{"problem":{"name":"E. A Colourful Prospect","description":{"content":"_Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement._ Little Tommy is watching a firework show. As circular shapes spread across the sky, a spl","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":1000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF934E"},"statements":[{"statement_type":"Markdown","content":"_Firecrackers scare Nian the monster, but they're wayyyyy too noisy! Maybe fireworks make a nice complement._\n\nLittle Tommy is watching a firework show. As circular shapes spread across the sky, a splendid view unfolds on the night of Lunar New Year's eve.\n\nA wonder strikes Tommy. How many regions are formed by the circles on the sky? We consider the sky as a flat plane. A region is a connected part of the plane with positive area, whose bound consists of parts of bounds of the circles and is a curve or several curves without self-intersections, and that does not contain any curve other than its boundaries. Note that exactly one of the regions extends infinitely.\n\n## Input\n\nThe first line of input contains one integer _n_ (1 ≤ _n_ ≤ 3), denoting the number of circles.\n\nThe following _n_ lines each contains three space-separated integers _x_, _y_ and _r_ ( - 10 ≤ _x_, _y_ ≤ 10, 1 ≤ _r_ ≤ 10), describing a circle whose center is (_x_, _y_) and the radius is _r_. No two circles have the same _x_, _y_ and _r_ at the same time.\n\n## Output\n\nPrint a single integer — the number of regions on the plane.\n\n[samples]\n\n## Note\n\nFor the first example,\n\nFor the second example,\n\nFor the third example,\n\n","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"鞭炮能吓跑年兽，但它们实在太吵了！也许烟花是更好的选择。\n\n小汤米正在观看一场烟花表演。当圆形图案在夜空中绽放时，农历新年前夜的天空展现出一幅壮丽的画卷。\n\n汤米产生了一个疑问：天空中的这些圆形成了多少个区域？我们将天空视为一个平面。一个区域是平面上一个连通的、面积为正的部分，其边界由圆的边界的一部分构成，是一条或若干条无自交的曲线，且不包含任何其他曲线。注意，恰好有一个区域是无限延伸的。\n\n输入的第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 3])，表示圆的数量。\n\n接下来的 #cf_span[n] 行，每行包含三个用空格分隔的整数 #cf_span[x]、#cf_span[y] 和 #cf_span[r] (#cf_span[ - 10 ≤ x, y ≤ 10]，#cf_span[1 ≤ r ≤ 10])，描述一个圆心为 #cf_span[(x, y)]、半径为 #cf_span[r] 的圆。任意两个圆的 #cf_span[x]、#cf_span[y] 和 #cf_span[r] 不会完全相同。\n\n请输出一个整数——平面上的区域数量。\n\n对于第一个示例，\n\n对于第二个示例，\n\n对于第三个示例，\n\n## Input\n\n输入的第一行包含一个整数 #cf_span[n] (#cf_span[1 ≤ n ≤ 3])，表示圆的数量。接下来的 #cf_span[n] 行，每行包含三个用空格分隔的整数 #cf_span[x]、#cf_span[y] 和 #cf_span[r] (#cf_span[ - 10 ≤ x, y ≤ 10]，#cf_span[1 ≤ r ≤ 10])，描述一个圆心为 #cf_span[(x, y)]、半径为 #cf_span[r] 的圆。任意两个圆的 #cf_span[x]、#cf_span[y] 和 #cf_span[r] 不会完全相同。\n\n## Output\n\n请输出一个整数——平面上的区域数量。\n\n[samples]\n\n## Note\n\n对于第一个示例，对于第二个示例，对于第三个示例，","is_translate":true,"language":"Chinese"},{"statement_type":"Markdown","content":"**Definitions** \nLet $ n \\in \\mathbb{Z} $ be the number of circles, with $ 1 \\leq n \\leq 3 $. \nFor each $ i \\in \\{1, \\dots, n\\} $, let $ C_i $ be a circle defined by center $ (x_i, y_i) \\in \\mathbb{R}^2 $ and radius $ r_i \\in \\mathbb{R}^+ $, with $ -10 \\leq x_i, y_i \\leq 10 $, $ 1 \\leq r_i \\leq 10 $, and all $ (x_i, y_i, r_i) $ distinct.\n\n**Constraints** \nNo two circles are identical.\n\n**Objective** \nCompute the number of connected regions $ R $ into which the plane is divided by the union of the circles $ \\bigcup_{i=1}^n C_i $, where each region is a maximal connected open set with boundary composed of arcs of the circles and no other curves.","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF934E","tags":["geometry","graphs"],"sample_group":[["3\n0 0 1\n2 0 1\n4 0 1","4"],["3\n0 0 2\n3 0 2\n6 0 2","6"],["3\n0 0 2\n2 0 2\n1 1 2","8"]],"created_at":"2026-03-03 11:00:39"}}