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Matej Focko
7ec83aafc4
* “A. To My Critics” * “B. Ten Words of Wisdom” * “C. Word on the Paper” * “D. Balanced Round” * “E. Cardboard for Pictures” * “F. We Were Both Children” * “G. The Morning Star” * “H. The Third Letter” Signed-off-by: Matej Focko <me@mfocko.xyz>
1345 lines
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">A. To My Critics</div><div class="time-limit"><div class="property-title">time limit per test</div>1 second</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>Suneet has three digits $$$a$$$, $$$b$$$, and $$$c$$$. </p><p>Since math isn't his strongest point, he asks you to determine if you can choose any two digits to make a sum greater or equal to $$$10$$$.</p><p>Output "<span class="tex-font-style-tt">YES</span>" if there is such a pair, and "<span class="tex-font-style-tt">NO</span>" otherwise.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.</p><p>The only line of each test case contains three digits $$$a$$$, $$$b$$$, $$$c$$$ ($$$0 \leq a, b, c \leq 9$$$).</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output "<span class="tex-font-style-tt">YES</span>" if such a pair exists, and "<span class="tex-font-style-tt">NO</span>" otherwise.</p><p>You can output the answer in any case (for example, the strings "<span class="tex-font-style-tt">yEs</span>", "<span class="tex-font-style-tt">yes</span>", "<span class="tex-font-style-tt">Yes</span>" and "<span class="tex-font-style-tt">YES</span>" will be recognized as a positive answer).</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">5</div><div class="test-example-line test-example-line-odd test-example-line-1">8 1 2</div><div class="test-example-line test-example-line-even test-example-line-2">4 4 5</div><div class="test-example-line test-example-line-odd test-example-line-3">9 9 9</div><div class="test-example-line test-example-line-even test-example-line-4">0 0 0</div><div class="test-example-line test-example-line-odd test-example-line-5">8 5 3</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
YES
|
||
NO
|
||
YES
|
||
NO
|
||
YES
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>For the first test case, by choosing the digits $$$8$$$ and $$$2$$$ we can obtain a sum of $$$8 + 2 = 10$$$ which satisfies the condition, thus the output should be "<span class="tex-font-style-tt">YES</span>".</p><p>For the second test case, any combination of chosen digits won't be at least $$$10$$$, thus the output should be "<span class="tex-font-style-tt">NO</span>" (note that we can not choose the digit on the same position twice).</p><p>For the third test case, any combination of chosen digits will have a sum equal to $$$18$$$, thus the output should be "<span class="tex-font-style-tt">YES</span>".</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">B. Ten Words of Wisdom</div><div class="time-limit"><div class="property-title">time limit per test</div>1 second</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>In the game show "Ten Words of Wisdom", there are $$$n$$$ participants numbered from $$$1$$$ to $$$n$$$, each of whom submits one response. The $$$i$$$-th response is $$$a_i$$$ words long and has quality $$$b_i$$$. No two responses have the same quality, and at least one response has length at most $$$10$$$.</p><p>The winner of the show is the response which has the highest quality out of all responses that are not longer than $$$10$$$ words. Which response is the winner?</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 100$$$) — the number of test cases.</p><p>The first line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 50$$$) — the number of responses.</p><p>Then $$$n$$$ lines follow, the $$$i$$$-th of which contains two integers $$$a_i$$$ and $$$b_i$$$ ($$$1 \leq a_i, b_i \leq 50$$$) — the number of words and the quality of the $$$i$$$-th response, respectively. </p><p><span class="tex-font-style-bf">Additional constraints on the input:</span> in each test case, at least one value of $$$i$$$ satisfies $$$a_i \leq 10$$$, and all values of $$$b_i$$$ are distinct.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output a single line containing one integer $$$x$$$ ($$$1 \leq x \leq n$$$) — the winner of the show, according to the rules given in the statement.</p><p>It can be shown that, according to the constraints in the statement, exactly one winner exists for each test case.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">3</div><div class="test-example-line test-example-line-odd test-example-line-1">5</div><div class="test-example-line test-example-line-odd test-example-line-1">7 2</div><div class="test-example-line test-example-line-odd test-example-line-1">12 5</div><div class="test-example-line test-example-line-odd test-example-line-1">9 3</div><div class="test-example-line test-example-line-odd test-example-line-1">9 4</div><div class="test-example-line test-example-line-odd test-example-line-1">10 1</div><div class="test-example-line test-example-line-even test-example-line-2">3</div><div class="test-example-line test-example-line-even test-example-line-2">1 2</div><div class="test-example-line test-example-line-even test-example-line-2">3 4</div><div class="test-example-line test-example-line-even test-example-line-2">5 6</div><div class="test-example-line test-example-line-odd test-example-line-3">1</div><div class="test-example-line test-example-line-odd test-example-line-3">1 43</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
4
|
||
3
|
||
1
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first test case, the responses provided are as follows:</p><ul> <li> Response 1: $$$7$$$ words, quality $$$2$$$ </li><li> Response 2: $$$12$$$ words, quality $$$5$$$ </li><li> Response 3: $$$9$$$ words, quality $$$3$$$ </li><li> Response 4: $$$9$$$ words, quality $$$4$$$ </li><li> Response 5: $$$10$$$ words, quality $$$1$$$ </li></ul><p>We can see that the responses with indices $$$1$$$, $$$3$$$, $$$4$$$, and $$$5$$$ have lengths not exceeding $$$10$$$ words. Out of these responses, the winner is the one with the highest quality.</p><p>Comparing the qualities, we find that: </p><ul> <li> Response 1 has quality $$$2$$$. </li><li> Response 3 has quality $$$3$$$. </li><li> Response 4 has quality $$$4$$$. </li><li> Response 5 has quality $$$1$$$. </li></ul><p>Among these responses, Response 4 has the highest quality.</p></div></div><p> </p></div>
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|
||
<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">C. Word on the Paper</div><div class="time-limit"><div class="property-title">time limit per test</div>1 second</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>On an $$$8 \times 8$$$ grid of dots, a word consisting of lowercase Latin letters is written vertically in one column, from top to bottom. What is it?</p></div><div class="input-specification"><div class="section-title">Input</div><p>The input consists of multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.</p><p>Each test case consists of $$$8$$$ lines, each containing $$$8$$$ characters. Each character in the grid is either $$$\texttt{.}$$$ (representing a dot) or a lowercase Latin letter ($$$\texttt{a}$$$–$$$\texttt{z}$$$). </p><p>The word lies entirely in a single column and is continuous from the beginning to the ending (without gaps). See the sample input for better understanding.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output a single line containing the word made up of lowercase Latin letters ($$$\texttt{a}$$$–$$$\texttt{z}$$$) that is written vertically in one column from top to bottom.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">5</div><div class="test-example-line test-example-line-odd test-example-line-1">........</div><div class="test-example-line test-example-line-odd test-example-line-1">........</div><div class="test-example-line test-example-line-odd test-example-line-1">........</div><div class="test-example-line test-example-line-odd test-example-line-1">........</div><div class="test-example-line test-example-line-odd test-example-line-1">...i....</div><div class="test-example-line test-example-line-odd test-example-line-1">........</div><div class="test-example-line test-example-line-odd test-example-line-1">........</div><div class="test-example-line test-example-line-odd test-example-line-1">........</div><div class="test-example-line test-example-line-even test-example-line-2">........</div><div class="test-example-line test-example-line-even test-example-line-2">.l......</div><div class="test-example-line test-example-line-even test-example-line-2">.o......</div><div class="test-example-line test-example-line-even test-example-line-2">.s......</div><div class="test-example-line test-example-line-even test-example-line-2">.t......</div><div class="test-example-line test-example-line-even test-example-line-2">........</div><div class="test-example-line test-example-line-even test-example-line-2">........</div><div class="test-example-line test-example-line-even test-example-line-2">........</div><div class="test-example-line test-example-line-odd test-example-line-3">........</div><div class="test-example-line test-example-line-odd test-example-line-3">........</div><div class="test-example-line test-example-line-odd test-example-line-3">........</div><div class="test-example-line test-example-line-odd test-example-line-3">........</div><div class="test-example-line test-example-line-odd test-example-line-3">......t.</div><div class="test-example-line test-example-line-odd test-example-line-3">......h.</div><div class="test-example-line test-example-line-odd test-example-line-3">......e.</div><div class="test-example-line test-example-line-odd test-example-line-3">........</div><div class="test-example-line test-example-line-even test-example-line-4">........</div><div class="test-example-line test-example-line-even test-example-line-4">........</div><div class="test-example-line test-example-line-even test-example-line-4">........</div><div class="test-example-line test-example-line-even test-example-line-4">........</div><div class="test-example-line test-example-line-even test-example-line-4">.......g</div><div class="test-example-line test-example-line-even test-example-line-4">.......a</div><div class="test-example-line test-example-line-even test-example-line-4">.......m</div><div class="test-example-line test-example-line-even test-example-line-4">.......e</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div><div class="test-example-line test-example-line-odd test-example-line-5">a.......</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
i
|
||
lost
|
||
the
|
||
game
|
||
aaaaaaaa
|
||
</pre></div></div></div></div><p> </p></div>
|
||
</div>
|
||
|
||
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|
||
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|
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|
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|
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|
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|
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|
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|
||
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|
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|
||
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|
||
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|
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|
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|
||
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|
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|
||
<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">D. Balanced Round</div><div class="time-limit"><div class="property-title">time limit per test</div>2 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>You are the author of a Codeforces round and have prepared $$$n$$$ problems you are going to set, problem $$$i$$$ having difficulty $$$a_i$$$. You will do the following process: </p><ul> <li> remove some (possibly zero) problems from the list; </li><li> rearrange the remaining problems in any order you wish. </li></ul><p>A round is considered <span class="tex-font-style-it">balanced</span> if and only if the absolute difference between the difficulty of any two consecutive problems is at most $$$k$$$ (less or equal than $$$k$$$).</p><p>What is the minimum number of problems you have to remove so that an arrangement of problems is balanced?</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.</p><p>The first line of each test case contains two positive integers $$$n$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$) and $$$k$$$ ($$$1 \leq k \leq 10^9$$$) — the number of problems, and the maximum allowed absolute difference between consecutive problems.</p><p>The second line of each test case contains $$$n$$$ space-separated integers $$$a_i$$$ ($$$1 \leq a_i \leq 10^9$$$) — the difficulty of each problem.</p><p>Note that the sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output a single integer — the minimum number of problems you have to remove so that an arrangement of problems is balanced.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">7</div><div class="test-example-line test-example-line-odd test-example-line-1">5 1</div><div class="test-example-line test-example-line-odd test-example-line-1">1 2 4 5 6</div><div class="test-example-line test-example-line-even test-example-line-2">1 2</div><div class="test-example-line test-example-line-even test-example-line-2">10</div><div class="test-example-line test-example-line-odd test-example-line-3">8 3</div><div class="test-example-line test-example-line-odd test-example-line-3">17 3 1 20 12 5 17 12</div><div class="test-example-line test-example-line-even test-example-line-4">4 2</div><div class="test-example-line test-example-line-even test-example-line-4">2 4 6 8</div><div class="test-example-line test-example-line-odd test-example-line-5">5 3</div><div class="test-example-line test-example-line-odd test-example-line-5">2 3 19 10 8</div><div class="test-example-line test-example-line-even test-example-line-6">3 4</div><div class="test-example-line test-example-line-even test-example-line-6">1 10 5</div><div class="test-example-line test-example-line-odd test-example-line-7">8 1</div><div class="test-example-line test-example-line-odd test-example-line-7">8 3 1 4 5 10 7 3</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
2
|
||
0
|
||
5
|
||
0
|
||
3
|
||
1
|
||
4
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>For the first test case, we can remove the first $$$2$$$ problems and construct a set using problems with the difficulties $$$[4, 5, 6]$$$, with difficulties between adjacent problems equal to $$$|5 - 4| = 1 \leq 1$$$ and $$$|6 - 5| = 1 \leq 1$$$.</p><p>For the second test case, we can take the single problem and compose a round using the problem with difficulty $$$10$$$.</p></div></div><p> </p></div>
|
||
</div>
|
||
|
||
<script>
|
||
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||
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let uuid = $problem.attr("data-uuid");
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let statementText = convertStatementToText($problem.find(".ttypography").get(0));
|
||
|
||
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|
||
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|
||
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|
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||
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|
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|
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|
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|
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|
||
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|
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|
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|
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|
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|
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|
||
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|
||
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|
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|
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|
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|
||
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|
||
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|
||
<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">E. Cardboard for Pictures</div><div class="time-limit"><div class="property-title">time limit per test</div>2 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>Mircea has $$$n$$$ pictures. The $$$i$$$-th picture is a square with a side length of $$$s_i$$$ centimeters. </p><p>He mounted each picture on a square piece of cardboard so that each picture has a border of $$$w$$$ centimeters of cardboard on all sides. In total, he used $$$c$$$ square centimeters of cardboard. Given the picture sizes and the value $$$c$$$, can you find the value of $$$w$$$?</p><center> <img class="tex-graphics" src="https://espresso.codeforces.com/9cc8c6f9c246d2c264aeb25fb11e193fda6b8f96.png" style="max-width: 100.0%;max-height: 100.0%;" /> <span class="tex-font-size-small">A picture of the first test case. Here $$$c = 50 = 5^2 + 4^2 + 3^2$$$, so $$$w=1$$$ is the answer.</span> </center><p>Please note that the piece of cardboard goes behind each picture, not just the border.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.</p><p>The first line of each test case contains two positive integers $$$n$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$) and $$$c$$$ ($$$1 \leq c \leq 10^{18}$$$) — the number of paintings, and the amount of used square centimeters of cardboard.</p><p>The second line of each test case contains $$$n$$$ space-separated integers $$$s_i$$$ ($$$1 \leq s_i \leq 10^4$$$) — the sizes of the paintings.</p><p>The sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$.</p><p><span class="tex-font-style-bf">Additional constraint on the input:</span> Such an integer $$$w$$$ exists for each test case.</p><p>Please note, that some of the input for some test cases won't fit into 32-bit integer type, so you should use at least 64-bit integer type in your programming language (like <span class="tex-font-style-tt">long long</span> for C++).</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output a single integer — the value of $$$w$$$ ($$$w \geq 1$$$) which was used to use exactly $$$c$$$ squared centimeters of cardboard.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre>
|
||
10
|
||
3 50
|
||
3 2 1
|
||
1 100
|
||
6
|
||
5 500
|
||
2 2 2 2 2
|
||
2 365
|
||
3 4
|
||
2 469077255466389
|
||
10000 2023
|
||
10 635472106413848880
|
||
9181 4243 7777 1859 2017 4397 14 9390 2245 7225
|
||
7 176345687772781240
|
||
9202 9407 9229 6257 7743 5738 7966
|
||
14 865563946464579627
|
||
3654 5483 1657 7571 1639 9815 122 9468 3079 2666 5498 4540 7861 5384
|
||
19 977162053008871403
|
||
9169 9520 9209 9013 9300 9843 9933 9454 9960 9167 9964 9701 9251 9404 9462 9277 9661 9164 9161
|
||
18 886531871815571953
|
||
2609 10 5098 9591 949 8485 6385 4586 1064 5412 6564 8460 2245 6552 5089 8353 3803 3764
|
||
</pre></div><div class="output"><div class="title">Output</div><pre>
|
||
1
|
||
2
|
||
4
|
||
5
|
||
7654321
|
||
126040443
|
||
79356352
|
||
124321725
|
||
113385729
|
||
110961227
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>The first test case is explained in the statement.</p><p>For the second test case, the chosen $$$w$$$ was $$$2$$$, thus the only cardboard covers an area of $$$c = (2 \cdot 2 + 6)^2 = 10^2 = 100$$$ squared centimeters.</p><p>For the third test case, the chosen $$$w$$$ was $$$4$$$, which obtains the covered area $$$c = (2 \cdot 4 + 2)^2 \times 5 = 10^2 \times 5 = 100 \times 5 = 500$$$ squared centimeters.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">F. We Were Both Children</div><div class="time-limit"><div class="property-title">time limit per test</div>3 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>Mihai and Slavic were looking at a group of $$$n$$$ frogs, numbered from $$$1$$$ to $$$n$$$, all initially located at point $$$0$$$. Frog $$$i$$$ has a hop length of $$$a_i$$$. </p><p>Each second, frog $$$i$$$ hops $$$a_i$$$ units forward. Before any frogs start hopping, Slavic and Mihai can place <span class="tex-font-style-bf">exactly one</span> trap in a coordinate in order to catch all frogs that will ever pass through the corresponding coordinate.</p><p>However, the children can't go far away from their home so they can only place a trap in the first $$$n$$$ points (that is, in a point with a coordinate between $$$1$$$ and $$$n$$$) and the children can't place a trap in point $$$0$$$ since they are scared of frogs.</p><p>Can you help Slavic and Mihai find out what is the maximum number of frogs they can catch using a trap?</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line of the input contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. The description of test cases follows.</p><p>The first line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$) — the number of frogs, which equals the distance Slavic and Mihai can travel to place a trap.</p><p>The second line of each test case contains $$$n$$$ integers $$$a_1, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^9$$$) — the lengths of the hops of the corresponding frogs.</p><p>It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case output a single integer — the maximum number of frogs Slavic and Mihai can catch using a trap.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">7</div><div class="test-example-line test-example-line-odd test-example-line-1">5</div><div class="test-example-line test-example-line-odd test-example-line-1">1 2 3 4 5</div><div class="test-example-line test-example-line-even test-example-line-2">3</div><div class="test-example-line test-example-line-even test-example-line-2">2 2 2</div><div class="test-example-line test-example-line-odd test-example-line-3">6</div><div class="test-example-line test-example-line-odd test-example-line-3">3 1 3 4 9 10</div><div class="test-example-line test-example-line-even test-example-line-4">9</div><div class="test-example-line test-example-line-even test-example-line-4">1 3 2 4 2 3 7 8 5</div><div class="test-example-line test-example-line-odd test-example-line-5">1</div><div class="test-example-line test-example-line-odd test-example-line-5">10</div><div class="test-example-line test-example-line-even test-example-line-6">8</div><div class="test-example-line test-example-line-even test-example-line-6">7 11 6 8 12 4 4 8</div><div class="test-example-line test-example-line-odd test-example-line-7">10</div><div class="test-example-line test-example-line-odd test-example-line-7">9 11 9 12 1 7 2 5 8 10</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
3
|
||
3
|
||
3
|
||
5
|
||
0
|
||
4
|
||
4
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first test case, the frogs will hop as follows: </p><ul> <li> Frog 1: $$$0 \to 1 \to 2 \to 3 \to \mathbf{\color{red}{4}} \to \cdots$$$ </li><li> Frog 2: $$$0 \to 2 \to \mathbf{\color{red}{4}} \to 6 \to 8 \to \cdots$$$ </li><li> Frog 3: $$$0 \to 3 \to 6 \to 9 \to 12 \to \cdots$$$ </li><li> Frog 4: $$$0 \to \mathbf{\color{red}{4}} \to 8 \to 12 \to 16 \to \cdots$$$ </li><li> Frog 5: $$$0 \to 5 \to 10 \to 15 \to 20 \to \cdots$$$ </li></ul> Therefore, if Slavic and Mihai put a trap at coordinate $$$4$$$, they can catch three frogs: frogs 1, 2, and 4. It can be proven that they can't catch any more frogs.<p>In the second test case, Slavic and Mihai can put a trap at coordinate $$$2$$$ and catch all three frogs instantly.</p></div></div><p> </p></div>
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||
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">G. The Morning Star</div><div class="time-limit"><div class="property-title">time limit per test</div>2 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>A compass points directly toward the morning star. It can only point in one of eight directions: the four cardinal directions (N, S, E, W) or some combination (NW, NE, SW, SE). Otherwise, it will break.</p><center> <img class="tex-graphics" src="https://espresso.codeforces.com/d4bc950948a1049d2c81071966d4259748cb892a.png" style="max-width: 100.0%;max-height: 100.0%;" /> <span class="tex-font-size-small">The directions the compass can point.</span> </center><p>There are $$$n$$$ distinct points with integer coordinates on a plane. How many ways can you put a compass at one point and the morning star at another so that the compass does not break?</p></div><div class="input-specification"><div class="section-title">Input</div><p>Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.</p><p>The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 2 \cdot 10^5$$$) — the number of points.</p><p>Then $$$n$$$ lines follow, each line containing two integers $$$x_i$$$, $$$y_i$$$ ($$$-10^9 \leq x_i, y_i \leq 10^9$$$) — the coordinates of each point, all points have distinct coordinates.</p><p>It is guaranteed that the sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output a single integer — the number of pairs of points that don't break the compass.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">5</div><div class="test-example-line test-example-line-odd test-example-line-1">3</div><div class="test-example-line test-example-line-odd test-example-line-1">0 0</div><div class="test-example-line test-example-line-odd test-example-line-1">-1 -1</div><div class="test-example-line test-example-line-odd test-example-line-1">1 1</div><div class="test-example-line test-example-line-even test-example-line-2">4</div><div class="test-example-line test-example-line-even test-example-line-2">4 5</div><div class="test-example-line test-example-line-even test-example-line-2">5 7</div><div class="test-example-line test-example-line-even test-example-line-2">6 9</div><div class="test-example-line test-example-line-even test-example-line-2">10 13</div><div class="test-example-line test-example-line-odd test-example-line-3">3</div><div class="test-example-line test-example-line-odd test-example-line-3">-1000000000 1000000000</div><div class="test-example-line test-example-line-odd test-example-line-3">0 0</div><div class="test-example-line test-example-line-odd test-example-line-3">1000000000 -1000000000</div><div class="test-example-line test-example-line-even test-example-line-4">5</div><div class="test-example-line test-example-line-even test-example-line-4">0 0</div><div class="test-example-line test-example-line-even test-example-line-4">2 2</div><div class="test-example-line test-example-line-even test-example-line-4">-1 5</div><div class="test-example-line test-example-line-even test-example-line-4">-1 10</div><div class="test-example-line test-example-line-even test-example-line-4">2 11</div><div class="test-example-line test-example-line-odd test-example-line-5">3</div><div class="test-example-line test-example-line-odd test-example-line-5">0 0</div><div class="test-example-line test-example-line-odd test-example-line-5">-1 2</div><div class="test-example-line test-example-line-odd test-example-line-5">1 -2</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
6
|
||
2
|
||
6
|
||
8
|
||
0
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first test case, any pair of points won't break the compass: </p><ul> <li> The compass is at $$$(0,0)$$$, the morning star is at $$$(-1,-1)$$$: the compass will point $$$\text{SW}$$$. </li><li> The compass is at $$$(0,0)$$$, the morning star is at $$$(1,1)$$$: the compass will point $$$\text{NE}$$$. </li><li> The compass is at $$$(-1,-1)$$$, the morning star is at $$$(0,0)$$$: the compass will point $$$\text{NE}$$$. </li><li> The compass is at $$$(-1,-1)$$$, the morning star is at $$$(1,1)$$$: the compass will point $$$\text{NE}$$$. </li><li> The compass is at $$$(1,1)$$$, the morning star is at $$$(0,0)$$$: the compass will point $$$\text{SW}$$$. </li><li> The compass is at $$$(1,1)$$$, the morning star is at $$$(-1,-1)$$$: the compass will point $$$\text{SW}$$$. </li></ul><p>In the second test case, only two pairs of points won't break the compass: </p><ul> <li> The compass is at $$$(6,9)$$$, the morning star is at $$$(10,13)$$$: the compass will point $$$\text{NE}$$$. </li><li> The compass is at $$$(10,13)$$$, the morning star is at $$$(6,9)$$$: the compass will point $$$\text{SW}$$$. </li></ul></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">H. The Third Letter</div><div class="time-limit"><div class="property-title">time limit per test</div>2 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file"><div class="property-title">input</div>standard input</div><div class="output-file"><div class="property-title">output</div>standard output</div></div><div><p>In order to win his toughest battle, Mircea came up with a great strategy for his army. He has $$$n$$$ soldiers and decided to arrange them in a certain way in camps. Each soldier has to belong to exactly one camp, and there is one camp at each integer point on the $$$x$$$-axis (at points $$$\cdots, -2, -1, 0, 1, 2, \cdots$$$).</p><p>The strategy consists of $$$m$$$ conditions. Condition $$$i$$$ tells that soldier $$$a_i$$$ should belong to a camp that is situated $$$d_i$$$ meters in front of the camp that person $$$b_i$$$ belongs to. (If $$$d_i < 0$$$, then $$$a_i$$$'s camp should be $$$-d_i$$$ meters behind $$$b_i$$$'s camp.)</p><p>Now, Mircea wonders if there exists a partition of soldiers that respects the condition and he asks for your help! Answer "<span class="tex-font-style-tt">YES</span>" if there is a partition of the $$$n$$$ soldiers that satisfies <span class="tex-font-style-bf">all</span> of the $$$m$$$ conditions and "<span class="tex-font-style-tt">NO</span>" otherwise.</p><p>Note that two different soldiers <span class="tex-font-style-bf">may</span> be placed in the same camp.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 100$$$) — the number of test cases.</p><p>The first line of each test case contains two positive integers $$$n$$$ and $$$m$$$ ($$$2 \leq n \leq 2 \cdot 10^5$$$; $$$1 \leq m \leq n$$$) — the number of soldiers, and the number of conditions respectively.</p><p>Then $$$m$$$ lines follow, each of them containing $$$3$$$ integers: $$$a_i$$$, $$$b_i$$$, $$$d_i$$$ ($$$a_i \neq b_i$$$; $$$1 \leq a_i, b_i \leq n$$$; $$$-10^9 \leq d_i \leq 10^9$$$) — denoting the conditions explained in the statement. Note that if $$$d_i$$$ is positive, $$$a_i$$$ should be $$$d_i$$$ meters in front of $$$b_i$$$ and if it is negative, $$$a_i$$$ should be $$$-d_i$$$ meters behind $$$b_i$$$.</p><p>Note that the sum of $$$n$$$ over all test cases doesn't exceed $$$2 \cdot 10^5$$$.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output "<span class="tex-font-style-tt">YES</span>" if there is an arrangement of the $$$n$$$ soldiers that satisfies <span class="tex-font-style-bf">all</span> of the $$$m$$$ conditions and "<span class="tex-font-style-tt">NO</span>" otherwise.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">4</div><div class="test-example-line test-example-line-odd test-example-line-1">5 3</div><div class="test-example-line test-example-line-odd test-example-line-1">1 2 2</div><div class="test-example-line test-example-line-odd test-example-line-1">2 3 4</div><div class="test-example-line test-example-line-odd test-example-line-1">4 2 -6</div><div class="test-example-line test-example-line-even test-example-line-2">6 5</div><div class="test-example-line test-example-line-even test-example-line-2">1 2 2</div><div class="test-example-line test-example-line-even test-example-line-2">2 3 4</div><div class="test-example-line test-example-line-even test-example-line-2">4 2 -6</div><div class="test-example-line test-example-line-even test-example-line-2">5 4 4</div><div class="test-example-line test-example-line-even test-example-line-2">3 5 100</div><div class="test-example-line test-example-line-odd test-example-line-3">2 2</div><div class="test-example-line test-example-line-odd test-example-line-3">1 2 5</div><div class="test-example-line test-example-line-odd test-example-line-3">1 2 4</div><div class="test-example-line test-example-line-even test-example-line-4">4 1</div><div class="test-example-line test-example-line-even test-example-line-4">1 2 3</div></pre></div><div class="output"><div class="title">Output</div><pre>
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YES
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NO
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NO
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YES
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</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>For the first test case, we can partition the soldiers into camps in the following way: soldier:</p><ul> <li> Soldier $$$1$$$ in the camp with the coordinate $$$x = 3$$$. </li><li> Soldier $$$2$$$ in the camp with the coordinate $$$x = 5$$$. </li><li> Soldier $$$3$$$ in the camp with the coordinate $$$x = 9$$$. </li><li> Soldier $$$4$$$ in the camp with the coordinate $$$x = 11$$$. </li></ul><p>For the second test case, there is no partition that can satisfy all the constraints at the same time.</p><p>For the third test case, there is no partition that satisfies all the constraints since we get contradictory information about the same pair.</p><p>For the fourth test case, in order to satisfy the only condition, a possible partition is:</p><ul> <li> Soldier $$$1$$$ in the camp with the coordinate $$$x = 10$$$. </li><li> Soldier $$$2$$$ in the camp with the coordinate $$$x = 13$$$. </li><li> Soldier $$$3$$$ in the camp with the coordinate $$$x = -2023$$$. </li><li> Soldier $$$4$$$ in the camp with the coordinate $$$x = -2023$$$. </li></ul></div></div><p> </p></div>
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