85 lines
11 KiB
JSON
85 lines
11 KiB
JSON
{
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"category_name": "main",
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"contest_code": "START209D",
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"contest_name": "Starters 209 (Rated)",
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"status": "success",
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"submit_error": "You need to login to submit.",
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"is_verified": false,
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"problem_code": "P2209",
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"contest_category": "9",
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"problem_name": "Divisible Duel",
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"intended_contest_code": "START209",
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"body": "This is an example problem statement in markdown, and a mini guide on writing statements. Please make sure to remove everything here before publishing your problem.\n\n- Codechef uses markdown for its problem statements. Markdown syntax can be found [here](https:\/\/github.com\/showdownjs\/showdown\/wiki\/Showdown's-Markdown-syntax). Note the `[text](link)` syntax to insert a hyperlink.\n- Codechef also uses $\\LaTeX$ to render mathematical expressions, and you are advised to make liberal use of it to make your statement look good.\n- Text can be made **bold** or *italicized*.\n- **Do not** use HTML tags (p, ul, li, pre, br, ...) in the statement.\n- To insert an image, first upload it to an online hosting service (for an official contest, ask a Codechef admin to do this for you \u2014 this is important) and then use the following syntax: ``.\n- If your problem doesn't contain subtasks, ensure that the Subtasks section below is disabled and **all content is deleted from it**.\n\nIf you face any issues, either contact a Codechef admin directly or send us an email at help@codechef.com.\n\nBelow is an example problem statement that uses some of the above-mentioned features.\n\n---------\n\nChef has a simple undirected graph $G$ with $N$ vertices and $M$ edges. A [subgraph](https:\/\/mathworld.wolfram.com\/Subgraph.html) $H$ of $G$ is called *good* if:\n- $H$ is connected\n- $H$ contains all $N$ vertices of $G$\n- There is a unique path between any two vertices in $H$, using only edges in $H$\n\nCount the number of *good* subgraphs of $G$. Since this number might be large, report it modulo $10^9 + 7$.\n\nIn other news, here's a completely unrelated image:\n\n.\n\n\n<aside style='background: #f8f8f8;padding: 10px 15px;'><div>All submissions for this problem are available.<\/div><\/aside>",
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"problemComponents": {
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"constraints": "- $1 \\leq T \\leq 10^4$\n- $1 \\leq X,Y \\leq 100$",
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"constraintsState": true,
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"subtasks": "- **Subtask 1 (10 points):** $1 \\leq M \\leq 10$\n- **Subtask 2 (20 points):** The sum of $N$ across all test cases won't exceed $20$.\n- **Subtask 3 (70 points):** No further constraints.",
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"subtasksState": false,
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"statement": "Alice loves working with numbers. One day, she takes two integers $X$ and $Y$ ($Y \\ge X$) and becomes curious about the numbers between them.\n\nShe looks at all integers between $X$ and $Y$ (inclusive) that are divisible by $X$. \nAmong these numbers, she separates them into even and odd numbers.\n\nLet the sum of all even numbers Alice has be $S_{even}$, and the sum of all odd numbers she has be $S_{odd}$.\n\nIf $S_{even} \\ge S_{odd}$, Alice will be happy; otherwise, she will be sad. \nAlice wants to know whether she will be happy or not.",
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"inputFormat": "- The first line contains a single integer $T$ \u2014 the number of test cases.\n- Each of the next $T$ lines contains two integers $X$ and $Y$, representing the lower and upper bound of integers Alice looked at.\n",
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"inputFormatState": true,
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"outputFormat": "For each test case, print the answer on a new line: `YES` if Alice will be happy, otherwise `NO`.\n\nYou may print each character of the answer in uppercase either or lowercase (for example, the strings `YES`, `yEs`, `yes`, and `yeS` will all be treated as identical).\n",
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"outputFormatState": true,
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"sampleTestCases": [
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{
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"id": "1",
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"input": "3\n1 4\n3 9\n4 100\n",
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"output": "YES\nNO\nYES",
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"explanation": "**Test case $1$:** Alice has $X = 1$ and $Y = 4$. \nThe multiples of $X$ in this range are $1, 2, 3, 4$. \nSo, $S_{even} = 2+4 = 6$, and $S_{odd} = 1+3 = 4$. \nSince $S_{even} \\ge S_{odd}$, Alice is happy.\n\n**Test case $2$:** Alice has $X = 3$ and $Y = 9$. \nThe multiples of $X$ in this range are $3, 6, 9$.\nSo, $S_{even} = 6$, and $S_{odd} = 3+9 = 12$. \nSince $S_{even} \\lt S_{odd}$, Alice is not happy.",
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"isDeleted": false
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}
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]
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},
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"gumlet_video_url": "",
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"video_editorial_url": "https:\/\/youtu.be\/DReIsGthZtY?si=4_3w50idMKuVf07I",
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"text_editorial_body": "<h1><a name=\"problem-link-1\" class=\"anchor\" href=\"#problem-link-1\"><\/a>PROBLEM LINK:<\/h1>\n<p><a href=\"https:\/\/www.codechef.com\/problems\/P2209\">Practice<\/a><br>\n<a href=\"https:\/\/www.codechef.com\/START209A\/problems\/P2209\">Contest: Division 1<\/a><br>\n<a href=\"https:\/\/www.codechef.com\/START209B\/problems\/P2209\">Contest: Division 2<\/a><br>\n<a href=\"https:\/\/www.codechef.com\/START209C\/problems\/P2209\">Contest: Division 3<\/a><br>\n<a href=\"https:\/\/www.codechef.com\/START209D\/problems\/P2209\">Contest: Division 4<\/a><\/p>\n<p><em><strong>Author:<\/strong><\/em> <a href=\"https:\/\/www.codechef.com\/users\/pols_agyi_pols\">pols_agyi_pols<\/a><br>\n<em><strong>Tester:<\/strong><\/em> <a href=\"https:\/\/www.codechef.com\/users\/kingmessi\">kingmessi<\/a><br>\n<em><strong>Editorialist:<\/strong><\/em> <a href=\"https:\/\/www.codechef.com\/users\/iceknight1093\">iceknight1093<\/a><\/p>\n<h1><a name=\"difficulty-2\" class=\"anchor\" href=\"#difficulty-2\"><\/a>DIFFICULTY:<\/h1>\n<p>Cakewalk<\/p>\n<h1><a name=\"prerequisites-3\" class=\"anchor\" href=\"#prerequisites-3\"><\/a>PREREQUISITES:<\/h1>\n<p>None<\/p>\n<h1><a name=\"problem-4\" class=\"anchor\" href=\"#problem-4\"><\/a>PROBLEM:<\/h1>\n<p>You\u2019re given <span class=\"math\">X<\/span> and <span class=\"math\">Y<\/span>.<br>\nLet <span class=\"math\">S_{even}<\/span> be the sum of all even multiples of <span class=\"math\">X<\/span> in the range <span class=\"math\">[X, Y]<\/span>, and <span class=\"math\">S_{odd}<\/span> be the sum of all odd multiples of <span class=\"math\">X<\/span> in this range.<br>\nChef will be happy only if <span class=\"math\">S_{even} \\ge S_{odd}<\/span>.<\/p>\n<p>Decide if Chef will be happy, given <span class=\"math\">X<\/span> and <span class=\"math\">Y<\/span>.<\/p>\n<h1><a name=\"explanation-5\" class=\"anchor\" href=\"#explanation-5\"><\/a>EXPLANATION:<\/h1>\n<p>Since <span class=\"math\">X<\/span> and <span class=\"math\">Y<\/span> are small (both being <span class=\"math\">\\le 100<\/span>), this problem can be solved by just brute-forcing it.<br>\nThat is, iterate over every integer <span class=\"math\">Z<\/span> in the range <span class=\"math\">[X, Y]<\/span>; and:<\/p>\n<ul>\n<li>If <span class=\"math\">Z<\/span> is not a multiple of <span class=\"math\">X<\/span>, ignore it.<\/li>\n<li>If <span class=\"math\">Z<\/span> is an even multiple of <span class=\"math\">X<\/span>, add <span class=\"math\">Z<\/span> to <span class=\"math\">S_{even}<\/span>.<\/li>\n<li>If <span class=\"math\">Z<\/span> is an odd multiple of <span class=\"math\">X<\/span>, add <span class=\"math\">Z<\/span> to <span class=\"math\">S_{odd}<\/span>.<\/li>\n<\/ul>\n<p>Finally, check if <span class=\"math\">S_{even} \\ge S_{odd}<\/span> or not.<\/p>\n<hr>\n<p>It\u2019s also possible to solve the problem mathematically.<\/p>\n<p>The multiples of in the range <span class=\"math\">[X, Y]<\/span> will be<br>\n<span class=\"math\">X, 2X, 3X, \\ldots, kX<\/span> for some integer <span class=\"math\">k \\ge 1<\/span>.<\/p>\n<p>If <span class=\"math\">X<\/span> is even, then all multiples of <span class=\"math\">X<\/span> will also be even; so <span class=\"math\">S_{odd}<\/span> will always equal <span class=\"math\">0<\/span>.<br>\nThus, <span class=\"math\">S_{even} \\ge S_{odd}<\/span> will always be true.<\/p>\n<p>If <span class=\"math\">X<\/span> is odd, then the multiples of <span class=\"math\">X<\/span> will alternate between odd and even.<br>\nIt can then be seen that:<\/p>\n<ul>\n<li>If <span class=\"math\">k<\/span> is odd, then <span class=\"math\">S_{odd} \\gt S_{even}<\/span>.<br>\n(Recall that <span class=\"math\">k<\/span> is the largest integer such that <span class=\"math\">k \\le Y<\/span>).<\/li>\n<li>If <span class=\"math\">k<\/span> is even, then <span class=\"math\">S_{odd} \\lt S_{even}<\/span>.<\/li>\n<\/ul>\n<p><span class=\"math\">k<\/span> can be computed mathematically by taking <span class=\"math\">\\frac{Y}{X}<\/span> and rounding it down to the nearest integer, so the problem is solved in constant time.<\/p>\n<h1><a name=\"time-complexity-6\" class=\"anchor\" href=\"#time-complexity-6\"><\/a>TIME COMPLEXITY:<\/h1>\n<p><span class=\"math\">\\mathcal{O}(Y)<\/span> or <span class=\"math\">\\mathcal{O}(1)<\/span> per testcase.<\/p>\n<h1><a name=\"code-7\" class=\"anchor\" href=\"#code-7\"><\/a>CODE:<\/h1>\n<details>\n<summary>\nEditorialist's code (PyPy3)<\/summary>\n<pre><code class=\"lang-python\">for _ in range(int(input())):\n x, y = map(int, input().split())\n k = y\/\/x\n \n if x%2 == 0 or k%2 == 0: print('Yes')\n else: print('No')\n<\/code><\/pre>\n<\/details>",
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"problem_author": "archit_adm",
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"problem_display_authors": ["archit_adm"],
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"problem_display_authors_html_handle": "<div class=\"multiple-usernames-container\"><a href='\/users\/archit_adm'>archit_adm<\/a><\/div>",
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"problem_tester": null,
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"problem_testers_usernames": ["kingmessi"],
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"problem_tester_html_handle": "<div class=\"multiple-usernames-container\"><a href='\/users\/kingmessi'><span \n class='rating' \n style='display: inline-block; \n font-size: 10px; \n background: #D0011B;\n padding: 0 3px; \n line-height: 1.3; \n color: white;\n margin-right: 2px;'>7★<\/span><span class='m-username--link'>kingmessi<\/span><\/a><\/div>",
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"problem_editorialist": "iceknight1093",
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"date_added": "21-10-2025",
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"user_tags": ["archit_adm", "cakewalk", "start209"],
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"difficulty_rating": "628",
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"problemDiscussURL": "https:\/\/discuss.codechef.com\/search?q=P2209",
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