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  1. #include <bits/stdc++.h>
  2.  
  3. using namespace std;
  4.  
  5. const double PI = 3.14159265358979323846;
  6. const double EPS = 1e-8;
  7. const int INF = 1e9 + 50;
  8.  
  9. int sign(double a) {
  10. if (abs(a) <= EPS) {
  11. return 0;
  12. } else if (a > EPS) {
  13. return 1;
  14. } else {
  15. return -1;
  16. }
  17. }
  18.  
  19.  
  20. struct Point {
  21. double x;
  22. double y;
  23. Point (double a = 0, double b = 0) {
  24. x = a;
  25. y = b;
  26. }
  27.  
  28. };
  29.  
  30.  
  31. struct Vector {
  32. double x;
  33. double y;
  34. Vector (double a = 0, double b = 0) {
  35. x = a;
  36. y = b;
  37. }
  38. Vector (Point A, Point B) {
  39. x = B.x - A.x;
  40. y = B.y - A.y;
  41. }
  42. double length() {
  43. return sqrt(x * x + y * y);
  44. }
  45. double length_sqared() {
  46. return (long long)x * x + (long long)y * y;
  47. }
  48. };
  49.  
  50.  
  51. struct Line {
  52. double a;
  53. double b;
  54. double c;
  55. Line (double x, double y, double z) {
  56. a = x;
  57. b = y;
  58. c = z;
  59. }
  60. Line (Point A, Point B) {
  61. b = B.x - A.x;
  62. a = A.y - B.y;
  63. c = -a * A.x - b * A.y;
  64. }
  65. Point GetPoint() {
  66. double x = - (a * c) / (a * a + b * b);
  67. double y = - (b * c) / (a * a + b * b);
  68. return Point(x, y);
  69. }
  70. };
  71.  
  72.  
  73. struct Circle {
  74. Point C;
  75. double r;
  76. Circle (Point A, double b) {
  77. C = A;
  78. r = b;
  79. }
  80. Circle (double x, double y, double b) {
  81. C = Point(x, y);
  82. r = b;
  83. }
  84. };
  85.  
  86.  
  87. double cross_product(Vector A, Vector B) {
  88. return A.x * B.y - B.x * A.y;
  89. }
  90.  
  91. double dot_product(Vector A, Vector B) {
  92. return A.x * B.x + A.y * B.y;
  93. }
  94.  
  95.  
  96. bool are_parallel_lines(Line a, Line b) {
  97. return abs(a.a * b.b - a.b * b.a) < EPS;
  98. }
  99.  
  100. bool are_similar_lines(Line a, Line b) {
  101. return abs(a.a * b.b - a.b * b.a) < EPS && abs(a.b * b.c - b.b * a.c) < EPS && abs(a.a * b.c - b.a * a.c) < EPS;
  102. }
  103.  
  104. Point cross_line_line(Line line1, Line line2) {
  105. double y = (line1.c * line2.a - line1.a * line2.c) / (line1.a * line2.b - line1.b * line2.a);
  106. double x = (line1.b * line2.c - line1.c * line2.b) / (line1.a * line2.b - line1.b * line2.a);
  107. return Point(x, y);
  108. }
  109.  
  110. bool is_segment_intersec(Point A, Point B, Point C, Point D) {
  111. Vector AB(A, B);
  112. Vector CD(C, D);
  113. Vector CA(C, A);
  114. Vector CB(C, B);
  115. Vector AC(A, C);
  116. Vector AD(A, D);
  117. Vector BC(B, C);
  118. Vector BD(B, D);
  119. Vector DA(D, A);
  120. Vector DB(D, B);
  121.  
  122. double prod1 = cross_product(AB, AC);
  123. double prod2 = cross_product(AB, AD);
  124. double prod3 = cross_product(CD, CA);
  125. double prod4 = cross_product(CD, CB);
  126.  
  127. if (abs(prod1) < EPS && abs(prod2) < EPS && abs(prod3) < EPS && abs(prod4) < EPS) { // все точки на одной прямой
  128.  
  129. bool b1 = abs(dot_product(AC, AD) + AC.length() * AD.length()) < EPS;
  130. bool b2 = abs(dot_product(BC, BD) + BC.length() * BD.length()) < EPS;
  131. bool b3 = abs(dot_product(CA, CB) + CA.length() * CB.length()) < EPS;
  132. bool b4 = abs(dot_product(DA, DB) + DA.length() * DB.length()) < EPS;
  133.  
  134. if (b1 || b2 || b3 || b4) {
  135. return true;
  136. } else {
  137. return false;
  138. }
  139.  
  140. } else if (prod1 * prod2 <= 0 && prod3 * prod4 <= 0) {
  141. return true;
  142. } else {
  143. return false;
  144. }
  145. }
  146.  
  147.  
  148. bool is_inside_triangle(Point P, Point A, Point B, Point C) {
  149. Vector AB(A, B);
  150. Vector BC(B, C);
  151. Vector CA(C, A);
  152. Vector AP(A, P);
  153. Vector BP(B, P);
  154. Vector CP(C, P);
  155. int prod1 = sign(cross_product(AB, AP));
  156. int prod2 = sign(cross_product(BC, BP));
  157. int prod3 = sign(cross_product(CA, CP));
  158. if (prod1 == 0 && prod2 == 0 && prod3 == 0) {
  159. double max_dist_in_segment = max(max(AB.length(), BC.length()), CA.length());
  160. double max_dist_to_point = max(max(AP.length(), BP.length()), CP.length());
  161. return max_dist_in_segment - max_dist_to_point > -EPS;
  162. } else if (prod1 >= 0 && prod2 >= 0 && prod3 >= 0) {
  163. return true;
  164. } else if (prod1 < 0 && prod2 < 0 && prod3 < 0) {
  165. return true;
  166. } else {
  167. return false;
  168. }
  169. }
  170.  
  171. bool is_inside_polygon(Point A, vector<Point> &polygon) {
  172. int pts_num = polygon.size();
  173. double ang = 0;
  174. for (int i = 0; i < pts_num; ++i) {
  175. Point pt1 = polygon[i];
  176. Point pt2 = polygon[(i + 1) % pts_num];
  177. Vector v1(A, pt1);
  178. Vector v2(A, pt2);
  179. if (abs(cross_product(v1, v2)) < EPS && dot_product(v1, v2) < EPS)
  180. return true;
  181. ang += atan2(cross_product(v1, v2), dot_product(v1, v2));
  182. }
  183. return abs(ang) > PI;
  184. }
  185.  
  186. bool is_inside_segment(Point A, Point B, Point P) {
  187. // точка на прямой АБ
  188. Vector AB(A, B);
  189. Vector BA(B, A);
  190. Vector AP(A, P);
  191. Vector BP(B, P);
  192. return abs(dot_product(AP, BP) + AP.length() * BP.length()) < EPS;
  193. }
  194.  
  195. bool is_inside_ray(Point A, Point B, Point P) {
  196. Vector AB(A, B);
  197. Vector AP(A, P);
  198. return abs(dot_product(AB, AP) - AB.length() * AP.length()) < EPS;
  199. }
  200.  
  201. bool is_inside_convex_polygon(Point A, vector<Point> &polygon) {
  202. int points_num = polygon.size();
  203. Vector left_bound(polygon[0], polygon[polygon.size() - 1]);
  204. Vector right_bound(polygon[0], polygon[1]);
  205. Vector to_point(polygon[0], A);
  206. if (cross_product(left_bound, to_point) < EPS && cross_product(right_bound, to_point) > -EPS) {
  207. int left = 0;
  208. int right = polygon.size() - 1;
  209. while (right - left > 1) {
  210. int mid = (left + right) / 2;
  211. Vector diag(polygon[0], polygon[mid]);
  212. if (cross_product(to_point, diag) > -EPS) {
  213. right = mid;
  214. } else {
  215. left = mid;
  216. }
  217. }
  218.  
  219. return is_inside_triangle(A, polygon[0], polygon[left], polygon[left + 1]);
  220. } else {
  221. return false;
  222. }
  223. }
  224.  
  225.  
  226. bool is_convex(vector<Point> &polygon) {
  227. int pts_num = polygon.size();
  228. bool has_negative = false;
  229. bool has_positive = false;
  230. for (int pt = 0; pt < pts_num; ++pt) {
  231. Point pt1 = polygon[pt];
  232. Point pt2 = polygon[(pt + 1) % pts_num];
  233. Point pt3 = polygon[(pt + 2) % pts_num];
  234. Vector v12(pt1, pt2);
  235. Vector v23(pt2, pt3);
  236. int turn = sign(cross_product(v12, v23));
  237. if (turn > 0)
  238. has_positive = true;
  239. else if (turn < 0)
  240. has_negative = true;
  241. }
  242. return !(has_negative && has_positive);
  243. }
  244.  
  245.  
  246.  
  247. istream& operator>>(istream& is, Point& p) {
  248. is >> p.x >> p.y;
  249. return is;
  250. }
  251.  
  252. ostream& operator<<(ostream& os, Point& p) {
  253. os << p.x << ' ' << p.y;
  254. return os;
  255. }
  256.  
  257. Point operator + (Point A, Vector B) {
  258. return Point(A.x + B.x, A.y + B.y);
  259. }
  260.  
  261. Point operator - (Point A, Vector B) {
  262. return Point(A.x - B.x, A.y - B.y);
  263. }
  264.  
  265. Vector operator + (Vector A, Vector B) {
  266. return Vector(A.x + B.x, A.y + B.y);
  267. }
  268.  
  269. Vector operator * (Vector A, double b) {
  270. return Vector(A.x * b, A.y * b);
  271. }
  272.  
  273. Vector operator * (double b, Vector A) {
  274. return Vector(A.x * b, A.y * b);
  275. }
  276.  
  277. Vector operator - (Vector A, Vector B) {
  278. return Vector(A.x - B.x, A.y - B.y);
  279. }
  280.  
  281. Vector operator / (Vector A, double b) {
  282. return Vector(A.x / b, A.y / b);
  283. }
  284.  
  285. bool operator < (const Point &A,const Point &B) {
  286. if (abs(A.x - B.x) < EPS) {
  287. return A.y < B.y;
  288. } else {
  289. return A.x < B.x;
  290. }
  291. }
  292.  
  293. bool operator == (const Point &A, const Point & B) {
  294. return abs(A.x - B.x) < EPS && abs(A.y - B.y) < EPS;
  295. }
  296.  
  297.  
  298. double distance_point_point (Point A, Point B) {
  299. return sqrt((A.x - B.x) * (A.x - B.x) + (A.y - B.y) * (A.y - B.y));
  300. }
  301.  
  302. double distance_segment_point(Point A, Point B, Point P) {
  303. Vector AB(A, B);
  304. Vector BA(B, A);
  305. Vector AP(A, P);
  306. Vector BP(B, P);
  307. Vector PB(P, B);
  308. Vector PA(P, A);
  309. double distA = distance_point_point(P, A);
  310. double distB = distance_point_point(P, B);
  311. if (dot_product(AB, AP) < EPS || dot_product(BA, BP) < EPS)
  312. return min(distA, distB);
  313. else
  314. return abs(cross_product(PA, PB) / AB.length());
  315. }
  316.  
  317. double distance_line_point(Line f, Point A) {
  318. return abs(f.a * A.x + f.b * A.y + f.c) / sqrt(f.a * f.a + f.b * f.b);
  319. }
  320.  
  321. double distance_ray_point(Point A, Point B, Point P) {
  322. Vector AB(A, B);
  323. Vector AP(A, P);
  324. if (dot_product(AB, AP) < EPS)
  325. return distance_point_point(P, A);
  326. else {
  327. return distance_line_point(Line(A, B), P);
  328. }
  329. }
  330.  
  331. double distance_line_line(Line a, Line b) {
  332. if (are_similar_lines(a, b))
  333. return 0;
  334. else if (are_parallel_lines(a, b)) {
  335. Point P = a.GetPoint();
  336. return distance_line_point(b, P);
  337. } else {
  338. return 0;
  339. }
  340.  
  341. }
  342.  
  343. double distance_segment_segment(Point A, Point B, Point C, Point D) {
  344. if (is_segment_intersec(A, B, C, D)) {
  345. return 0;
  346. } else {
  347. return min(min(distance_segment_point(A, B, C), distance_segment_point(A, B, D)),
  348. min(distance_segment_point(C, D, A), distance_segment_point(C, D, B)));
  349. }
  350. }
  351.  
  352. double distance_line_segment(Line f, Point A, Point B) {
  353. Line g(A, B);
  354. if (are_similar_lines(f, g))
  355. return 0;
  356. else if (are_parallel_lines(f, g)) {
  357. return distance_line_point(f, A);
  358. } else {
  359. Point P = cross_line_line(f, g);
  360. if (is_inside_segment(A, B, P))
  361. return 0;
  362. else
  363. return min(distance_line_point(f, A), distance_line_point(f, B));
  364. }
  365.  
  366. }
  367.  
  368. double distance_line_ray(Line f, Point A, Point B) {
  369. Line g(A, B);
  370. if (are_parallel_lines(f, g))
  371. return distance_line_point(f, A);
  372. else {
  373. Point P = cross_line_line(f, g);
  374. if (is_inside_ray(A, B, P))
  375. return 0;
  376. else
  377. return distance_line_point(f, A);
  378. }
  379. }
  380.  
  381. double distance_ray_ray(Point A, Point B, Point C, Point D) {
  382. Line f(A, B);
  383. Line g(C, D);
  384. Vector AB(A, B);
  385. Vector CD(C, D);
  386. Vector CA(C, A);
  387. Vector AC(A, C);
  388. double edge_dist = min(distance_ray_point(C, D, A), distance_ray_point(A, B, C));
  389. if (are_similar_lines(f, g)) {
  390. if (abs(dot_product(AB, AC) - AB.length() * AC.length()) < EPS || abs(dot_product(CD, CA) - CD.length() * CA.length()) < EPS)
  391. return 0;
  392. else {
  393. return edge_dist;
  394. }
  395. } else if (are_parallel_lines(f, g)) {
  396. return edge_dist;
  397. } else {
  398. Point P = cross_line_line(f, g);
  399. if (is_inside_ray(A, B, P) && is_inside_ray(C, D, P))
  400. return 0;
  401. else
  402. return edge_dist;
  403. }
  404. }
  405.  
  406. double distance_ray_segment(Point A, Point B, Point C, Point D) {
  407. Line f(A, B);
  408. Line g(C, D);
  409. Vector AB(A, B);
  410. Vector CD(C, D);
  411. Vector CA(C, A);
  412. Vector AC(A, C);
  413. double edge_dist = min(min(distance_ray_point(A, B, C), distance_ray_point(A, B, D)), distance_segment_point(C, D, A));
  414. if (are_similar_lines(f, g)) {
  415. if (is_inside_ray(A, B, C) || is_inside_ray(A, B, D))
  416. return 0;
  417. else
  418. return edge_dist;
  419. } else if (are_parallel_lines(f, g)) {
  420. return edge_dist;
  421. } else {
  422. Point P = cross_line_line(f, g);
  423. if (is_inside_ray(A, B, P) && is_inside_segment(C, D, P))
  424. return 0;
  425. else {
  426. return edge_dist;
  427. }
  428. }
  429. }
  430.  
  431.  
  432. bool cmp(Point A, Point B) {
  433. if (abs(A.y * B.x - A.x * B.y) < EPS) {
  434. return A.x * A.x + A.y * A.y < B.x * B.x + B.y * B.y;
  435. } else {
  436. double ang1 = atan2(A.y, A.x);
  437. double ang2 = atan2(B.y, B.x);
  438. return ang1 < ang2;
  439. }
  440. }
  441.  
  442. void build_convex_hull(vector<Point> &polygon, vector<Point> &hull) {
  443. // в центре самая нижняя левая точка, для целых чисел правильно работает, потом - хз
  444. long long min_y = INF;
  445. long long min_x = INF;
  446.  
  447. for (auto pt: polygon) {
  448. if (pt.y == min_y)
  449. min_x = min((long long)pt.x, min_x);
  450. else if (pt.y < min_y) {
  451. min_y = pt.y;
  452. min_x = pt.x;
  453. }
  454. }
  455. for (int i = 0; i < polygon.size(); ++i) {
  456. polygon[i].x -= min_x;
  457. polygon[i].y -= min_y;
  458. }
  459.  
  460. sort(polygon.begin(), polygon.end(), cmp);
  461. hull.push_back(polygon[0]);
  462. hull.push_back(polygon[1]);
  463. for (int i = 2; i < polygon.size(); ++i) {
  464. while (hull.size() > 1 && cross_product(Vector(hull[hull.size() - 2], hull.back()), Vector(hull.back(), polygon[i])) < -EPS) {
  465. hull.pop_back();
  466. }
  467. hull.push_back(polygon[i]);
  468. }
  469. for (int i = 0; i < polygon.size(); ++i) {
  470. polygon[i].x += min_x;
  471. polygon[i].y += min_y;
  472. }
  473.  
  474. }
  475.  
  476. void restore_convex_polygon(vector<Point> &polygon) {
  477. double min_x, min_y = 1e6;
  478. for (Point pt : polygon) {
  479. if (abs(pt.y - min_y) < EPS) {
  480. if (pt.x < min_x) min_x = pt.x;
  481. } else if (pt.y < min_y) {
  482. min_y = pt.y;
  483. min_x = pt.x;
  484. }
  485. }
  486. for (int i = 0; i < polygon.size(); ++i) {
  487. polygon[i].y -= min_y;
  488. polygon[i].x -= min_x;
  489. if (abs(polygon[i].y) < EPS) polygon[i].y = 0;
  490. if (abs(polygon[i].x ) < EPS) polygon[i].x = 0;
  491. }
  492. sort(polygon.begin(), polygon.end(), cmp);
  493. for (int i = 0; i < polygon.size(); ++i) {
  494. polygon[i].y += min_y;
  495. polygon[i].x += min_x;
  496.  
  497. }
  498.  
  499. }
  500.  
  501. double polygon_perimeter(vector<Point> &polygon) {
  502. double ans = 0;
  503. for (int i = 0; i < polygon.size(); ++i)
  504. ans += distance_point_point(polygon[i], polygon[(i + 1) % polygon.size()]);
  505. return ans;
  506. }
  507.  
  508. double polygon_area_by_Gauss_formula(vector<Point> &polygon) {
  509. vector<double> x_coord, y_coord;
  510. for (Point p : polygon) {
  511. x_coord.push_back(p.x);
  512. y_coord.push_back(p.y);
  513. }
  514.  
  515. long long s = 0;
  516. for (int i = 0; i < x_coord.size() - 1; ++i) {
  517. s += x_coord[i] * y_coord[i + 1];
  518. }
  519. s += x_coord[x_coord.size() - 1] * y_coord[0];
  520.  
  521. for (int i = 0; i < y_coord.size() - 1; ++i) {
  522. s -= y_coord[i] * x_coord[i + 1];
  523. }
  524. s -= y_coord[y_coord.size() - 1] * x_coord[0];
  525.  
  526. return abs(double(s) / 2);
  527. }
  528.  
  529. void pinpoint_turn(Line f, vector<Point> &polygon, vector<Point> &left, vector<Point> &right) {
  530. for (Point pt : polygon) {
  531. double eq = f.a * pt.x + f.b * pt.y + f.c;
  532. if (eq > EPS)
  533. right.emplace_back(pt);
  534. else if (eq < -EPS)
  535. left.emplace_back(pt);
  536. }
  537. }
  538.  
  539. vector<Point> cross_line_polygon(Line f, vector<Point> &polygon) {
  540. vector<Point> cross_pts;
  541. for (int i = 0; i < polygon.size(); ++i) {
  542. Point A = polygon[i];
  543. Point B = polygon[(i + 1) % polygon.size()];
  544. Line AB(A, B);
  545. if (abs(AB.a * f.b - f.a * AB.b) > EPS) {
  546. Point inter = cross_line_line(f, AB);
  547. Vector AP(A, inter);
  548. Vector BP(B, inter);
  549. if ( abs(AP.length() * BP.length() + dot_product(AP, BP)) < EPS && (abs(inter.y - A.y) > EPS|| abs(inter.x - A.x) > EPS))
  550. cross_pts.push_back(inter);
  551. }
  552. }
  553. return cross_pts;
  554. }
  555.  
  556. Point rotate_point(Point A, double angle) {
  557. double x = A.x * cos(angle) - A.y * sin(angle);
  558. double y = A.x * sin(angle) + A.y * cos(angle);
  559. return Point(x, y);
  560. }
  561.  
  562. Vector rotate_vector(Vector A, double angle) {
  563. double x = A.x * cos(angle) - A.y * sin(angle);
  564. double y = A.x * sin(angle) + A.y * cos(angle);
  565. return Vector(x, y);
  566. }
  567.  
  568. Point height_point(Line f, Point A) {
  569. double dist = distance_line_point(f, A);
  570. double x = A.x - dist * f.a / sqrt(f.a * f.a + f.b * f.b);
  571. double y = A.y - dist * f.b / sqrt(f.a * f.a + f.b * f.b);
  572. return Point(x, y);
  573. }
  574.  
  575. vector<Point> cross_circle_circle(Circle O1, Circle O2) {
  576. double r1 = O1.r;
  577. double r2 = O2.r;
  578. vector<Point> intersection;
  579. double d = distance_point_point(O1.C, O2.C);
  580. if (r1 < r2) {
  581. swap(O1, O2);
  582. swap(r1, r2);
  583. }
  584.  
  585. if (O1.C == O2.C && abs(r1 - r2) < EPS) {
  586. // бесконечность
  587. intersection.push_back({O1.C.x, O1.C.y - r1});
  588. intersection.push_back({O1.C.x, O1.C.y + r1});
  589. intersection.push_back({O1.C.x + r1, O1.C.y});
  590.  
  591. } else if (abs(d - r1 - r2) < EPS || abs(r1 - d - r2) < EPS) {
  592. // одна
  593. Vector O1O2(O1.C, O2.C);
  594. Vector resized = O1O2 / O1O2.length() * r1;
  595. Point P = O1.C + resized;
  596. intersection.push_back(P);
  597.  
  598. } else if (d - r1 - r2 > EPS || r1 - d - r2 > EPS) {
  599. // ноль
  600.  
  601. } else {
  602. // две
  603. Vector O1O2(O1.C, O2.C);
  604. Vector resized = O1O2 / O1O2.length() * r1;
  605. double ang = acos((r1 * r1 + d * d - r2 * r2) / (2 * r1 * d));
  606. Point P1 = O1.C + rotate_vector(resized, ang);
  607. Point P2 = O1.C + rotate_vector(resized, -ang);
  608. intersection.push_back(P1);
  609. intersection.push_back(P2);
  610. }
  611. return intersection;
  612. }
  613.  
  614. vector<Line> tangent_point_circle (Point P, Circle O) {
  615. vector<Line> ans;
  616. double dist = distance_point_point(P, O.C);
  617. if (O.r < EPS) {
  618. // окуржность - точка
  619. ans.push_back(Line(P, O.C));
  620. } else if (abs(O.r - dist) < EPS) {
  621. // точка на покужности
  622. Line f(O.C, P);
  623. double a = -f.b;
  624. double b = f.a;
  625. double c = - a * P.x - b * P.y;
  626. ans.push_back(Line(a, b, c));
  627. } else if (dist + EPS < O.r) {
  628. // точка в окружности
  629. } else {
  630. // норм
  631. double l = sqrt(dist * dist - O.r * O.r);
  632. double ang = asin( O.r / dist);
  633. Vector PC(P, O.C);
  634. Vector norm = PC / PC.length() * l;
  635. Vector norm_turn_1 = rotate_vector(norm, ang);
  636. Vector norm_turn_2 = rotate_vector(norm, -ang);
  637. Point T1 = P + norm_turn_1;
  638. Point T2 = P + norm_turn_2;
  639. ans.push_back(Line(P, T1));
  640. ans.push_back(Line(P, T2));
  641. }
  642. return ans;
  643. }
  644.  
  645. vector<Point> cross_line_circle(Line f, Circle O) {
  646. Point C = O.C;
  647. double r = O.r;
  648. vector<Point> ans;
  649. double dist = distance_line_point(f, C);
  650.  
  651. if (abs(dist - r) < EPS) {
  652. ans.push_back(height_point(f, C));
  653. } else if (abs(dist) < r) {
  654. double l = sqrt(r * r - dist * dist);
  655. Point H = height_point(f, C);
  656. Vector to_point(-f.b, f.a);
  657. to_point = to_point / to_point.length();
  658. to_point = to_point * l;
  659. Point A = H + to_point;
  660. Point B = H - to_point;
  661. if (B < A)
  662. swap(B, A);
  663. ans.push_back(A);
  664. ans.push_back(B);
  665. }
  666. return ans;
  667. }
  668.  
  669.  
  670. int main() {
  671. ios_base::sync_with_stdio(false);
  672. cin.tie(0);
  673. cout.precision(18);
  674.  
  675. }
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