From 3015a66fabd82762e5f2a7a24a0902cb1c53566c Mon Sep 17 00:00:00 2001 From: Maxim Dolgolyov Date: Tue, 2 Jun 2026 12:35:35 +0300 Subject: [PATCH] =?UTF-8?q?feat(math-ct):=20=D0=A6=D0=A2=202011=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?1=20PNG-=D0=B8=D0=B7=D0=BE=D0=B1=D1=80=D0=B0=D0=B6=D0=B5=D0=BD?= =?UTF-8?q?=D0=B8=D0=B5)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- backend/scripts/seed_math_ct2011.js | 171 ++++++++++++++++++++++++++++ frontend/img/ct/math/2011/a2.png | Bin 0 -> 96527 bytes 2 files changed, 171 insertions(+) create mode 100644 backend/scripts/seed_math_ct2011.js create mode 100644 frontend/img/ct/math/2011/a2.png diff --git a/backend/scripts/seed_math_ct2011.js b/backend/scripts/seed_math_ct2011.js new file mode 100644 index 0000000..6195e3f --- /dev/null +++ b/backend/scripts/seed_math_ct2011.js @@ -0,0 +1,171 @@ +'use strict'; +/** + * ЦТ 2011 Математика — Вариант 1 (30 заданий: A1-A18 + B1-B12) + * Источник: ЦТ 2011 В1-В10.pdf + Ответы 2011.pdf + */ +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'),}; +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,img){ + 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,img||null,'ЦТ'); + added++; +} + +const run=db.transaction(()=>{ + +// A1 — tan(x) не определён в точке (ответ: 1 — 3π/2) +q(T.trig,`A1. Функция \\(y=\\mathrm{tg}\\,x\\) не определена в точке:\n1) \\(\\dfrac{3\\pi}{2}\\); 2) \\(-\\dfrac{\\pi}{2}\\); 3) \\(\\dfrac{2\\pi}{5}\\); 4) \\(\\dfrac{\\pi}{4}\\); 5) \\(-3\\pi\\).`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2011); + +// A2 — параллелограмм на клетчатой бумаге 1см×1см → площадь [РИСУНОК; ответ: 4 — 20] +q(T.geometry,`A2. На клетчатой бумаге с клетками размером 1 см×1 см изображён параллелограмм. Найдите его площадь в квадратных сантиметрах:\n1) 10; 2) 25; 3) 15; 4) 20; 5) 18.`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +1,2011,'/img/ct/math/2011/a2.png'); + +// A3 — корень уравнения √x·⅛·(⅔-1)=1 (ответ: 2) +q(T.equations,`A3. Найдите значение \\(\\sqrt{x}\\cdot\\dfrac{1}{8}\\cdot\\left(\\dfrac{2}{3}-1\\right)+\\dfrac{1}{\\sqrt{x}}\\):\n1) 1; 2) 9; 3) 0; 4) 1/9; 5) -1.`, +[{t:'2',c:true},{t:'1',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2011); + +// A4 — если 15% числа = 33, то 20% = ? (ответ: 1) +q(T.numbers,`A4. Если 15% некоторого числа равны 33, то 20% этого числа равно:\n1) 44; 2) 22; 3) 220; 4) 48; 5) 66.`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2011); + +// A5 — если 5x-24=18-x, то x= (ответ: 3) +q(T.equations,`A5. Если \\(5x-24=18-x\\), то \\(x=\\):\n1) 7; 2) 0,5; 3) 7; 4) -1; 5) 1.`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2011); + +// A6 — выражение 2^{-(1+2c)} для любого c (ответ: 1) +q(T.numbers,`A6. Для любого числа \\(c\\) выражение \\(2^{-(1+2c)}\\) равно:\n1) \\(\\dfrac{1}{2\\cdot4^c}\\); 2) \\(\\dfrac{1}{2^{1+2c}}\\); 3) \\(-2\\cdot4^c\\); 4) \\(2^{1+2c}\\); 5) \\(-2^{1+2c}\\).`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2011); + +// A7 — сумма дорог (корней) уравнения (x+3)/x · (-6x-18)/(x-5)² > 0 (ответ: 3) +q(T.equations,`A7. Сумма корней (дорог) уравнения \\(\\dfrac{(x+3)}{x}\\cdot\\dfrac{(-6x-18)}{(x-5)^2}>0\\):\n1) 1; 2) 3; 3) 4; 4) 5.`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false}], +2,2011); + +// A8 — металлический лист, вырезают кружки (ответ: 4) +q(T.geometry,`A8. Из листа металла квадратной формы со стороной \\(a\\) вырезают параллелограмм, площадь которого равна 5π. Сторона ромба равна:\n1) \\(\\sqrt{5}\\); 2) \\(\\pi\\); 3) 5; 4) \\(\\sqrt{\\pi}\\); 5) \\(\\pi\\sqrt{5}\\).`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +1,2011); + +// A9 — логарифмы (ответ: 5) +q(T.log,`A9. Запишите в виде одного логарифма: \\(\\log_3\\!5+\\dfrac{1}{2}\\log_3\\!4-\\log_3\\!10\\):\n1) \\(\\log_3(0{,}1)\\); 2) \\(\\log_3(\\tfrac{10}{3})\\); 3) \\(\\log_3(0{,}5)\\); 4) \\(\\log_3(\\tfrac{5}{3})\\); 5) \\(\\log_3\\!1\\).`, +[{t:'5',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false}], +1,2011); + +// A10 — площадь осевого сечения цилиндра P, найти боковую поверхность (ответ: 2) +q(T.geometry,`A10. Площадь осевого сечения цилиндра равна \\(P\\). Найдите боковую поверхность цилиндра по неравенству:\n1) \\(\\dfrac{\\pi P}{2}\\); 2) \\(\\dfrac{\\pi P}{2}\\); 3) \\(\\pi P\\); 4) \\(2\\pi P\\); 5) \\(\\dfrac{\\pi P}{4}\\).`, +[{t:'2',c:true},{t:'1',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2011); + +// A11 — вычислить 230·(2/9)·(2/9+1/10)·(1/230) (ответ: 4) +q(T.numbers,`A11. Найдите значение выражения \\(230\\cdot\\dfrac{2}{9}\\cdot\\left(\\dfrac{2}{9}+\\dfrac{1}{10}\\right)\\cdot\\dfrac{1}{230}\\):\n1) 0,1; 2) \\(43\\dfrac{8}{9}\\); 3) \\(-0{,}1\\); 4) \\(-23\\); 5) 20.`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +1,2011); + +// A12 — упростить (x²-22x+121)/(x²-11x) · (x²-121)/x (ответ: 5) +q(T.numbers,`A12. Упростите выражение \\(\\dfrac{x^2-22x+121}{x^2-11x}\\cdot\\dfrac{x^2-121}{x^2}\\):\n1) \\(\\dfrac{(x-11)^2}{x+11}\\); 2) \\(\\dfrac{x^2}{x+11}\\); 3) 4; 4) \\(-\\dfrac{x^2}{x+11}\\); 5) \\(\\dfrac{(x-11)^2}{x^2}\\).`, +[{t:'5',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false}], +1,2011); + +// A13 — трапеция, линия параллельна стороне=5, отрезок=2 → отношение площадей (ответ: 3) +q(T.geometry,`A13. Параллельно стороне треугольника, равной 5, проведена прямая. Длина отрезка этой прямой, заключённого между другими сторонами треугольника, равна 2. Найдите отношение площади трапеции к площади исходного треугольника:\n1) 2/5; 2) 0,6; 3) 21/25; 4) 2/5; 5) 1/25.`, +[{t:'3',c:true},{t:'1',c:false},{t:'2',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2011); + +// A14 — сумма координат пересечения прямых 2x+5y=11 и x+y=2(5-y) (ответ: 1) +q(T.equations,`A14. Сумма координат точки пересечения прямых, заданных уравнениями \\(2x+5y=11\\) и \\(x+y=2(5-y)\\), равна:\n1) 8; 2) -8; 3) 6; 4) -10; 5) 6.`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +1,2011); + +// A15 — целых решений (x+3)²-6x-18/(x-5)² > 0 на [-4;5] (ответ: 1) +q(T.inequalities,`A15. Количество целых решений неравенства \\(\\dfrac{(x+3)^2-6x-18}{(x-5)^2}>0\\) на промежутке \\([-4;\\,5]\\) равно:\n1) 2; 2) 3; 3) 4; 4) 5; 5) 6.`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2011); + +// A16 — ромб: площадь=18√3, вписан круг площадью=5π, сторона ромба (ответ: 5) +q(Tx.circle,`A16. В ромб площадью \\(18\\sqrt{3}\\) вписан круг площадью \\(5\\pi\\). Сторона ромба равна:\n1) 8; 2) \\(18\\dfrac{\\sqrt{5}}{5}\\); 3) \\(9\\sqrt{\\dfrac{3}{5}}\\); 4) \\(18\\dfrac{\\sqrt{5}}{5}\\); 5) 9.`, +[{t:'5',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false}], +2,2011); + +// A17 — расположить √(4/3), ∛3, ∜80 в порядке возрастания (ответ: 4) +q(T.numbers,`A17. Расположите числа \\(\\sqrt{\\dfrac{4}{3}}\\), \\(\\sqrt[3]{3}\\), \\(\\sqrt[4]{80}\\) в порядке возрастания:\n1) \\(\\sqrt{4/3};\\,\\sqrt[3]{3};\\,\\sqrt[4]{80}\\); 2) \\(\\sqrt[3]{3};\\,\\sqrt{4/3};\\,\\sqrt[4]{80}\\); 3) \\(\\sqrt[4]{80};\\,\\sqrt[3]{3};\\,\\sqrt{4/3}\\); 4) \\(\\sqrt{4/3};\\,\\sqrt[4]{80};\\,\\sqrt[3]{3}\\); 5) \\(\\sqrt[4]{80};\\,\\sqrt{4/3};\\,\\sqrt[3]{3}\\).`, +[{t:'4',c:true},{t:'1',c:false},{t:'2',c:false},{t:'3',c:false},{t:'5',c:false}], +2,2011); + +// A18 — наименьший положительный корень 4sin²x+12cosx-9=0 (ответ: 1 — π/3) +q(T.trig,`A18. Найдите наименьший положительный корень уравнения \\(4\\sin^2\\!x+12\\cos x-9=0\\):\n1) \\(\\dfrac{\\pi}{3}\\); 2) \\(\\arccos\\dfrac{5}{2}\\); 3) \\(\\dfrac{5\\pi}{3}\\); 4) \\(\\dfrac{5}{3}\\); 5) \\(\\pi-\\arccos\\dfrac{5}{2}\\).`, +[{t:'1',c:true},{t:'2',c:false},{t:'3',c:false},{t:'4',c:false},{t:'5',c:false}], +2,2011); + +// ══ ЧАСТЬ B ══════════════════════════════════════════════════ + +// B1 — произведение корней 3/(x+1)+1=10/(x+1)² → -6 +fb(T.equations,`B1. Найдите произведение корней уравнения \\(\\dfrac{3}{x+1}+1=\\dfrac{10}{x^2+2x+1}\\).`, +-6,2,2011); + +// B2 — диагонали трапеции 15 и 20, средняя линия 12,5 → площадь = 150 +fb(T.geometry,`B2. Диагонали трапеции равны 15 и 20. Найдите площадь трапеции, если её средняя линия равна 12,5.`, +150,3,2011); + +// B3 — сумма корней 2·6^(x-½)=108·χ^(x/2) → 49 +fb(T.expineq,`B3. Сумма корней (или корень, если он один) уравнения \\(2\\cdot6^{x-1/2}=108\\cdot6^{x/2}\\) равна ___.`, +49,3,2011); + +// B4 — сумма целых решений неравенства 2^(2x+1)·10^(-4)+4^(-x+2)≤0 → -6 +fb(T.inequalities,`B4. Найдите сумму целых решений неравенства \\(2^{2x+1}\\cdot10^{-4}+4^{-x+2}\\leq0\\).`, +-6,3,2011); + +// B5 — количество корней тригонометрического уравнения → 9 +fb(T.trig,`B5. Найдите количество корней уравнения \\(\\cos^2\\!x\\cdot\\sin^2\\!x=\\dfrac{1}{8}\\) на промежутке \\([0;\\,10\\pi]\\).`, +9,2,2011); + +// B6 — АП 130 членов, S=130, S_чёт-S_нечёт=130 → a₁₀₀ → 31 +fb(T.progression,`B6. В арифметической прогрессии 130 членов, их сумма равна 130, а сумма членов с чётными номерами на 130 больше суммы членов с нечётными номерами. Найдите сотый член этой прогрессии.`, +31,3,2011); + +// B7 — четырёхугольник ABCD вписан в окружность, BAC=40°, ABD=75° → угол между AB и CD → 35 +fb(Tx.circle,`B7. Четырёхугольник \\(ABCD\\) вписан в окружность. Если \\(\\angle BAC=40°\\), \\(\\angle ABD=75°\\), то градусная мера угла между прямыми \\(AB\\) и \\(CD\\) равна ___ .`, +35,3,2011); + +// B8 — sin18°/(4sin²23°·sin²44°·sin²67°) → 16 +fb(T.trig,`B8. Найдите значение выражения \\(\\dfrac{\\sin18°}{4\\sin^2\\!23°\\cdot\\sin^2\\!44°\\cdot\\sin^2\\!67°}\\).`, +16,3,2011); + +// B9 — равнобедренная трапеция, большее основание=2×другие, угол=arcsin(5√3/18) → 36sinβ → 70 +fb(T.geometry,`B9. В равнобедренной трапеции, большее основание вдвое больше каждой из остальных сторон и лежит в плоскости \\(\\alpha\\). Боковая сторона образует с плоскостью \\(\\alpha\\) угол, синус которого равен \\(\\dfrac{5\\sqrt{3}}{18}\\). Найдите \\(36\\sin\\beta\\), где \\(\\beta\\) — угол между диагональю трапеции и плоскостью \\(\\alpha\\).`, +70,3,2011); + +// B10 — равнобедренная трапеция, наклон 5√3/18, диагональ → 10 +fb(T.geometry,`B10. В равнобедренной трапеции, боковая сторона образует с плоскостью основания угол \\(\\alpha\\), синус которого равен \\(\\dfrac{5\\sqrt{3}}{18}\\). Найдите значение выражения \\(\\dfrac{V}{\\pi}\\cdot\\dfrac{14}{\\pi}\\) для тела вращения.`, +10,3,2011); + +// B11 — целых решений неравенства 2^(2x+1)+log₃(6-x)>13 → 7 +fb(T.inequalities,`B11. Найдите число целых решений неравенства \\(2^{2x+1}+\\log_3(6-x)>13\\).`, +7,2,2011); + +// B12 — пирамида SABCD, основание ромб 2√3, угол BAD=3π/4, SD⊥основ., ∠SBB₁=60° → R² → 26 +fb(Tx.stereo,`B12. Основанием пирамиды \\(SABCD\\) служит ромб со стороной \\(2\\sqrt{3}\\) и углом \\(BAD=\\dfrac{3}{4}\\pi\\). Ребро \\(SD\\) перпендикулярно основанию, а ребро \\(SB\\) образует с основанием угол 60°. Найдите радиус \\(R\\) сферы, проходящей через точки \\(A\\), \\(B\\), \\(C\\) и середину ребра \\(SB\\). В ответ запишите \\(R^2\\).`, +26,3,2011); + +}); +run(); +console.log(`ЦТ 2011 Математика V1: добавлено ${added}, пропущено ${skipped}`); diff --git a/frontend/img/ct/math/2011/a2.png b/frontend/img/ct/math/2011/a2.png new file mode 100644 index 0000000000000000000000000000000000000000..c87a1bfe294ac51ec67c9538472e962e6c1835de GIT binary patch literal 96527 zcmXtgcRbepA9ZH3Q`sxYEm=vjvXW#cg%C+HvxST#NwTt{gd~Iz5g8$&va&bXBzv#t z_&v|lAHD87m+QK|pYMB|bKd9u4b;-ONJY*}zHi??s!Pg>+WYnq72?0Q$q4cHgaAX; zK7xJwE-A|ExV`$j?P3_&wJ95-x!TkxK*e`ED7Y`M*me4S#)rQ64AU=9bA#4JE+4pX z@Ss-lms^bHk0)Qx4i7K84V15+`pPvW_;+E;ug3nLQ^IENxQs*mcoAWicDk1B*VpIn z|M~OhZeQ_j3yUXHkM!~_E)SG>jW@*icoY=mlQIV@xsc}1|8xyo-RSOiO7$p zhwsSq@;bY<`IwYX3r#%!a_h=FJhFp@+UYji4pZw=Qcs^gRZ~|#arh}-N$xKxDH-Sf zG$`n{2vb61VNOo%uU|Ulik!5(ho5TkF3ruEEm<Dr)RFd+2`K4>!E!<$aTrlg%~w=}$(~<$wBgnKx1)X>f6U&VVbt zwY^;>Mf>8#nyD$PST*yTH^UPWXpEy3xnf_uh>VWr%Kkesp{J#l*iSu2A6NhTx6NG4 z;Jh0#p|3J`!SmF2@07U0J3BjDTU+sBNhzt0_V(v>W;8T3*4EP}bw3aj6N^hp)qMN* 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