From cf21c5797c957758ec940ff6c131a8c7bfee2ee7 Mon Sep 17 00:00:00 2001 From: Maxim Dolgolyov Date: Tue, 2 Jun 2026 10:32:49 +0300 Subject: [PATCH] =?UTF-8?q?feat(math-ct):=20=D0=A6=D0=A2=202016=20V1=20?= =?UTF-8?q?=E2=80=94=2030=20=D0=B7=D0=B0=D0=B4=D0=B0=D0=BD=D0=B8=D0=B9=20(?= =?UTF-8?q?5=20=D1=81=20PNG-=D0=B8=D0=B7=D0=BE=D0=B1=D1=80=D0=B0=D0=B6?= =?UTF-8?q?=D0=B5=D0=BD=D0=B8=D1=8F=D0=BC=D0=B8)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- backend/scripts/seed_math_ct2016.js | 173 +++++++++++++++++++++++++++ frontend/img/ct/math/2016_v1_a11.png | Bin 0 -> 144945 bytes frontend/img/ct/math/2016_v1_a2.png | Bin 0 -> 152404 bytes frontend/img/ct/math/2016_v1_a3.png | Bin 0 -> 68928 bytes frontend/img/ct/math/2016_v1_a7.png | Bin 0 -> 91600 bytes frontend/img/ct/math/2016_v1_a8.png | Bin 0 -> 142266 bytes 6 files changed, 173 insertions(+) create mode 100644 backend/scripts/seed_math_ct2016.js create mode 100644 frontend/img/ct/math/2016_v1_a11.png create mode 100644 frontend/img/ct/math/2016_v1_a2.png create mode 100644 frontend/img/ct/math/2016_v1_a3.png create mode 100644 frontend/img/ct/math/2016_v1_a7.png create mode 100644 frontend/img/ct/math/2016_v1_a8.png diff --git a/backend/scripts/seed_math_ct2016.js b/backend/scripts/seed_math_ct2016.js new file mode 100644 index 0000000..7f7e62c --- /dev/null +++ b/backend/scripts/seed_math_ct2016.js @@ -0,0 +1,173 @@ +'use strict'; +/** + * ЦТ 2016 Математика — Вариант 1 (30 заданий: A1-A18 + B1-B12) + * Источник: ЦТ 2016.pdf, страница ответов 31 + */ +const db = require('../src/db/db'); +const MATH_ID = 3; +const T = {arithmetic:16,word:17,numbers:18,trig:19,quadratic:20,progression:21,inequalities:22,geometry:23,functions:24,log:25,expineq:26,equations:27,stats:28}; +function getTopic(n){const e=db.prepare('SELECT id FROM topics WHERE subject_id=? AND LOWER(name)=LOWER(?)').get(MATH_ID,n);if(e)return e.id;return Number(db.prepare('INSERT INTO topics (subject_id,name) VALUES (?,?)').run(MATH_ID,n).lastInsertRowid);} +const Tx={stereo:getTopic('Стереометрия'),circle:getTopic('Окружность и круг'),sets:getTopic('Числовые промежутки'),}; +const ex=new Set(db.prepare('SELECT text FROM questions WHERE subject_id=3').all().map(q=>q.text.slice(0,80).trim())); +let added=0,skipped=0; +const insQ=db.prepare(`INSERT INTO questions (subject_id,topic_id,text,type,difficulty,year,explanation,correct_text,image,source_type) VALUES (?,?,?,?,?,?,?,?,?,?)`); +const insO=db.prepare(`INSERT INTO options (question_id,text,is_correct,order_index) VALUES (?,?,?,?)`); +function q(tid,text,opts,diff,year,img){ + const key=text.slice(0,80).trim();if(ex.has(key)){skipped++;return;}ex.add(key); + const r=insQ.run(MATH_ID,tid,text,'single',diff,year||null,null,null,img||null,'ЦТ'); + const id=r.lastInsertRowid;opts.forEach((o,i)=>insO.run(id,o.t,o.c?1:0,i));added++; +} +function fb(tid,text,ans,diff,year){ + const a=String(ans); + const key=text.slice(0,80).trim();if(ex.has(key)){skipped++;return;}ex.add(key); + insQ.run(MATH_ID,tid,text,'fill-blank',diff,year||null,null,a,null,'ЦТ'); + added++; +} + +const run=db.transaction(()=>{ + +// ══ ЧАСТЬ A ══════════════════════════════════════════════════ + +// A1 — наименьшее натуральное число, кратное 2, при делении на 15 с остатком — неполное частное 3 → 46 (ответ: 5) +q(T.numbers,`Определите наименьшее натуральное число, кратное 2, которое при делении на 15 с остатком даёт неполное частное, равное 3:\n1) 44; 2) 50; 3) 48; 4) 18; 5) 46.`, +[{t:'5',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false}], +1,2016); + +// A2 — треугольник ABC, угол ACB=38°, угол AMN=109°, найти угол BAC [РИСУНОК; ответ: 1 — 33°] +q(T.geometry,`A2. На рисунке изображён треугольник \\(ABC\\), в котором \\(\\angle ACB=38°\\), \\(\\angle AMN=109°\\). Используя данные рисунка, найдите градусную меру угла \\(BAC\\):\n1) 33°; 2) 52°; 3) 26°; 4) 30°; 5) 60°.`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2016,'/img/ct/math/2016_v1_a2.png'); + +// A3 — числовая прямая 0 < k < t, верное утверждение [РИСУНОК; ответ: 4 — k/(-3)>t/(-3)] +q(T.inequalities,`A3. Используя рисунок, определите верное утверждение и укажите его номер:\n1) \\(-3k<-3t\\); 2) \\(\\dfrac{1}{t}>\\dfrac{1}{k}\\); 3) \\(3k>3t\\); 4) \\(\\dfrac{k}{-3}>\\dfrac{t}{-3}\\); 5) \\(k>t\\).`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +1,2016,'/img/ct/math/2016_v1_a3.png'); + +// A4 — значение выражения (степени) → 125/81 (ответ: 4) +q(T.numbers,`Значение выражения \\(3^{-4}\\cdot5^3\\) равно:\n1) \\(\\dfrac{27}{125}\\); 2) \\(\\dfrac{4}{5}\\); 3) \\(\\dfrac{16}{81}\\); 4) \\(\\dfrac{125}{81}\\); 5) \\(\\dfrac{125}{32}\\).`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +1,2016); + +// A5 — формула n-го члена АП (a₁=2, a₂=5) → aₙ=3n-1 (ответ: 3) +q(T.progression,`Укажите формулу для нахождения n-го члена арифметической прогрессии \\((a_n)\\), если \\(a_1=2\\), \\(a_2=5\\):\n1) \\(a_n=-3n+5\\); 2) \\(a_n=3n+5\\); 3) \\(a_n=3n-1\\); 4) \\(a_n=3n+2\\); 5) \\(a_n=5n+8\\).`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2016); + +// A6 — прямая пропорциональность, таблица a,b; найти неизвестное a=32 (ответ: 1) +q(T.functions,`Величины \\(a\\) и \\(b\\) являются прямо пропорциональными. Используя данные таблицы \\([a:19\\,|\\,?\\,;\\; b:108\\,|\\,7{,}6]\\), найдите неизвестное значение \\(a\\):\n1) 32; 2) 27; 3) \\(\\dfrac{4}{14}\\); 4) 13; 5) 56.`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2016); + +// A7 — площадь фигуры на рисунке → 35,5 см² (ответ: 1) [РИСУНОК] +q(T.geometry,`A7. Найдите площадь фигуры, изображённой на рисунке (в кв. сантиметрах):\n1) 35,5 см²; 2) 28 см²; 3) 36 см²; 4) 49 см²; 5) 35 см².`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2016,'/img/ct/math/2016_v1_a7.png'); + +// A8 — сумма целых значений f(x) на (-5;5) по графику (ответ: 3 — 7) [РИСУНОК] +q(T.functions,`A8. Найдите сумму всех целых значений функции \\(y=f(x)\\), заданной графиком на промежутке \\((-5;\\,5)\\) (см. рис.):\n1) 12; 2) 14; 3) 7; 4) 10; 5) 11.`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2016,'/img/ct/math/2016_v1_a8.png'); + +// A9 — вычислить НОК(12,18,36)+НОД(38,52) (ответ: 5) +q(T.numbers,`Найдите значение выражения \\(\\text{НОК}(12,\\,18,\\,36)+\\text{НОД}(38,\\,52)\\):\n1) 26; 2) 50; 3) 48; 4) 72; 5) 38.`, +[{t:'5',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false}], +1,2016); + +// A10 — прямая пересекает плоскость α в точке A под углом 60°, B лежит на прямой, AB=6√2; расстояние от B до α (ответ: 2) +q(T.geometry,`Прямая \\(a\\) пересекает плоскость \\(\\alpha\\) в точке \\(A\\) и образует с плоскостью угол 60°. Точка \\(B\\) лежит на прямой \\(a\\), причём \\(AB=6\\sqrt{2}\\). Найдите расстояние от точки \\(B\\) до плоскости \\(\\alpha\\):\n1) \\(2\\sqrt{3}/6\\); 2) \\(3\\sqrt{6}\\); 3) \\(3\\sqrt{3}\\); 4) \\(6\\sqrt{6}\\); 5) \\(3\\sqrt{2}\\).`, +[{t:'2',c:true},{t:'1',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2016); + +// A11 — круговая диаграмма: гречиха → 150 га (ответ: 2) [РИСУНОК] +q(T.stats,`A11. На круговой диаграмме показано распределение посевных площадей под зерновые культуры в агрохозяйстве. Сколько гектаров отведено под гречиху, если овсом засеяно на 390 га больше, чем рожью?\n1) 110 га; 2) 150 га; 3) 120 га; 4) 160 га; 5) 180 га.`, +[{t:'2',c:true},{t:'1',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2016,'/img/ct/math/2016_v1_a11.png'); + +// A12 — стороны треугольника — целые числа, одна сторона 1, другая 3; периметр (ответ: 3) +q(T.geometry,`Длины всех сторон треугольника являются целыми числами. Если длина одной стороны треугольника равна 1, а другой — 3, то периметр треугольника равен:\n1) 7; 2) 14; 3) 7; 4) 6; 5) 8.`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2016); + +// A13 — сократить дробь (x²-9)/(8x²-23x-3) (ответ: 4) +q(T.equations,`Сократите дробь \\(\\dfrac{x^2-9}{8x^2-23x-3}\\):\n1) \\(\\dfrac{x-3}{8x+1}\\); 2) \\(\\dfrac{x+3}{8x+1}\\); 3) \\(\\dfrac{x+3}{8x-1}\\); 4) \\(\\dfrac{x+3}{8x+1}\\); 5) \\(\\dfrac{x-3}{8x-1}\\).`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +1,2016); + +// A14 — два автомобиля, расстояние 160 км, встречное движение, скорости a и 50 км/ч; расстояние от A (ответ: 5) +q(T.word,`Из пунктов \\(A\\) и \\(B\\), расстояние между которыми 160 км, одновременно навстречу друг другу выехали два автомобиля с постоянными и неравными скоростями: из пункта \\(A\\) — со скоростью \\(a\\) км/ч, из пункта \\(B\\) — со скоростью 50 км/ч. Через некоторое время автомобили встретились. Составьте выражение, определяющее расстояние (в км) от пункта \\(A\\) до места встречи:\n1) \\(\\dfrac{160a}{a+b}\\); 2) \\(\\dfrac{160}{a+50}\\); 3) \\(\\dfrac{160a}{1600}\\); 4) \\(\\dfrac{1600}{a+50}\\); 5) \\(\\dfrac{160a}{a+50}\\).`, +[{t:'5',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false}], +1,2016); + +// A15 — три точки A,B,C лежат на большой окружности сферы, ABC — равносторонний, AB=3√6; площадь треугольника (ответ: 3) +q(Tx.stereo,`Точки \\(A\\), \\(B\\), \\(C\\) лежат на большой окружности сферы так, что треугольник \\(ABC\\) — равносторонний. Если \\(AB=3\\sqrt{6}\\), то площадь треугольника \\(ABC\\) равна:\n1) \\(144\\pi\\); 2) \\(72\\pi\\); 3) \\(\\dfrac{81\\sqrt{3}}{4}\\); 4) \\(18\\pi\\); 5) \\(6\\sqrt{3}\\).`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2016); + +// A16 — упростить 5cos(2x+a)·sin(11x/2-a) (ответ: 1) +q(T.trig,`Упростите выражение \\(5\\cos(2x+a)+\\sin\\dfrac{11x}{2}-a\\):\n1) \\(6\\cos a\\); 2) \\(-6\\sin a\\); 3) \\(-4\\cos a\\); 4) \\(4\\sin a\\); 5) \\(4\\cos a\\).`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2016); + +// A17 — график y=kx+b симметричен относительно начала координат, проходит через A(4;12) (ответ: 3) +q(T.functions,`График функции, заданной формулой \\(y=kx+b\\), симметричен относительно начала координат и проходит через точку \\(A(4;\\,12)\\). Значение выражения \\(k+b\\) равно:\n1) \\(-8\\); 2) 2; 3) 3; 4) \\(-1\\); 5) 1.`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2016); + +// A18 — сумма всех натуральных решений неравенства (6-x)(x+4)²(x-13)²≥0 (ответ: 4) +q(T.inequalities,`Сумма всех натуральных решений неравенства \\((6-x)(x+4)^2(x-13)^2\\geq0\\) равна:\n1) 11; 2) 19; 3) 21; 4) 34; 5) 49.`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +2,2016); + +// ══ ЧАСТЬ B ══════════════════════════════════════════════════ + +// B1 — краска: минимальная стоимость покупки для площади 175 м² → 960 000 руб. +fb(T.word,`Для покраски стен общей площадью 175 м² планируется закупка краски. Объём и стоимость банок с краской приведены в таблице:\nОбъём банки (в литрах): 2,5 | 10. Стоимость банки с краской (в рублях): 75 000 | 270 000.\nКакую минимальную сумму (в рублях) потратят на покупку необходимого количества краски, если её расход составляет 0,2 л/м²?`, +960000,2,2016); + +// B2 — корень уравнения 2√(x+√24x)=x+√(24x) → 9 +fb(T.equations,`Найдите сумму корней (корень, если он единственный) уравнения \\(2\\sqrt{x+\\sqrt{24x}}=x+\\sqrt{24x}\\).`, +9,2,2016); + +// B3 — равнобедренная трапеция с вписанной окружностью, сумма двух углов 60°, найти периметр → 38 +fb(T.geometry,`В равнобедренную трапецию, площадь которой равна \\(\\dfrac{36\\sqrt{3}}{5}\\), вписана окружность. Сумма двух углов трапеции равна 60°. Найдите периметр трапеции.`, +38,2,2016); + +// B4 — система уравнений {6x-7=5y-x, 5x²-xy+x²-12=12}, найти 5y-x → 23 +fb(T.equations,`Пусть \\((x;\\,y)\\) — решение системы уравнений \\(\\begin{cases}6x-7=5y-x\\\\5x^2-xy+x^2-12=12\\end{cases}\\). Найдите значение выражения \\(5y-x\\).`, +23,2,2016); + +// B5 — значение выражения 2(√(5√5-√36√6))·(√3+√6)-4√30 → 68 +fb(T.numbers,`Найдите значение выражения \\(2\\left(\\sqrt{5}\\cdot\\sqrt{5}-\\sqrt{36}\\cdot\\sqrt{6}\\right)\\cdot\\left(\\sqrt{3}+\\sqrt{6}\\right)-4\\sqrt{30}\\).`, +-68,3,2016); + +// B6 — сумма корней уравнения (x-64)^(4/3)·(4^3+3·2^x·2^x-81)=0 → 69 +fb(T.expineq,`Найдите сумму корней уравнения \\((x-64)^{4/3}\\cdot(4^3+3\\cdot2^x-2^x-81)=0\\).`, +69,2,2016); + +// B7 — площадь боковой поверхности правильной треугольной пирамиды (биссектриса 4√3, угол при вершине arcctg(2/3)) → 60 +fb(Tx.stereo,`Найдите площадь боковой поверхности правильной треугольной пирамиды, если длина биссектрисы \\(4\\sqrt{3}\\) и плоский угол при вершине равен 2arcctg\\(\\dfrac{2}{3}\\).`, +60,3,2016); + +// B8 — сумма наименьшего и наибольшего целых решений log½·log₂·log₃(x+15)>0 → 712 +fb(T.log,`Найдите сумму наименьшего и наибольшего целых решений неравенства \\(\\log_\\frac{1}{2}\\log_2\\log_3(x+15)>0\\).`, +712,3,2016); + +// B9 — (в градусах) сумма корней уравнения 10sin5cos5x+5sin10cos2x=0 на [110°;150°] → -34 +fb(T.trig,`Найдите (в градусах) сумму корней уравнения \\(10\\sin5x\\cos5x+5\\sin10x\\cos2x=0\\) на промежутке \\([110°;\\,150°]\\).`, +-34,3,2016); + +// B10 — произведение наименьшего и наибольшего целых решений |5-2x|+|2x+3|≤|4x+5| → 58 +fb(T.inequalities,`Найдите произведение наименьшего и наибольшего целых решений неравенства \\(|5-2x|+|2x+3|\\leq|4x+5|\\).`, +58,2,2016); + +// B11 — по периметру треугольника KMP движется точка, скорость в 6 раз больше, чем у A → 112 +fb(T.word,`Точка \\(A\\) движется по периметру треугольника \\(KMP\\). Точки \\(K_1\\), \\(M_1\\), \\(P_1\\) лежат на медианах треугольника \\(KMP\\) и делят их в отношении 11:3, считая от вершины. По периметру треугольника \\(K_1M_1P_1\\) движется точка \\(B\\) со скоростью в 6 раз больше, чем у точки \\(A\\). Сколько раз точка \\(B\\) обойдёт по периметру \\(K_1M_1P_1\\) за то время, когда точка \\(A\\) дважды обойдёт по периметру \\(KMP\\)?`, +112,3,2016); + +// B12 — объём меньшей части параллелепипеда ABCDA₁B₁C₁D₁ (ребро CC₁, CP·PC₁=2:1, секущая плоскость) → 724 +fb(Tx.stereo,`Объём прямоугольного параллелепипеда \\(ABCDA_1B_1C_1D_1\\) равен 1728. Через точку \\(P\\) бокового ребра \\(CC_1\\), такую что \\(CP:PC_1=2:1\\). Через точку \\(P\\) перпендикулярно \\(D\\) к середине бокового ребра \\(AA_1\\) проведена секущая плоскость, которая делит прямоугольный параллелепипед на две части. Найдите объём меньшей из частей.`, +724,3,2016); + +}); +run(); +console.log(`ЦТ 2016 V1: добавлено ${added}, пропущено (дубликаты) ${skipped}`); diff --git a/frontend/img/ct/math/2016_v1_a11.png b/frontend/img/ct/math/2016_v1_a11.png new file mode 100644 index 0000000000000000000000000000000000000000..4cf2029a2298ee182c34cb579e8133c1aa8bbe13 GIT binary patch literal 144945 zcmYg&2RxQ<8+J){vWk=y3RxABkdeKGgeci7i9+@&D{pu1}q$?aXcc6X5x z;X4DYG?Tk_@7kquK~~Q_`PbGRL;AMEf1{44C-~j>+gt3{-D3J6HIwyjD}`+Fnc_3Q z>h8RL!JxA{n`?lkIB;%CRw3paEscV-Z2HXSkKJsZyyZT+&M19LrzYv!Wo^MygB>1NX#cUJUo045mA!o zL9YMzDrEce?5WJxrlBFP&NSK9*4DU#t%b&w{QUgAeSNJ?`r6tfi|(<*94GkrM&unF zer^5x`!?c`{M0VigDx&ECV~lX$8>lY^!4@m;*E55j|vG1(U375IH0Dg8YGt<5fSn7 z<*C#Tp%W(pz0<92ZF7^8lWS`w6WS)0*FSvtu(q~&KDy~VxyrIEJB^;69uE(XNax!6 z`nr#gPrgB6Mww%8Z|~2J)bG8e?1bBvUs(5IjZ#WVAzZ#Ajjp-9UQ<)kE7xf`TU%%G zWL&pyu3m}pYF*iSI=kUG0?$ILy zd|BT&G$bh{RhN~O<^AsucF22efy1l6zh6^XS@_JEosCudn>SB7c&|+9R5}euL`G(2 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