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Номер 5, страница 240 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Задачи с практическим содержанием - номер 5, страница 240.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745543 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи с практическим содержанием" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "240" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Book {#1048 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:2 [ 0 => App\Models\Term {#1052 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:5 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1066 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Term {#1068 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1072 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #original: array:6 [ "id" => 41 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "базовый и углублённый" "field_cases" => array:6 [ …6] "field_translit" => "bazovyy i uglublennyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1073 #items: array:1 [ 0 => App\Models\Term {#1074 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #original: array:10 [ "id" => 6706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "ФГОС (старый)" "field_cases" => array:6 [ …6] "field_color" => null "field_end_year" => "2022" "field_start_year" => "2019" "field_translit" => "fgos-old" "field_type" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\Term {#1076 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Term {#1082 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #original: array:6 [ "id" => 51 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "коричневый" "field_cases" => array:6 [ …6] "field_translit" => "korichnevyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1062} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_country" => Illuminate\Database\Eloquent\Collection {#1065} "field_city" => Illuminate\Database\Eloquent\Collection {#1067} "field_series" => Illuminate\Database\Eloquent\Collection {#1069} "field_umk" => Illuminate\Database\Eloquent\Collection {#1070} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1071} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1073} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1075} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1077} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1078} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1079} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1081} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи с практическим содержанием" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "240" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1085} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1083} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:4 [ 0 => App\Models\Element {#1091 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 941632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1092 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745543" "type" => "task" ] "text" => "<p> <b>5.</b> По четырём дорогам, никакие две из которых не параллельны и никакие три не проходят через одну точку, с постоянными скоростями идут 4 пешехода. Известно, что первый пешеход встретился со вторым, третьим и четвёртым, а второй — с третьим и четвёртым. Докажите, что третий пешеход встретился с четвёртым. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "979" "height" => 401 "path" => "/media/geometrija_10/atanasjan-u23/0-zad/5-1.webp?ts=1739437115" ] ] ] #original: array:7 [ "id" => 941632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "745543" "type" => "task" ] "text" => "<p> <b>5.</b> По четырём дорогам, никакие две из которых не параллельны и никакие три не проходят через одну точку, с постоянными скоростями идут 4 пешехода. Известно, что первый пешеход встретился со вторым, третьим и четвёртым, а второй — с третьим и четвёртым. Докажите, что третий пешеход встретился с четвёртым. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "979" "height" => 401 "path" => "/media/geometrija_10/atanasjan-u23/0-zad/5-1.webp?ts=1739437115" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1106 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 946854 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Edition {#1116 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2316 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Пелеванюк" "field_moderator" => "filin" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2316 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Пелеванюк" "field_moderator" => "filin" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745543" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "1955" "height" => 2848 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-1.webp?ts=1739522765" ] 1 => array:5 [ "name" => "5-2.jpg" "alt" => null "width" => "2117" "height" => 3240 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-2.webp?ts=1739522765" ] 2 => array:5 [ "name" => "5-3.jpg" "alt" => null "width" => "1950" "height" => 3111 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-3.webp?ts=1739522765" ] 3 => array:5 [ "name" => "5-4.jpg" "alt" => null "width" => "1980" "height" => 1202 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-4.webp?ts=1739522765" ] ] ] #original: array:6 [ "id" => 946854 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117} "task" => array:2 [ "refs" => "745543" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "1955" "height" => 2848 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-1.webp?ts=1739522765" ] 1 => array:5 [ "name" => "5-2.jpg" "alt" => null "width" => "2117" "height" => 3240 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-2.webp?ts=1739522765" ] 2 => array:5 [ "name" => "5-3.jpg" "alt" => null "width" => "1950" "height" => 3111 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-3.webp?ts=1739522765" ] 3 => array:5 [ "name" => "5-4.jpg" "alt" => null "width" => "1980" "height" => 1202 "path" => "/media/geometrija_10/atanasjan-u23/1-zad/5-4.webp?ts=1739522765" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1098 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 943081 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1099 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745543" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "5-1.png" "alt" => null "width" => "694" "height" => 2613 "path" => "/media/geometrija_10/atanasjan-u23/2-zad/5-1.webp?ts=1739441947" ] ] ] #original: array:6 [ "id" => 943081 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "745543" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "5-1.png" "alt" => null "width" => "694" "height" => 2613 "path" => "/media/geometrija_10/atanasjan-u23/2-zad/5-1.webp?ts=1739441947" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1109 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1352856 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Edition {#1110 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "745543" "type" => "task" ] "text" => "<p>Обозначим положение $i$-го пешехода в момент времени $t$ как радиус-вектор $\vec{r_i}(t)$. Поскольку пешеходы движутся с постоянными скоростями, закон их движения можно записать в виде: $$ \vec{r_i}(t) = \vec{p_i} + \vec{v_i}t $$ где $\vec{p_i}$ — это начальное положение пешехода (в $t=0$), а $\vec{v_i}$ — его постоянный вектор скорости. Дорога, по которой движется $i$-й пешеход, представляет собой прямую линию, заданную этим уравнением.</p><p>Условие, что два пешехода, $i$-й и $j$-й, встретились, означает, что в некоторый момент времени $t_{ij}$ их положения совпали: $$ \vec{r_i}(t_{ij}) = \vec{r_j}(t_{ij}) $$ Из этого следует: $$ \vec{p_i} + \vec{v_i}t_{ij} = \vec{p_j} + \vec{v_j}t_{ij} $$ $$ \vec{p_i} - \vec{p_j} = (\vec{v_j} - \vec{v_i})t_{ij} $$ Это уравнение имеет решение для $t_{ij}$ тогда и только тогда, когда вектор начального относительного положения $(\vec{p_i} - \vec{p_j})$ коллинеарен (параллелен) вектору относительной скорости $(\vec{v_i} - \vec{v_j})$. В двумерном пространстве условие коллинеарности двух векторов $\vec{a}$ и $\vec{b}$ можно записать с помощью векторного (псевдоскалярного) произведения как $[\vec{a}, \vec{b}] = 0$. Таким образом, условие встречи $i$-го и $j$-го пешеходов имеет вид: $$ [(\vec{p_i} - \vec{p_j}), (\vec{v_i} - \vec{v_j})] = 0 $$</p><p>Для решения задачи удобно перейти в систему отсчёта, связанную с первым пешеходом. В этой системе отсчёта положение и скорость $j$-го пешехода будут: $$ \vec{r'_j}(t) = \vec{r_j}(t) - \vec{r_1}(t) = (\vec{p_j} - \vec{p_1}) + (\vec{v_j} - \vec{v_1})t $$ Обозначим новые начальные положения и скорости как $\vec{p'_j} = \vec{p_j} - \vec{p_1}$ и $\vec{v'_j} = \vec{v_j} - \vec{v_1}$. Первый пешеход в этой системе покоится в начале координат ($\vec{p'_1} = \vec{0}$, $\vec{v'_1} = \vec{0}$).</p><p>Рассмотрим условия задачи в новой системе отсчёта.</p><p>По условию, первый пешеход встретился со вторым, третьим и четвёртым. Встреча 1-го и $j$-го пешехода ($j=2,3,4$) означает, что в некоторый момент времени $t_{1j}$ их относительное положение равно нулю: $\vec{r'_j}(t_{1j}) = \vec{0}$. $$ \vec{p'_j} + \vec{v'_j}t_{1j} = \vec{0} \implies \vec{p'_j} = -t_{1j}\vec{v'_j} $$ Это означает, что для $j=2, 3, 4$ вектор начального положения $\vec{p'_j}$ коллинеарен вектору скорости $\vec{v'_j}$. Таким образом, в новой системе отсчёта второй, третий и четвёртый пешеходы движутся по прямым, проходящим через начало координат (положение первого пешехода).</p><p>Далее, по условию, второй пешеход встретился с третьим. Условие их встречи инвариантно относительно перехода в другую инерциальную систему отсчёта: $$ [(\vec{p_2} - \vec{p_3}), (\vec{v_2} - \vec{v_3})] = [(\vec{p'_2} - \vec{p'_3}), (\vec{v'_2} - \vec{v'_3})] = 0 $$ Мы знаем, что существуют скаляры $k_j = -t_{1j}$ такие, что $\vec{p'_j} = k_j \vec{v'_j}$. Подставим это в уравнение: $$ [k_2 \vec{v'_2} - k_3 \vec{v'_3}, \vec{v'_2} - \vec{v'_3}] = 0 $$ Раскроем скобки, используя свойства векторного произведения ($[\vec{a}, \vec{a}]=0$ и $[\vec{a}, \vec{b}] = -[\vec{b}, \vec{a}]$): $$ k_2 [\vec{v'_2}, \vec{v'_2}] - k_2 [\vec{v'_2}, \vec{v'_3}] - k_3 [\vec{v'_3}, \vec{v'_2}] + k_3 [\vec{v'_3}, \vec{v'_3}] = 0 $$ $$ -k_2 [\vec{v'_2}, \vec{v'_3}] + k_3 [\vec{v'_2}, \vec{v'_3}] = 0 $$ $$ (k_3 - k_2)[\vec{v'_2}, \vec{v'_3}] = 0 $$ Это уравнение означает, что либо $k_2 = k_3$, либо $[\vec{v'_2}, \vec{v'_3}] = 0$.</p><p>Рассмотрим случай $k_2 = k_3$. Это значит, что $t_{12} = t_{13}$. То есть, первый пешеход встретился со вторым и третьим в один и тот же момент времени $t^*=t_{12}=t_{13}$. В этот момент времени $\vec{r_1}(t^*) = \vec{r_2}(t^*)$ и $\vec{r_1}(t^*) = \vec{r_3}(t^*)$. Следовательно, $\vec{r_1}(t^*)=\vec{r_2}(t^*)=\vec{r_3}(t^*)$, и все три пешехода оказались в одной точке пространства. Это означает, что их пути (дороги) пересекаются в одной точке. Но по условию задачи "никакие три дороги не проходят через одну точку". Следовательно, этот случай невозможен, и $k_2 \neq k_3$.</p><p>Таким образом, из равенства $(k_3 - k_2)[\vec{v'_2}, \vec{v'_3}] = 0$ следует, что $[\vec{v'_2}, \vec{v'_3}] = 0$. Это означает, что векторы относительных скоростей $\vec{v'_2}$ и $\vec{v'_3}$ коллинеарны: $\vec{v'_2} \parallel \vec{v'_3}$.</p><p>Аналогично, из условия встречи 2-го и 4-го пешеходов получаем $[(\vec{p'_2} - \vec{p'_4}), (\vec{v'_2} - \vec{v'_4})] = 0$, что приводит к уравнению: $$ (k_4 - k_2)[\vec{v'_2}, \vec{v'_4}] = 0 $$ По тем же причинам, что и выше, случай $k_2=k_4$ невозможен, так как это означало бы, что дороги 1, 2 и 4 пересекаются в одной точке. Следовательно, $[\vec{v'_2}, \vec{v'_4}] = 0$, то есть $\vec{v'_2} \parallel \vec{v'_4}$.</p><p>Из того, что $\vec{v'_2} \parallel \vec{v'_3}$ и $\vec{v'_2} \parallel \vec{v'_4}$, следует, что все три вектора относительных скоростей коллинеарны друг другу: $\vec{v'_2} \parallel \vec{v'_3} \parallel \vec{v'_4}$. В частности, $\vec{v'_3} \parallel \vec{v'_4}$, а значит $[\vec{v'_3}, \vec{v'_4}] = 0$.</p><p>Теперь докажем, что третий пешеход встретился с четвёртым. Для этого нужно показать, что выполняется условие их встречи, которое в системе отсчёта первого пешехода выглядит так: $$ [(\vec{p'_3} - \vec{p'_4}), (\vec{v'_3} - \vec{v'_4})] = 0 $$ Подставим выражения $\vec{p'_3} = k_3 \vec{v'_3}$ и $\vec{p'_4} = k_4 \vec{v'_4}$: $$ [k_3 \vec{v'_3} - k_4 \vec{v'_4}, \vec{v'_3} - \vec{v'_4}] $$ Раскрывая скобки, получаем: $$ k_3 [\vec{v'_3}, \vec{v'_3}] - k_3 [\vec{v'_3}, \vec{v'_4}] - k_4 [\vec{v'_4}, \vec{v'_3}] + k_4 [\vec{v'_4}, \vec{v'_4}] = (k_4 - k_3)[\vec{v'_3}, \vec{v'_4}] $$ Поскольку мы уже установили, что $[\vec{v'_3}, \vec{v'_4}] = 0$, то всё выражение равно нулю: $$ (k_4 - k_3) \cdot 0 = 0 $$ Условие встречи третьего и четвёртого пешеходов выполняется, что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение доказано. Третий пешеход встретился с четвёртым.</p>" ] #original: array:6 [ "id" => 1352856 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1126} "task" => array:2 [ "refs" => "745543" "type" => "task" ] "text" => "<p>Обозначим положение $i$-го пешехода в момент времени $t$ как радиус-вектор $\vec{r_i}(t)$. Поскольку пешеходы движутся с постоянными скоростями, закон их движения можно записать в виде: $$ \vec{r_i}(t) = \vec{p_i} + \vec{v_i}t $$ где $\vec{p_i}$ — это начальное положение пешехода (в $t=0$), а $\vec{v_i}$ — его постоянный вектор скорости. Дорога, по которой движется $i$-й пешеход, представляет собой прямую линию, заданную этим уравнением.</p><p>Условие, что два пешехода, $i$-й и $j$-й, встретились, означает, что в некоторый момент времени $t_{ij}$ их положения совпали: $$ \vec{r_i}(t_{ij}) = \vec{r_j}(t_{ij}) $$ Из этого следует: $$ \vec{p_i} + \vec{v_i}t_{ij} = \vec{p_j} + \vec{v_j}t_{ij} $$ $$ \vec{p_i} - \vec{p_j} = (\vec{v_j} - \vec{v_i})t_{ij} $$ Это уравнение имеет решение для $t_{ij}$ тогда и только тогда, когда вектор начального относительного положения $(\vec{p_i} - \vec{p_j})$ коллинеарен (параллелен) вектору относительной скорости $(\vec{v_i} - \vec{v_j})$. В двумерном пространстве условие коллинеарности двух векторов $\vec{a}$ и $\vec{b}$ можно записать с помощью векторного (псевдоскалярного) произведения как $[\vec{a}, \vec{b}] = 0$. Таким образом, условие встречи $i$-го и $j$-го пешеходов имеет вид: $$ [(\vec{p_i} - \vec{p_j}), (\vec{v_i} - \vec{v_j})] = 0 $$</p><p>Для решения задачи удобно перейти в систему отсчёта, связанную с первым пешеходом. В этой системе отсчёта положение и скорость $j$-го пешехода будут: $$ \vec{r'_j}(t) = \vec{r_j}(t) - \vec{r_1}(t) = (\vec{p_j} - \vec{p_1}) + (\vec{v_j} - \vec{v_1})t $$ Обозначим новые начальные положения и скорости как $\vec{p'_j} = \vec{p_j} - \vec{p_1}$ и $\vec{v'_j} = \vec{v_j} - \vec{v_1}$. Первый пешеход в этой системе покоится в начале координат ($\vec{p'_1} = \vec{0}$, $\vec{v'_1} = \vec{0}$).</p><p>Рассмотрим условия задачи в новой системе отсчёта.</p><p>По условию, первый пешеход встретился со вторым, третьим и четвёртым. Встреча 1-го и $j$-го пешехода ($j=2,3,4$) означает, что в некоторый момент времени $t_{1j}$ их относительное положение равно нулю: $\vec{r'_j}(t_{1j}) = \vec{0}$. $$ \vec{p'_j} + \vec{v'_j}t_{1j} = \vec{0} \implies \vec{p'_j} = -t_{1j}\vec{v'_j} $$ Это означает, что для $j=2, 3, 4$ вектор начального положения $\vec{p'_j}$ коллинеарен вектору скорости $\vec{v'_j}$. Таким образом, в новой системе отсчёта второй, третий и четвёртый пешеходы движутся по прямым, проходящим через начало координат (положение первого пешехода).</p><p>Далее, по условию, второй пешеход встретился с третьим. Условие их встречи инвариантно относительно перехода в другую инерциальную систему отсчёта: $$ [(\vec{p_2} - \vec{p_3}), (\vec{v_2} - \vec{v_3})] = [(\vec{p'_2} - \vec{p'_3}), (\vec{v'_2} - \vec{v'_3})] = 0 $$ Мы знаем, что существуют скаляры $k_j = -t_{1j}$ такие, что $\vec{p'_j} = k_j \vec{v'_j}$. Подставим это в уравнение: $$ [k_2 \vec{v'_2} - k_3 \vec{v'_3}, \vec{v'_2} - \vec{v'_3}] = 0 $$ Раскроем скобки, используя свойства векторного произведения ($[\vec{a}, \vec{a}]=0$ и $[\vec{a}, \vec{b}] = -[\vec{b}, \vec{a}]$): $$ k_2 [\vec{v'_2}, \vec{v'_2}] - k_2 [\vec{v'_2}, \vec{v'_3}] - k_3 [\vec{v'_3}, \vec{v'_2}] + k_3 [\vec{v'_3}, \vec{v'_3}] = 0 $$ $$ -k_2 [\vec{v'_2}, \vec{v'_3}] + k_3 [\vec{v'_2}, \vec{v'_3}] = 0 $$ $$ (k_3 - k_2)[\vec{v'_2}, \vec{v'_3}] = 0 $$ Это уравнение означает, что либо $k_2 = k_3$, либо $[\vec{v'_2}, \vec{v'_3}] = 0$.</p><p>Рассмотрим случай $k_2 = k_3$. Это значит, что $t_{12} = t_{13}$. То есть, первый пешеход встретился со вторым и третьим в один и тот же момент времени $t^*=t_{12}=t_{13}$. В этот момент времени $\vec{r_1}(t^*) = \vec{r_2}(t^*)$ и $\vec{r_1}(t^*) = \vec{r_3}(t^*)$. Следовательно, $\vec{r_1}(t^*)=\vec{r_2}(t^*)=\vec{r_3}(t^*)$, и все три пешехода оказались в одной точке пространства. Это означает, что их пути (дороги) пересекаются в одной точке. Но по условию задачи "никакие три дороги не проходят через одну точку". Следовательно, этот случай невозможен, и $k_2 \neq k_3$.</p><p>Таким образом, из равенства $(k_3 - k_2)[\vec{v'_2}, \vec{v'_3}] = 0$ следует, что $[\vec{v'_2}, \vec{v'_3}] = 0$. Это означает, что векторы относительных скоростей $\vec{v'_2}$ и $\vec{v'_3}$ коллинеарны: $\vec{v'_2} \parallel \vec{v'_3}$.</p><p>Аналогично, из условия встречи 2-го и 4-го пешеходов получаем $[(\vec{p'_2} - \vec{p'_4}), (\vec{v'_2} - \vec{v'_4})] = 0$, что приводит к уравнению: $$ (k_4 - k_2)[\vec{v'_2}, \vec{v'_4}] = 0 $$ По тем же причинам, что и выше, случай $k_2=k_4$ невозможен, так как это означало бы, что дороги 1, 2 и 4 пересекаются в одной точке. Следовательно, $[\vec{v'_2}, \vec{v'_4}] = 0$, то есть $\vec{v'_2} \parallel \vec{v'_4}$.</p><p>Из того, что $\vec{v'_2} \parallel \vec{v'_3}$ и $\vec{v'_2} \parallel \vec{v'_4}$, следует, что все три вектора относительных скоростей коллинеарны друг другу: $\vec{v'_2} \parallel \vec{v'_3} \parallel \vec{v'_4}$. В частности, $\vec{v'_3} \parallel \vec{v'_4}$, а значит $[\vec{v'_3}, \vec{v'_4}] = 0$.</p><p>Теперь докажем, что третий пешеход встретился с четвёртым. Для этого нужно показать, что выполняется условие их встречи, которое в системе отсчёта первого пешехода выглядит так: $$ [(\vec{p'_3} - \vec{p'_4}), (\vec{v'_3} - \vec{v'_4})] = 0 $$ Подставим выражения $\vec{p'_3} = k_3 \vec{v'_3}$ и $\vec{p'_4} = k_4 \vec{v'_4}$: $$ [k_3 \vec{v'_3} - k_4 \vec{v'_4}, \vec{v'_3} - \vec{v'_4}] $$ Раскрывая скобки, получаем: $$ k_3 [\vec{v'_3}, \vec{v'_3}] - k_3 [\vec{v'_3}, \vec{v'_4}] - k_4 [\vec{v'_4}, \vec{v'_3}] + k_4 [\vec{v'_4}, \vec{v'_4}] = (k_4 - k_3)[\vec{v'_3}, \vec{v'_4}] $$ Поскольку мы уже установили, что $[\vec{v'_3}, \vec{v'_4}] = 0$, то всё выражение равно нулю: $$ (k_4 - k_3) \cdot 0 = 0 $$ Условие встречи третьего и четвёртого пешеходов выполняется, что и требовалось доказать.</p><p><strong>Ответ:</strong> Утверждение доказано. Третий пешеход встретился с четвёртым.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "745544" "type" => "task" ] "previous" => array:2 [ "refs" => "745542" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\BookPage {#1041 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 934022 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-240" "field_display_title" => "240" "field_folder" => "1" "field_image_name" => "240" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1120 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Element {#1134 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1108286 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1108286 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "934023" "type" => "book_page" ] "previous" => array:2 [ "refs" => "934021" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:10 [ 0 => App\Models\Task {#1227 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745539 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1229 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1230 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1231 …2} "content" => Illuminate\Database\Eloquent\Collection {#1233 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1242 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745539 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-1" "field_display_title" => "1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1229 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1230 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1231 …2} "content" => Illuminate\Database\Eloquent\Collection {#1233 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1242 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1234 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745540 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1232 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1240 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1239 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1241 …2} "content" => Illuminate\Database\Eloquent\Collection {#1235 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1258 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745540 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1232 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1240 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1239 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1241 …2} "content" => Illuminate\Database\Eloquent\Collection {#1235 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1258 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1238 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745541 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1237 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1256 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1255 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1257 …2} "content" => Illuminate\Database\Eloquent\Collection {#1236 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1274 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745541 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-3" "field_display_title" => "3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1237 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1256 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1255 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1257 …2} "content" => Illuminate\Database\Eloquent\Collection {#1236 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1274 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1254 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745542 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1253 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1272 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1271 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1252 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1290 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745542 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-4" "field_display_title" => "4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1253 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1272 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1271 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1252 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1290 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1270 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745543 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1288 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1287 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1289 …2} "content" => Illuminate\Database\Eloquent\Collection {#1285 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1268 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745543 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1288 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1287 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1289 …2} "content" => Illuminate\Database\Eloquent\Collection {#1285 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1268 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1286 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 745544 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1284 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1300 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1301 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1302 …2} "content" => Illuminate\Database\Eloquent\Collection {#1304 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1313 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 745544 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1284 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1300 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1301 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1302 …2} "content" => Illuminate\Database\Eloquent\Collection {#1304 …2} "next" => array:2 [ …2] …3 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1305 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1309 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 8 => App\Models\Task {#1325 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 9 => App\Models\Task {#1341 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 934022 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/page-240" "field_display_title" => "240" "field_folder" => "1" "field_image_name" => "240" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1084} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1043} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1120} "content" => Illuminate\Database\Eloquent\Collection {#1135} "next" => array:2 [ "refs" => "934023" "type" => "book_page" ] "previous" => array:2 [ "refs" => "934021" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1211} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 745543 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "240" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/zad-5" "field_display_title" => "5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1038} "content" => Illuminate\Database\Eloquent\Collection {#1034} "next" => array:2 [ "refs" => "745544" "type" => "task" ] "previous" => array:2 [ "refs" => "745542" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1089} "page" => array:2 [ "refs" => "934022" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№5 (с. 240)
Условие. №5 (с. 240)
скриншот условия
5. По четырём дорогам, никакие две из которых не параллельны и никакие три не проходят через одну точку, с постоянными скоростями идут 4 пешехода. Известно, что первый пешеход встретился со вторым, третьим и четвёртым, а второй — с третьим и четвёртым. Докажите, что третий пешеход встретился с четвёртым.
Решение 1. №5 (с. 240)
Решение 2. №5 (с. 240)
Решение 6. №5 (с. 240)
Обозначим положение $i$-го пешехода в момент времени $t$ как радиус-вектор $\vec{r_i}(t)$. Поскольку пешеходы движутся с постоянными скоростями, закон их движения можно записать в виде: $$ \vec{r_i}(t) = \vec{p_i} + \vec{v_i}t $$ где $\vec{p_i}$ — это начальное положение пешехода (в $t=0$), а $\vec{v_i}$ — его постоянный вектор скорости. Дорога, по которой движется $i$-й пешеход, представляет собой прямую линию, заданную этим уравнением.
Условие, что два пешехода, $i$-й и $j$-й, встретились, означает, что в некоторый момент времени $t_{ij}$ их положения совпали: $$ \vec{r_i}(t_{ij}) = \vec{r_j}(t_{ij}) $$ Из этого следует: $$ \vec{p_i} + \vec{v_i}t_{ij} = \vec{p_j} + \vec{v_j}t_{ij} $$ $$ \vec{p_i} - \vec{p_j} = (\vec{v_j} - \vec{v_i})t_{ij} $$ Это уравнение имеет решение для $t_{ij}$ тогда и только тогда, когда вектор начального относительного положения $(\vec{p_i} - \vec{p_j})$ коллинеарен (параллелен) вектору относительной скорости $(\vec{v_i} - \vec{v_j})$. В двумерном пространстве условие коллинеарности двух векторов $\vec{a}$ и $\vec{b}$ можно записать с помощью векторного (псевдоскалярного) произведения как $[\vec{a}, \vec{b}] = 0$. Таким образом, условие встречи $i$-го и $j$-го пешеходов имеет вид: $$ [(\vec{p_i} - \vec{p_j}), (\vec{v_i} - \vec{v_j})] = 0 $$
Для решения задачи удобно перейти в систему отсчёта, связанную с первым пешеходом. В этой системе отсчёта положение и скорость $j$-го пешехода будут: $$ \vec{r'_j}(t) = \vec{r_j}(t) - \vec{r_1}(t) = (\vec{p_j} - \vec{p_1}) + (\vec{v_j} - \vec{v_1})t $$ Обозначим новые начальные положения и скорости как $\vec{p'_j} = \vec{p_j} - \vec{p_1}$ и $\vec{v'_j} = \vec{v_j} - \vec{v_1}$. Первый пешеход в этой системе покоится в начале координат ($\vec{p'_1} = \vec{0}$, $\vec{v'_1} = \vec{0}$).
Рассмотрим условия задачи в новой системе отсчёта.
По условию, первый пешеход встретился со вторым, третьим и четвёртым. Встреча 1-го и $j$-го пешехода ($j=2,3,4$) означает, что в некоторый момент времени $t_{1j}$ их относительное положение равно нулю: $\vec{r'_j}(t_{1j}) = \vec{0}$. $$ \vec{p'_j} + \vec{v'_j}t_{1j} = \vec{0} \implies \vec{p'_j} = -t_{1j}\vec{v'_j} $$ Это означает, что для $j=2, 3, 4$ вектор начального положения $\vec{p'_j}$ коллинеарен вектору скорости $\vec{v'_j}$. Таким образом, в новой системе отсчёта второй, третий и четвёртый пешеходы движутся по прямым, проходящим через начало координат (положение первого пешехода).
Далее, по условию, второй пешеход встретился с третьим. Условие их встречи инвариантно относительно перехода в другую инерциальную систему отсчёта: $$ [(\vec{p_2} - \vec{p_3}), (\vec{v_2} - \vec{v_3})] = [(\vec{p'_2} - \vec{p'_3}), (\vec{v'_2} - \vec{v'_3})] = 0 $$ Мы знаем, что существуют скаляры $k_j = -t_{1j}$ такие, что $\vec{p'_j} = k_j \vec{v'_j}$. Подставим это в уравнение: $$ [k_2 \vec{v'_2} - k_3 \vec{v'_3}, \vec{v'_2} - \vec{v'_3}] = 0 $$ Раскроем скобки, используя свойства векторного произведения ($[\vec{a}, \vec{a}]=0$ и $[\vec{a}, \vec{b}] = -[\vec{b}, \vec{a}]$): $$ k_2 [\vec{v'_2}, \vec{v'_2}] - k_2 [\vec{v'_2}, \vec{v'_3}] - k_3 [\vec{v'_3}, \vec{v'_2}] + k_3 [\vec{v'_3}, \vec{v'_3}] = 0 $$ $$ -k_2 [\vec{v'_2}, \vec{v'_3}] + k_3 [\vec{v'_2}, \vec{v'_3}] = 0 $$ $$ (k_3 - k_2)[\vec{v'_2}, \vec{v'_3}] = 0 $$ Это уравнение означает, что либо $k_2 = k_3$, либо $[\vec{v'_2}, \vec{v'_3}] = 0$.
Рассмотрим случай $k_2 = k_3$. Это значит, что $t_{12} = t_{13}$. То есть, первый пешеход встретился со вторым и третьим в один и тот же момент времени $t^*=t_{12}=t_{13}$. В этот момент времени $\vec{r_1}(t^*) = \vec{r_2}(t^*)$ и $\vec{r_1}(t^*) = \vec{r_3}(t^*)$. Следовательно, $\vec{r_1}(t^*)=\vec{r_2}(t^*)=\vec{r_3}(t^*)$, и все три пешехода оказались в одной точке пространства. Это означает, что их пути (дороги) пересекаются в одной точке. Но по условию задачи "никакие три дороги не проходят через одну точку". Следовательно, этот случай невозможен, и $k_2 \neq k_3$.
Таким образом, из равенства $(k_3 - k_2)[\vec{v'_2}, \vec{v'_3}] = 0$ следует, что $[\vec{v'_2}, \vec{v'_3}] = 0$. Это означает, что векторы относительных скоростей $\vec{v'_2}$ и $\vec{v'_3}$ коллинеарны: $\vec{v'_2} \parallel \vec{v'_3}$.
Аналогично, из условия встречи 2-го и 4-го пешеходов получаем $[(\vec{p'_2} - \vec{p'_4}), (\vec{v'_2} - \vec{v'_4})] = 0$, что приводит к уравнению: $$ (k_4 - k_2)[\vec{v'_2}, \vec{v'_4}] = 0 $$ По тем же причинам, что и выше, случай $k_2=k_4$ невозможен, так как это означало бы, что дороги 1, 2 и 4 пересекаются в одной точке. Следовательно, $[\vec{v'_2}, \vec{v'_4}] = 0$, то есть $\vec{v'_2} \parallel \vec{v'_4}$.
Из того, что $\vec{v'_2} \parallel \vec{v'_3}$ и $\vec{v'_2} \parallel \vec{v'_4}$, следует, что все три вектора относительных скоростей коллинеарны друг другу: $\vec{v'_2} \parallel \vec{v'_3} \parallel \vec{v'_4}$. В частности, $\vec{v'_3} \parallel \vec{v'_4}$, а значит $[\vec{v'_3}, \vec{v'_4}] = 0$.
Теперь докажем, что третий пешеход встретился с четвёртым. Для этого нужно показать, что выполняется условие их встречи, которое в системе отсчёта первого пешехода выглядит так: $$ [(\vec{p'_3} - \vec{p'_4}), (\vec{v'_3} - \vec{v'_4})] = 0 $$ Подставим выражения $\vec{p'_3} = k_3 \vec{v'_3}$ и $\vec{p'_4} = k_4 \vec{v'_4}$: $$ [k_3 \vec{v'_3} - k_4 \vec{v'_4}, \vec{v'_3} - \vec{v'_4}] $$ Раскрывая скобки, получаем: $$ k_3 [\vec{v'_3}, \vec{v'_3}] - k_3 [\vec{v'_3}, \vec{v'_4}] - k_4 [\vec{v'_4}, \vec{v'_3}] + k_4 [\vec{v'_4}, \vec{v'_4}] = (k_4 - k_3)[\vec{v'_3}, \vec{v'_4}] $$ Поскольку мы уже установили, что $[\vec{v'_3}, \vec{v'_4}] = 0$, то всё выражение равно нулю: $$ (k_4 - k_3) \cdot 0 = 0 $$ Условие встречи третьего и четвёртого пешеходов выполняется, что и требовалось доказать.
Ответ: Утверждение доказано. Третий пешеход встретился с четвёртым.
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