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5b:T7fc,Заметим, что $ a_1 \geqslant a_1 $ и что $ a_1 + \dfrac{a_2}{2} = \dfrac{a_1 + a_1}{2} + \dfrac{a_2}{2} \geqslant a_2 $. Предположим, для каждого $ k \leqslant n $ справедливо неравенство \[ \sum_{j=1}^{k} \frac{a_j}{j} \geqslant a_k. \] Просуммируем эти неравенства по $ k = 1, 2, \ldots, n $. Получим \[ na_1 + (n-1)\frac{a_2}{2} + (n-2)\frac{a_3}{3} + \ldots + 2\frac{a_{n-1}}{n-1} + \frac{a_n}{n} \geqslant a_1 + a_2 + \ldots + a_n. \] Прибавим к обеим частям $ a_1 + a_2 + \ldots + a_n $. Получим \[ (n+1)\left(a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \ldots + \frac{a_n}{n}\right) \geqslant 2(a_1 + a_2 + \ldots + a_n). \] Заметим, что $ 2(a_1 + a_2 + \ldots + a_n) = (a_1 + a_n) + (a_2 + a_{n-1}) + \ldots + (a_n + a_1) \geqslant na_{n+1} $. Стало быть, \[ a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \ldots + \frac{a_n}{n} \geqslant \frac{na_{n+1}}{n+1} = a_{n+1} - \frac{a_{n+1}}{n+1}, \] то есть справедливо также неравенство \[ \sum_{j=1}^{n+1} \frac{a_j}{j} \geqslant a_{n+1}. \] Тогда $ a_1 + \ldots + \dfrac{a_7}{7} \geqslant a_7 $, то есть значение исходного выражения не меньше 1. Равенство достигается, например, при $ a_k = k $ ($ k = 1, \ldots, 7 $): все неравенства $ a_i + a_j \geqslant a_{i+j} $ обращаются в равенства. Минимум равен 1. **Примечание.** В официальном решении в качестве примера достижения минимума указан набор $ a_1 = \ldots = a_7 = 1 $, однако для него значение выражения равно $ 1 + \tfrac{1}{2} + \ldots + \tfrac{1}{7} \approx 2{,}59 $. Корректный пример — $ a_k = k $; оценка снизу и ответ (1) в официальном решении верны.41:[["$","script",null,{"id":"collection-schema","type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"Course\",\"name\":\"ДВИ МГУ 2025 — вариант 253\",\"description\":\"Коллекция \\\"ДВИ МГУ 2025 — вариант 253\\\" для подготовки к экзамену. Содержит 7 задач. 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Далее,\n\\[ 14 < \\sqrt{224} < 15, \\qquad 5^2 < 29 < 15 + \\sqrt{224} < 30 < 6^2, \\qquad 5 < \\sqrt{7} + \\sqrt{8} < 6. \\]","explanation":null,"score":10,"part":"2","metadata":{"dvi":true,"ingest_batch":"dvi-2020-2025-v1","position_in_variant":1,"solutions_url":"https://pk.math.msu.ru/wp-content/uploads/2026/03/dvi_2025_answers.pdf","variant_key":"2025-v3","variant_label":"253","year":2025},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-08-31T18:36:47.303772Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"a4c5b1b9-850a-422b-b0b4-723ad67fab1c","name":"Вычисления и оценки","description":null},"tags":[{"id":"31d4e4b7-a2c1-4cbd-85ff-1aa0dec4e314","code":"tag-45","name":"Сравнение чисел"},{"id":"33fabf53-5145-4325-a6fc-cb998f32f4d5","code":"tag-252","name":"Оценки и случай равенства"}]}},{"problem_id":"90a901e0-c8eb-4439-830b-4deae29fb32a","collection_id":"4eb335bf-34bb-42d7-b115-251cdab1df9d","display_order":2,"metadata":{},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-06-10T20:49:51.310551Z","problem":{"id":"90a901e0-c8eb-4439-830b-4deae29fb32a","short_id":17719,"task_number":15,"topic_id":"aaf1e2e8-a5c8-4dae-880f-0853b25f4b34","exam_id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","difficulty":4,"content_text":"Дана последовательность \$ a_1, a_2, a_3, \\ldots \$ действительных чисел, удовлетворяющих при каждом натуральном \$ n \\geqslant 3 \$ равенству\n\\[ a_n = (-1)^n \\cdot 3 \\cdot a_{n-1} + \\frac{a_{n-1}^2}{a_{n-2}}. \\]\nНайдите \$ \\sqrt[2024]{a_{2025}} \$, если известно, что \$ a_1 = 1 \$ и \$ a_2 = 4 \$.","answer_text":"\$2\$","answer":"\$2\$","solution_text":"Положим \$ b_n = a_n/a_{n-1} \$. Тогда \$ b_2 = 4 \$ и \$ b_n = b_{n-1} + (-1)^n \\cdot 3 \$ при \$ n \\geqslant 3 \$. Стало быть, \$ b_{2k} = 4 \$ и \$ b_{2k+1} = 1 \$, откуда ввиду равенства \$ a_n = b_n a_{n-1} \$ получаем, что \$ a_{2k+1} = a_{2k} \$ и \$ a_{2k+2} = 4a_{2k+1} \$. Но тогда \$ a_{2k} = a_{2k+1} = 4^k = 2^{2k} \$. То есть \$ a_{2025} = 2^{2024} \$ и \$ \\sqrt[2024]{a_{2025}} = 2 \$.","explanation":null,"score":10,"part":"2","metadata":{"dvi":true,"ingest_batch":"dvi-2020-2025-v1","position_in_variant":2,"solutions_url":"https://pk.math.msu.ru/wp-content/uploads/2026/03/dvi_2025_answers.pdf","variant_key":"2025-v3","variant_label":"253","year":2025},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-08-31T18:36:47.303772Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"aaf1e2e8-a5c8-4dae-880f-0853b25f4b34","name":"Последовательности","description":null},"tags":[{"id":"6352f8cc-638a-4f87-beeb-1608b2daf4f4","code":"tag-249","name":"Замена переменной"},{"id":"cac715c7-756d-4aa2-91ef-8ac2fd2653b6","code":"tag-193","name":"Последовательности и прогрессии"}]}},{"problem_id":"3b615534-ce61-4e5c-9a0e-b5a1d073076b","collection_id":"4eb335bf-34bb-42d7-b115-251cdab1df9d","display_order":3,"metadata":{},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-06-10T20:49:51.310551Z","problem":{"id":"3b615534-ce61-4e5c-9a0e-b5a1d073076b","short_id":17720,"task_number":16,"topic_id":"72c9fcf0-a99f-4128-b5bb-edf7f779b53c","exam_id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","difficulty":5,"content_text":"Решите неравенство\n\\[ \\sqrt{\\log_2^2\\left(x^{\\sqrt{2}}\\right) - 3\\log_4 x^6 + 13} > \\log_2\\left(\\frac{x}{2}\\right). \\]","answer_text":"\$ x \\in (0, 8) \\cup (16, \\infty) \$","answer":"\$ x \\in (0, 8) \\cup (16, \\infty) \$","solution_text":"Положим \$ y = \\log_2 x \$ и заметим, что \$ 2y^2 - 9y + 13 > 0 \$, ибо \$ 9^2 - 4 \\cdot 2 \\cdot 13 < 0 \$. Тогда\n\\[ \\sqrt{\\log_2^2\\left(x^{\\sqrt{2}}\\right) - 3\\log_4 x^6 + 13} > \\log_2\\left(\\frac{x}{2}\\right) \\Longleftrightarrow \\sqrt{2y^2 - 9y + 13} > y - 1 \\Longleftrightarrow \\]\n\\[ \\Longleftrightarrow \\left[\\begin{array}{l} y < 1 \\\\ 2y^2 - 9y + 13 > y^2 - 2y + 1 \\end{array}\\right. \\Longleftrightarrow \\left[\\begin{array}{l} y < 1 \\\\ y^2 - 7y + 12 > 0 \\end{array}\\right. \\Longleftrightarrow \\left[\\begin{array}{l} y < 1 \\\\ (y-3)(y-4) > 0 \\end{array}\\right. \\Longleftrightarrow \\]\n\\[ \\Longleftrightarrow \\left[\\begin{array}{l} y < 3 \\\\ y > 4 \\end{array}\\right. \\Longleftrightarrow \\left[\\begin{array}{l} \\log_2 x < 3 \\\\ \\log_2 x > 4 \\end{array}\\right. \\Longleftrightarrow \\left[\\begin{array}{l} 0 < x < 8 \\\\ x > 16 \\end{array}\\right. \\]","explanation":null,"score":10,"part":"2","metadata":{"dvi":true,"ingest_batch":"dvi-2020-2025-v1","position_in_variant":3,"solutions_url":"https://pk.math.msu.ru/wp-content/uploads/2026/03/dvi_2025_answers.pdf","variant_key":"2025-v3","variant_label":"253","year":2025},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-08-31T18:36:47.303772Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"72c9fcf0-a99f-4128-b5bb-edf7f779b53c","name":"Логарифмические неравенства","description":null},"tags":[{"id":"2c55495d-5b54-461b-8a9b-3567966b9d28","code":"tag-40","name":"Метод интервалов"},{"id":"3d83bcf4-0d09-4a5d-b894-e2c684238e58","code":"tag-58","name":"Неравенство содержащее радикал"},{"id":"6352f8cc-638a-4f87-beeb-1608b2daf4f4","code":"tag-249","name":"Замена переменной"}]}},{"problem_id":"74ab71a6-35e0-4a7d-b96e-1b6a7979ff2b","collection_id":"4eb335bf-34bb-42d7-b115-251cdab1df9d","display_order":4,"metadata":{},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-06-10T20:49:51.310551Z","problem":{"id":"74ab71a6-35e0-4a7d-b96e-1b6a7979ff2b","short_id":17721,"task_number":17,"topic_id":"56ee582a-3c52-4fd3-85c9-966671ec7454","exam_id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","difficulty":5,"content_text":"Решите уравнение \$ \\dfrac{1}{\\cos^2 x} + \\dfrac{12}{\\sin^2 2x} + \\dfrac{12}{\\sin x \\sin 2x} = \\dfrac{3}{\\sin^2 x} \$.","answer_text":"\$ x = \\pm\\frac{2\\pi}{3} + 2k\\pi,\\ k \\in \\mathbb{Z} \$","answer":"\$ x = \\pm\\frac{2\\pi}{3} + 2k\\pi,\\ k \\in \\mathbb{Z} \$","solution_text":"\\[ \\frac{1}{\\cos^2 x} + \\frac{12}{\\sin^2 2x} + \\frac{12}{\\sin x \\sin 2x} = \\frac{3}{\\sin^2 x} \\Longleftrightarrow \\frac{\\sin^2 x + 3 + 6\\cos x - 3\\cos^2 x}{\\sin^2 x \\cos^2 x} = 0 \\Longleftrightarrow \\]\n\\[ \\Longleftrightarrow \\frac{4 + 6\\cos x - 4\\cos^2 x}{\\sin^2 x \\cos^2 x} = 0 \\Longleftrightarrow \\frac{2(2 - \\cos x)(1 + 2\\cos x)}{\\sin^2 x \\cos^2 x} = 0 \\Longleftrightarrow \\cos x = -\\frac{1}{2} \\Longleftrightarrow \\]\n\\[ \\Longleftrightarrow x = \\pm\\frac{2\\pi}{3} + 2k\\pi,\\ k \\in \\mathbb{Z} \\]","explanation":null,"score":10,"part":"2","metadata":{"dvi":true,"ingest_batch":"dvi-2020-2025-v1","position_in_variant":4,"solutions_url":"https://pk.math.msu.ru/wp-content/uploads/2026/03/dvi_2025_answers.pdf","variant_key":"2025-v3","variant_label":"253","year":2025},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-08-31T18:36:47.303772Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"56ee582a-3c52-4fd3-85c9-966671ec7454","name":"Тригонометрические уравнения","description":null},"tags":[{"id":"9f2c32f9-1c81-45d4-bfd9-70a4620bebf8","code":"tag-142","name":"Тригонометрические уравнения, решаемые разложением на множители"},{"id":"b44bdba8-2ba0-4cd7-a541-f58966163e08","code":"tag-170","name":"Преобразования тригонометрических выражений"}]}},{"problem_id":"29fce52d-37de-46aa-9569-d23025204d0c","collection_id":"4eb335bf-34bb-42d7-b115-251cdab1df9d","display_order":5,"metadata":{},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-06-10T20:49:51.310551Z","problem":{"id":"29fce52d-37de-46aa-9569-d23025204d0c","short_id":17722,"task_number":18,"topic_id":"1148d06d-7ecc-4ba9-b2d9-947ac91b9c9d","exam_id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","difficulty":7,"content_text":"На сторонах \$ AB \$, \$ BC \$, \$ AC \$ треугольника \$ ABC \$ отмечены точки \$ D, E, F \$ соответственно. На \$ BD \$ и на \$ FC \$ как на диаметрах построены окружности. Эти окружности касаются отрезка \$ AE \$ в одной и той же точке. Найдите \$ DF \$, если известно, что \$ AB : AC = 2 : 3 \$, \$ BD : FC = 1 : 2 \$ и что \$ BC = 12 \$.","answer_text":"\$3\$","answer":"\$3\$","solution_text":"Пусть \$ T \$ — точка касания окружностей с отрезком \$ AE \$. Тогда \$ AB \\cdot AD = AT^2 = AC \\cdot AF \$, откуда следует, что треугольники \$ ABC \$ и \$ AFD \$ подобны. Найдём отношение \$ BD : AF \$. Пусть \$ BD = d \$, \$ AF = x \$. Тогда \$ FC = 2d \$, \$ AD = AF \\cdot AC/AB = \\frac{3}{2}x \$. Получаем\n\\[ \\frac{3}{2}x\\left(\\frac{3}{2}x + d\\right) = x(x + 2d). \\]\nСокращая на \$ x \$, приходим к равенству \$ \\frac{5}{4}x = \\frac{1}{2}d \$, откуда\n\\[ \\frac{BD}{AF} = \\frac{d}{x} = \\frac{5}{2}. \\]\nИтак,\n\\[ DF = BC \\cdot \\frac{AF}{AB} = BC \\cdot \\frac{AF}{AD + BD} = BC \\cdot \\frac{1}{\\frac{AD}{AF} + \\frac{BD}{AF}} = 12 \\cdot \\frac{1}{\\frac{3}{2} + \\frac{5}{2}} = 3. \\]","explanation":null,"score":10,"part":"2","metadata":{"dvi":true,"ingest_batch":"dvi-2020-2025-v1","position_in_variant":5,"solutions_url":"https://pk.math.msu.ru/wp-content/uploads/2026/03/dvi_2025_answers.pdf","variant_key":"2025-v3","variant_label":"253","year":2025},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-08-31T18:36:47.303772Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"1148d06d-7ecc-4ba9-b2d9-947ac91b9c9d","name":"Окружность","description":null},"tags":[{"id":"91f89d3c-fa9c-4ac7-8db7-bedf8feba097","code":"tag-258","name":"Степень точки"},{"id":"a7109bee-7556-446b-98b9-32d957694fec","code":"tag-155","name":"Подобие"}]}},{"problem_id":"234c4c3b-0ca5-4ecf-8928-4493909bf2a9","collection_id":"4eb335bf-34bb-42d7-b115-251cdab1df9d","display_order":6,"metadata":{},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-06-10T20:49:51.310551Z","problem":{"id":"234c4c3b-0ca5-4ecf-8928-4493909bf2a9","short_id":17723,"task_number":19,"topic_id":"e463f8eb-fd15-4d94-acb7-4092de9c6b54","exam_id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","difficulty":9,"content_text":"Положительные действительные числа \$ a_1, a_2, a_3, \\ldots, a_7 \$ удовлетворяют неравенствам\n\\[ a_i + a_j \\geqslant a_{i+j} \\]\nпри всех натуральных \$ i, j \$, таких что \$ i + j \\leqslant 7 \$. Найдите наименьшее возможное значение выражения\n\\[ \\frac{a_1 + \\dfrac{a_2}{2} + \\dfrac{a_3}{3} + \\dfrac{a_4}{4} + \\dfrac{a_5}{5} + \\dfrac{a_6}{6} + \\dfrac{a_7}{7}}{a_7}. \\]","answer_text":"\$1\$","answer":"\$1\$","solution_text":"$5b","explanation":null,"score":10,"part":"2","metadata":{"dvi":true,"erratum_resolved":"2026-06-11, вердикт founder","ingest_batch":"dvi-2020-2025-v1","position_in_variant":6,"solutions_url":"https://pk.math.msu.ru/wp-content/uploads/2026/03/dvi_2025_answers.pdf","variant_key":"2025-v3","variant_label":"253","year":2025},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-08-31T18:36:47.303772Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"e463f8eb-fd15-4d94-acb7-4092de9c6b54","name":"Экстремальные задачи и оценки","description":null},"tags":[{"id":"33fabf53-5145-4325-a6fc-cb998f32f4d5","code":"tag-252","name":"Оценки и случай равенства"},{"id":"420a3b65-1300-4f78-a2c0-bc14831078a7","code":"tag-259","name":"Телескопические суммы"}]}},{"problem_id":"d8f6b03a-9dc5-4d6f-bd1d-4795fa34f322","collection_id":"4eb335bf-34bb-42d7-b115-251cdab1df9d","display_order":7,"metadata":{},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-06-10T20:49:51.310551Z","problem":{"id":"d8f6b03a-9dc5-4d6f-bd1d-4795fa34f322","short_id":17724,"task_number":20,"topic_id":"b37141ac-ec06-41ef-977b-401cb8faa903","exam_id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","difficulty":6,"content_text":"Радиус сферы, вписанной в правильную треугольную пирамиду, равен 1. Радиус окружности, вписанной в основание этой пирамиды, равен \$ \\dfrac{1 + \\sqrt{5}}{2} \$. Найдите радиус сферы, описанной около этой пирамиды.","answer_text":"\$ 1 + \\sqrt{5} \$","answer":"\$ 1 + \\sqrt{5} \$","solution_text":"Положим для краткости \$ \\varphi = \\frac{1 + \\sqrt{5}}{2} \$. Пусть \$ O \$ — центр вписанной сферы, \$ M \$ — середина одного из рёбер основания, \$ H \$ — основание высоты пирамиды и \$ S \$ — её вершина. Тогда \$ \\tg\\angle OMH = OH/HM = 1/\\varphi \$, откуда\n\\[ \\frac{SH}{HM} = \\tg(2\\angle OMH) = \\frac{2\\tg\\angle OMH}{1 - \\tg^2\\angle OMH} = \\frac{2\\varphi}{\\varphi^2 - 1}, \\]\nоткуда \$ SH = \\frac{2\\varphi^2}{\\varphi^2 - 1} = 2\\varphi = 2HM \$, что равно радиусу окружности, описанной около основания пирамиды. Стало быть, точка \$ H \$ равноудалена от всех вершин пирамиды и искомый радиус равен \$ 2\\varphi = 1 + \\sqrt{5} \$.","explanation":null,"score":10,"part":"2","metadata":{"dvi":true,"ingest_batch":"dvi-2020-2025-v1","position_in_variant":7,"solutions_url":"https://pk.math.msu.ru/wp-content/uploads/2026/03/dvi_2025_answers.pdf","variant_key":"2025-v3","variant_label":"253","year":2025},"created_at":"2026-06-10T20:49:51.310551Z","updated_at":"2026-08-31T18:36:47.303772Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"b37141ac-ec06-41ef-977b-401cb8faa903","name":"Сферы","description":null},"tags":[{"id":"c7a9f11c-aee5-4ebf-811d-3d1a226c0716","code":"tag-188","name":"Описанный шар"},{"id":"ce7c2741-ac03-4ffe-afe8-d71552c8416c","code":"tag-196","name":"Синус и косинус двойного угла"}]}}],"total":7},"serverExams":[{"id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","code":"dvi-math","name":"ДВИ МГУ (математика)","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","program_id":"1aed6121-06cc-4a5a-87f5-77de484ebb36","program_code":"dvi","exam_type":"ДВИ","description":"Дополнительное вступительное испытание МГУ им. М.В. Ломоносова по математике. С 2026 года экзамен состоит из двух частей: лёгкий тест с кратким ответом и традиционная письменная часть с полным решением.","has_part2":true,"max_task_number":20,"exam_date":"2026-07-11","metadata":{"duration_minutes":180,"max_secondary_score":100,"mock_exam_enabled":true,"part2_tasks":6,"parts":[{"answer_format":"detailed","part":1,"scale":"linear","slots_in_variant":10,"tasks":[1,2,3,4,5,6,7,8,9,10,11,12,13],"weight":30},{"answer_format":"detailed","part":2,"scale":"score_conversion","slots_in_variant":6,"tasks":[14,15,16,17,18,19,20],"weight":70}],"score_conversion":[2,30,30,35,35,35,35,35,40,40,40,40,40,45,45,45,45,45,50,50,50,50,50,55,55,55,55,55,60,60,60,60,60,65,65,65,65,65,70,70,70,70,70,75,75,75,75,75,80,80,80,80,80,85,85,85,85,85,90,90,90,90,90,95,95,95,95,95,100,100,100],"scoring_table":{"1":1,"2":1,"3":1,"4":1,"5":1,"6":1,"7":1,"8":1,"9":1,"10":1,"11":1,"12":1,"13":1,"14":10,"15":10,"16":10,"17":10,"18":10,"19":10,"20":10},"structure_2026":{"consultation_video_url":"https://disk.360.yandex.ru/d/NbP139HUZpCPDA","exam_2026":{"exam_date":"2026-07-11","note":"Реальный основной день; в образце Ш12622 тест был из 15 задач.","test_task_count":10,"written_task_count":6},"exam_portal":"https://exam.msu.ru/","official_exam_page":"https://cpk.msu.ru/exam/101","parts":["base","written"],"test_task_count":15},"task_count":7,"task_numbering":"part_relative","year":2026},"created_at":"2026-06-10T20:49:34.08312Z","updated_at":"2026-09-01T00:33:23.788959Z","problem_count":1008,"theory_count":36,"system_collection_count":190},{"id":"3a8471c5-2a11-4932-ba3e-7ab7d1ae7a1d","code":"ege-base","name":"Математика (база) ЕГЭ","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","old_code":"math_base","program_id":"175d2aa1-bdcb-4df2-9710-7a61057ddd66","program_code":"ege","exam_type":"ЕГЭ","description":"Базовый ЕГЭ по математике для получения аттестата","has_part2":false,"max_task_number":21,"exam_date":"2026-06-04","metadata":{"duration_minutes":180,"max_score":21,"max_secondary_score":5,"mock_exam_enabled":true,"part1_tasks":21,"part2_tasks":0,"score_conversion":[2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5],"scoring_table":{"1":1,"2":1,"3":1,"4":1,"5":1,"6":1,"7":1,"8":1,"9":1,"10":1,"11":1,"12":1,"13":1,"14":1,"15":1,"16":1,"17":1,"18":1,"19":1,"20":1,"21":1},"year":2026},"created_at":"2025-11-09T12:48:12.440483Z","updated_at":"2025-11-09T12:48:12.440483Z","problem_count":8561,"theory_count":85,"system_collection_count":111},{"id":"23f00ba7-bd19-4ca8-9349-5134d88397cf","code":"ege-prof","name":"Математика (профиль) ЕГЭ","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","old_code":"math_prof","program_id":"175d2aa1-bdcb-4df2-9710-7a61057ddd66","program_code":"ege","exam_type":"ЕГЭ","description":"Профильный ЕГЭ по математике для поступающих в технические вузы","has_part2":true,"max_task_number":20,"exam_date":"2026-06-08","metadata":{"duration_minutes":235,"max_score":33,"max_secondary_score":100,"max_task_number":20,"mock_exam_enabled":true,"part1_tasks":13,"part2_tasks":7,"score_conversion":[0,6,11,17,22,27,34,40,46,52,58,64,70,72,74,76,78,80,82,84,86,88,90,92,94,95,96,97,98,99,100,100,100,100],"scoring_table":{"1":1,"2":1,"3":1,"4":1,"5":1,"6":1,"7":1,"8":1,"9":1,"10":1,"11":1,"12":1,"13":1,"14":2,"15":3,"16":2,"17":2,"18":3,"19":4,"20":4},"year":2026},"created_at":"2025-11-09T12:48:12.440483Z","updated_at":"2026-08-30T23:51:20.728833Z","problem_count":2491,"theory_count":123,"system_collection_count":255},{"id":"b0db9506-ddcf-4200-8815-7a709ea0dc4e","code":"oge-math","name":"ОГЭ (математика)","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","program_id":"ff8f63b8-bfcc-47f3-9d55-bc666ee30237","program_code":"oge","exam_type":"ОГЭ","description":"Основной государственный экзамен по математике (9 класс): 25 заданий, 235 минут. 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Максимум 31 первичный балл, итоговая отметка 2-5.","has_part2":true,"max_task_number":25,"metadata":{"duration_minutes":235,"geometry_min_points":2,"geometry_min_points_note":"Для отметки выше 2 необходимо >=2 первичных баллов за геометрию (задания 15-19, 23-25); кодируется в скоринге пробника через geometry_tasks + geometry_min_points (0423)","geometry_tasks":[15,16,17,18,19,23,24,25],"hidden":false,"max_secondary_score":5,"mock_exam_enabled":true,"part1_tasks":19,"part2_tasks":[20,21,22,23,24,25],"score_conversion":[2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5],"scoring_table":{"1":1,"2":1,"3":1,"4":1,"5":1,"6":1,"7":1,"8":1,"9":1,"10":1,"11":1,"12":1,"13":1,"14":1,"15":1,"16":1,"17":1,"18":1,"19":1,"20":2,"21":2,"22":2,"23":2,"24":2,"25":2},"task_count":25,"year":2026},"created_at":"2026-07-18T14:59:37.743898Z","updated_at":"2026-07-18T14:59:37.743898Z","problem_count":2181,"theory_count":104,"system_collection_count":89}]}]]

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