLONNES-376x141.png",376,141,true]},"rttpg_author":{"display_name":"St\u00e9phanie CROQUELOIS","author_link":"https:\/\/math.univ-lyon1.fr\/dream\/?author=3"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?cat=107\" rel=\"category\">Situation Probl\u00e8me<\/a>","rttpg_excerpt":"Dans un champ infini de colonnes, quelles trajectoires d\u2019un laser depuis les axes \u00e9vitent toutes les colonnes\u202f? Comment les d\u00e9crire et le d\u00e9montrer\u202f?","_links":{"self":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/2840","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2840"}],"version-history":[{"count":8,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/2840\/revisions"}],"predecessor-version":[{"id":3791,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/2840\/revisions\/3791"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/media\/2841"}],"wp:attachment":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2840"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2840"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2840"},{"taxonomy":"niveau","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fniveau&post=2840"},{"taxonomy":"type_situation","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype_situation&post=2840"},{"taxonomy":"theme_maths","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftheme_maths&post=2840"},{"taxonomy":"probleme_associe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fprobleme_associe&post=2840"},{"taxonomy":"type-expe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype-expe&post=2840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}