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Matej Focko
9c74f2847d
* “A. Escalator Conversations” * “B. Parity Sort” * “C. Tiles Comeback” Signed-off-by: Matej Focko <me@mfocko.xyz>
1279 lines
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">A. Escalator Conversations</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>One day, Vlad became curious about who he can have a conversation with on the escalator in the subway. There are a total of $$$n$$$ passengers. The escalator has a total of $$$m$$$ steps, all steps indexed from $$$1$$$ to $$$m$$$ and $$$i$$$-th step has height $$$i \cdot k$$$.</p><p>Vlad's height is $$$H$$$ centimeters. Two people with heights $$$a$$$ and $$$b$$$ can have a conversation on the escalator if they are standing on <span class="tex-font-style-bf">different</span> steps and the height difference between them is equal to the height difference between the steps.</p><p>For example, if two people have heights $$$170$$$ and $$$180$$$ centimeters, and $$$m = 10, k = 5$$$, then they can stand on steps numbered $$$7$$$ and $$$5$$$, where the height difference between the steps is equal to the height difference between the two people: $$$k \cdot 2 = 5 \cdot 2 = 10 = 180 - 170$$$. There are other possible ways.</p><p>Given an array $$$h$$$ of size $$$n$$$, where $$$h_i$$$ represents the height of the $$$i$$$-th person. Vlad is interested in how many people he can have a conversation with on the escalator <span class="tex-font-style-bf">individually</span>.</p><p>For example, if $$$n = 5, m = 3, k = 3, H = 11$$$, and $$$h = [5, 4, 14, 18, 2]$$$, Vlad can have a conversation with the person with height $$$5$$$ (Vlad will stand on step $$$1$$$, and the other person will stand on step $$$3$$$) and with the person with height $$$14$$$ (for example, Vlad can stand on step $$$3$$$, and the other person will stand on step $$$2$$$). Vlad cannot have a conversation with the person with height $$$2$$$ because even if they stand on the extreme steps of the escalator, the height difference between them will be $$$6$$$, while their height difference is $$$9$$$. Vlad cannot have a conversation with the rest of the people on the escalator, so the answer for this example is $$$2$$$.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases.</p><p>Then the descriptions of the test cases follow.</p><p>The first line of each test case contains integers: $$$n, m, k, H$$$ ($$$1 \le n,m \le 50$$$, $$$1 \le k,H \le 10^6$$$). Here, $$$n$$$ is the number of people, $$$m$$$ is the number of steps, $$$k$$$ is the height difference between neighboring steps, and $$$H$$$ is Vlad's height.</p><p>The second line contains $$$n$$$ integers: $$$h_1, h_2, \ldots, h_n$$$ ($$$1 \le h_i \le 10^6$$$). Here, $$$h_i$$$ represents the height of the $$$i$$$-th person.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, output a single integer — the number of people Vlad can have a conversation with on the escalator <span class="tex-font-style-bf">individually</span>.</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 3 3 11</div><div class="test-example-line test-example-line-odd test-example-line-1">5 4 14 18 2</div><div class="test-example-line test-example-line-even test-example-line-2">2 9 5 6</div><div class="test-example-line test-example-line-even test-example-line-2">11 9</div><div class="test-example-line test-example-line-odd test-example-line-3">10 50 3 11</div><div class="test-example-line test-example-line-odd test-example-line-3">43 44 74 98 62 60 99 4 11 73</div><div class="test-example-line test-example-line-even test-example-line-4">4 8 8 49</div><div class="test-example-line test-example-line-even test-example-line-4">68 58 82 73</div><div class="test-example-line test-example-line-odd test-example-line-5">7 1 4 66</div><div class="test-example-line test-example-line-odd test-example-line-5">18 66 39 83 48 99 79</div><div class="test-example-line test-example-line-even test-example-line-6">9 1 1 13</div><div class="test-example-line test-example-line-even test-example-line-6">26 23 84 6 60 87 40 41 25</div><div class="test-example-line test-example-line-odd test-example-line-7">6 13 3 28</div><div class="test-example-line test-example-line-odd test-example-line-7">30 70 85 13 1 55</div></pre></div><div class="output"><div class="title">Output</div><pre>
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2
|
||
1
|
||
4
|
||
1
|
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0
|
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0
|
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3
|
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</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>The first example is explained in the problem statement.</p><p>In the second example, Vlad can have a conversation with the person with height $$$11$$$.</p><p>In the third example, Vlad can have a conversation with people with heights: $$$44, 74, 98, 62$$$. Therefore, the answer is $$$4$$$.</p><p>In the fourth example, Vlad can have a conversation with the person with height $$$73$$$.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">B. Parity Sort</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 have an array of integers $$$a$$$ of length $$$n$$$. You can apply the following operation to the given array: </p><ul> <li> Swap two elements $$$a_i$$$ and $$$a_j$$$ such that $$$i \neq j$$$, $$$a_i$$$ and $$$a_j$$$ are either <span class="tex-font-style-bf">both</span> even or <span class="tex-font-style-bf">both</span> odd. </li></ul><p> Determine whether it is possible to sort the array in non-decreasing order by performing the operation any number of times (possibly zero).</p><p>For example, let $$$a$$$ = [$$$7, 10, 1, 3, 2$$$]. Then we can perform $$$3$$$ operations to sort the array: </p><ol> <li> Swap $$$a_3 = 1$$$ and $$$a_1 = 7$$$, since $$$1$$$ and $$$7$$$ are odd. We get $$$a$$$ = [$$$1, 10, 7, 3, 2$$$]; </li><li> Swap $$$a_2 = 10$$$ and $$$a_5 = 2$$$, since $$$10$$$ and $$$2$$$ are even. We get $$$a$$$ = [$$$1, 2, 7, 3, 10$$$]; </li><li> Swap $$$a_4 = 3$$$ and $$$a_3 = 7$$$, since $$$3$$$ and $$$7$$$ are odd. We get $$$a$$$ = [$$$1, 2, 3, 7, 10$$$]. </li></ol></div><div class="input-specification"><div class="section-title">Input</div><p>The first line of input data contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.</p><p>The description of the test cases follows.</p><p>The first line of each test case contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the length of array $$$a$$$.</p><p>The second line of each test case contains exactly $$$n$$$ positive integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$) — the elements of array $$$a$$$.</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 on a separate line: </p><ul> <li> <span class="tex-font-style-tt">YES</span> if the array can be sorted by applying the operation to it some number of times; </li><li> <span class="tex-font-style-tt">NO</span> otherwise. </li></ul><p>You can output <span class="tex-font-style-tt">YES</span> and <span class="tex-font-style-tt">NO</span> in any case (for example, 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 positive response).</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">6</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 10 1 3 2</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">11 9 3 5</div><div class="test-example-line test-example-line-odd test-example-line-3">5</div><div class="test-example-line test-example-line-odd test-example-line-3">11 3 15 3 2</div><div class="test-example-line test-example-line-even test-example-line-4">6</div><div class="test-example-line test-example-line-even test-example-line-4">10 7 8 1 2 3</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">5</div><div class="test-example-line test-example-line-even test-example-line-6">6 6 4 1 6</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
YES
|
||
YES
|
||
NO
|
||
NO
|
||
YES
|
||
NO</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>The first test case is explained in the problem statement.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">C. Tiles Comeback</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>Vlad remembered that he had a series of $$$n$$$ tiles and a number $$$k$$$. The tiles were numbered from left to right, and the $$$i$$$-th tile had colour $$$c_i$$$.</p><p>If you stand on the <span class="tex-font-style-bf">first</span> tile and start jumping any number of tiles <span class="tex-font-style-bf">right</span>, you can get a path of length $$$p$$$. The length of the path is the number of tiles you stood on.</p><p>Vlad wants to see if it is possible to get a path of length $$$p$$$ such that: </p><ul> <li> it ends at tile with index $$$n$$$; </li><li> $$$p$$$ is divisible by $$$k$$$ </li><li> the path is divided into blocks of length exactly $$$k$$$ each; </li><li> tiles in each block have the same colour, the colors in adjacent blocks are not necessarily different. </li></ul><p>For example, let $$$n = 14$$$, $$$k = 3$$$.</p><p>The colours of the tiles are contained in the array $$$c$$$ = [$$$\color{red}{1}, \color{violet}{2}, \color{red}{1}, \color{red}{1}, \color{violet}{2}, \color{violet}{2}, \color{green}{3}, \color{green}{3}, \color{red}{1}, \color{green}{3}, \color{blue}{4}, \color{blue}{4}, \color{violet}{2}, \color{blue}{4}$$$]. Then we can construct a path of length $$$6$$$ consisting of $$$2$$$ blocks:</p><p>$$$\color{red}{c_1} \rightarrow \color{red}{c_3} \rightarrow \color{red}{c_4} \rightarrow \color{blue}{c_{11}} \rightarrow \color{blue}{c_{12}} \rightarrow \color{blue}{c_{14}}$$$</p><p>All tiles from the $$$1$$$-st block will have colour $$$\color{red}{\textbf{1}}$$$, from the $$$2$$$-nd block will have colour $$$\color{blue}{\textbf{4}}$$$.</p><p>It is also possible to construct a path of length $$$9$$$ in this example, in which all tiles from the $$$1$$$-st block will have colour $$$\color{red}{\textbf{1}}$$$, from the $$$2$$$-nd block will have colour $$$\color{green}{\textbf{3}}$$$, and from the $$$3$$$-rd block will have colour $$$\color{blue}{\textbf{4}}$$$.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line of input data contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.</p><p>The description of the test cases follows.</p><p>The first line of each test case ontains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 2 \cdot 10^5$$$)—the number of tiles in the series and the length of the block.</p><p>The second line of each test case contains $$$n$$$ integers $$$c_1, c_2, c_3, \dots, c_n$$$ ($$$1 \le c_i \le n$$$) — the colours of the tiles.</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 on a separate line: </p><ul> <li> <span class="tex-font-style-tt">YES</span> if you can get a path that satisfies these conditions; </li><li> <span class="tex-font-style-tt">NO</span> otherwise. </li></ul><p>You can output <span class="tex-font-style-tt">YES</span> and <span class="tex-font-style-tt">NO</span> in any case (for example, 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 positive response).</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">10</div><div class="test-example-line test-example-line-odd test-example-line-1">4 2</div><div class="test-example-line test-example-line-odd test-example-line-1">1 1 1 1</div><div class="test-example-line test-example-line-even test-example-line-2">14 3</div><div class="test-example-line test-example-line-even test-example-line-2">1 2 1 1 7 5 3 3 1 3 4 4 2 4</div><div class="test-example-line test-example-line-odd test-example-line-3">3 3</div><div class="test-example-line test-example-line-odd test-example-line-3">3 1 3</div><div class="test-example-line test-example-line-even test-example-line-4">10 4</div><div class="test-example-line test-example-line-even test-example-line-4">1 2 1 2 1 2 1 2 1 2</div><div class="test-example-line test-example-line-odd test-example-line-5">6 2</div><div class="test-example-line test-example-line-odd test-example-line-5">1 3 4 1 6 6</div><div class="test-example-line test-example-line-even test-example-line-6">2 2</div><div class="test-example-line test-example-line-even test-example-line-6">1 1</div><div class="test-example-line test-example-line-odd test-example-line-7">4 2</div><div class="test-example-line test-example-line-odd test-example-line-7">2 1 1 1</div><div class="test-example-line test-example-line-even test-example-line-8">2 1</div><div class="test-example-line test-example-line-even test-example-line-8">1 2</div><div class="test-example-line test-example-line-odd test-example-line-9">3 2</div><div class="test-example-line test-example-line-odd test-example-line-9">2 2 2</div><div class="test-example-line test-example-line-even test-example-line-10">4 1</div><div class="test-example-line test-example-line-even test-example-line-10">1 1 2 2</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
YES
|
||
YES
|
||
NO
|
||
NO
|
||
YES
|
||
YES
|
||
NO
|
||
YES
|
||
YES
|
||
YES
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first test case, you can jump from the first tile to the last tile;</p><p>The second test case is explained in the problem statement.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">D. Prefix Permutation Sums</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>Your friends have an array of $$$n$$$ elements, calculated its array of prefix sums and passed it to you, accidentally losing one element during the transfer. Your task is to find out if the given array can matches <span class="tex-font-style-bf">permutation</span>.</p><p>A permutation of $$$n$$$ elements is an array of $$$n$$$ numbers from $$$1$$$ to $$$n$$$ such that each number occurs exactly <span class="tex-font-style-bf">one</span> times in it.</p><p>The array of prefix sums of the array $$$a$$$ — is such an array $$$b$$$ that $$$b_i = \sum_{j=1}^i a_j, 1 \le i \le n$$$.</p><p>For example, the original permutation was $$$[1, 5, 2, 4, 3]$$$. Its array of prefix sums — $$$[1, 6, 8, 12, 15]$$$. Having lost one element, you can get, for example, arrays $$$[6, 8, 12, 15]$$$ or $$$[1, 6, 8, 15]$$$.</p><p>It can also be shown that the array $$$[1, 2, 100]$$$ does not correspond to any permutation.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a positive number $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.</p><p>The first line of the description of each test case contains a positive number $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$) — the size of the initial array.</p><p>The second line of the description of each test case contains $$$n - 1$$$ positive number $$$a_i$$$ ($$$1 \le a_i \le 10^{18}$$$), $$$a_{i-1} < a_i$$$ — elements of the array of prefix sums.</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 "<span class="tex-font-style-tt">YES</span>" if such a permutation exists, and "<span class="tex-font-style-tt">NO</span>" otherwise.</p><p>You can output "<span class="tex-font-style-tt">YES</span>" and "<span class="tex-font-style-tt">NO</span>" in any case (for example, the strings "<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 response).</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">12</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">6 8 12 15</div><div class="test-example-line test-example-line-even test-example-line-2">5</div><div class="test-example-line test-example-line-even test-example-line-2">1 6 8 15</div><div class="test-example-line test-example-line-odd test-example-line-3">4</div><div class="test-example-line test-example-line-odd test-example-line-3">1 2 100</div><div class="test-example-line test-example-line-even test-example-line-4">4</div><div class="test-example-line test-example-line-even test-example-line-4">1 3 6</div><div class="test-example-line test-example-line-odd test-example-line-5">2</div><div class="test-example-line test-example-line-odd test-example-line-5">2</div><div class="test-example-line test-example-line-even test-example-line-6">3</div><div class="test-example-line test-example-line-even test-example-line-6">1 2</div><div class="test-example-line test-example-line-odd test-example-line-7">4</div><div class="test-example-line test-example-line-odd test-example-line-7">3 7 10</div><div class="test-example-line test-example-line-even test-example-line-8">5</div><div class="test-example-line test-example-line-even test-example-line-8">5 44 46 50</div><div class="test-example-line test-example-line-odd test-example-line-9">4</div><div class="test-example-line test-example-line-odd test-example-line-9">1 9 10</div><div class="test-example-line test-example-line-even test-example-line-10">5</div><div class="test-example-line test-example-line-even test-example-line-10">13 21 36 42</div><div class="test-example-line test-example-line-odd test-example-line-11">5</div><div class="test-example-line test-example-line-odd test-example-line-11">1 2 3 1000000000000000000</div><div class="test-example-line test-example-line-even test-example-line-12">9</div><div class="test-example-line test-example-line-even test-example-line-12">9 11 12 20 25 28 30 33</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
YES
|
||
YES
|
||
NO
|
||
YES
|
||
YES
|
||
NO
|
||
YES
|
||
NO
|
||
NO
|
||
NO
|
||
NO
|
||
NO
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the fourth example, for example, the permutation $$$[1, 2, 3, 4]$$$ is suitable. In the fifth example, for example, the permutation $$$[1, 2]$$$ is suitable. In the seventh example, for example, the permutation $$$[1, 2, 4, 3]$$$ is suitable.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">E. Nastya and Potions</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>Alchemist Nastya loves mixing potions. There are a total of $$$n$$$ types of potions, and one potion of type $$$i$$$ can be bought for $$$c_i$$$ coins.</p><p>Any kind of potions can be obtained in no more than one way, by mixing from several others. The potions used in the mixing process will be <span class="tex-font-style-bf">consumed</span>. Moreover, no potion can be obtained from itself through one or more mixing processes.</p><p>As an experienced alchemist, Nastya has an <span class="tex-font-style-bf">unlimited</span> supply of $$$k$$$ types of potions $$$p_1, p_2, \dots, p_k$$$, but she doesn't know which one she wants to obtain next. To decide, she asks you to find, for each $$$1 \le i \le n$$$, the minimum number of coins she needs to spend to obtain a potion of type $$$i$$$ next.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line of each test contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.</p><p>Each test case is described as follows:</p><p>The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k < n \le 2 \cdot 10^5$$$) — the total number of potion types and the number of potion types Nastya already has.</p><p>The second line contains $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$1 \le c_i \le 10^9$$$) — the costs of buying the potions.</p><p>The third line contains $$$k$$$ distinct integers $$$p_1, p_2, \dots, p_k$$$ ($$$1 \le p_i \le n$$$) — the indices of potions Nastya already has an <span class="tex-font-style-bf">unlimited</span> supply of.</p><p>This is followed by $$$n$$$ lines describing ways to obtain potions by mixing.</p><p>Each line starts with the integer $$$m_i$$$ ($$$0 \le m_i < n$$$) — the number of potions required to mix a potion of the type $$$i$$$ ($$$1 \le i \le n$$$).</p><p>Then line contains $$$m_i$$$ distinct integers $$$e_1, e_2, \dots, e_{m_i}$$$ ($$$1 \le e_j \le n$$$, $$$e_j \ne i$$$) — the indices of potions needed to mix a potion of the type $$$i$$$. If this list is empty, then a potion of the type $$$i$$$ can only be bought.</p><p>It is guaranteed that no potion can be obtained from itself through one or more mixing processes.</p><p>It is guaranteed that the sum of all $$$n$$$ values across all test cases does not exceed $$$2 \cdot 10^5$$$. Similarly, it is guaranteed that the sum of all $$$m_i$$$ values across 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 $$$n$$$ integers — the minimum number of coins Nastya needs to spend to obtain a potion of each type.</p></div><div class="sample-tests"><div class="section-title">Examples</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 1</div><div class="test-example-line test-example-line-odd test-example-line-1">30 8 3 5 10</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">3 2 4 5</div><div class="test-example-line test-example-line-odd test-example-line-1">0 </div><div class="test-example-line test-example-line-odd test-example-line-1">0 </div><div class="test-example-line test-example-line-odd test-example-line-1">2 3 5</div><div class="test-example-line test-example-line-odd test-example-line-1">0 </div><div class="test-example-line test-example-line-even test-example-line-2">3 2</div><div class="test-example-line test-example-line-even test-example-line-2">5 143 3</div><div class="test-example-line test-example-line-even test-example-line-2">1 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">0 </div><div class="test-example-line test-example-line-even test-example-line-2">2 1 2</div><div class="test-example-line test-example-line-odd test-example-line-3">5 1</div><div class="test-example-line test-example-line-odd test-example-line-3">5 4 1 3 4</div><div class="test-example-line test-example-line-odd test-example-line-3">2</div><div class="test-example-line test-example-line-odd test-example-line-3">2 4 5</div><div class="test-example-line test-example-line-odd test-example-line-3">3 3 5 4</div><div class="test-example-line test-example-line-odd test-example-line-3">2 1 4</div><div class="test-example-line test-example-line-odd test-example-line-3">1 5</div><div class="test-example-line test-example-line-odd test-example-line-3">0 </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">1 1 5 4</div><div class="test-example-line test-example-line-even test-example-line-4">2 4</div><div class="test-example-line test-example-line-even test-example-line-4">3 2 4 3</div><div class="test-example-line test-example-line-even test-example-line-4">0 </div><div class="test-example-line test-example-line-even test-example-line-4">2 2 4</div><div class="test-example-line test-example-line-even test-example-line-4">1 2</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
23 8 0 5 10
|
||
0 143 0
|
||
5 0 1 3 4
|
||
0 0 0 0
|
||
</pre></div><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">6 3</div><div class="test-example-line test-example-line-odd test-example-line-1">5 5 4 5 2 2</div><div class="test-example-line test-example-line-odd test-example-line-1">3 4 5</div><div class="test-example-line test-example-line-odd test-example-line-1">2 2 5</div><div class="test-example-line test-example-line-odd test-example-line-1">1 5</div><div class="test-example-line test-example-line-odd test-example-line-1">3 4 1 6</div><div class="test-example-line test-example-line-odd test-example-line-1">4 2 6 1 5</div><div class="test-example-line test-example-line-odd test-example-line-1">0 </div><div class="test-example-line test-example-line-odd test-example-line-1">0 </div><div class="test-example-line test-example-line-even test-example-line-2">6 2</div><div class="test-example-line test-example-line-even test-example-line-2">1 4 4 1 5 2</div><div class="test-example-line test-example-line-even test-example-line-2">3 6</div><div class="test-example-line test-example-line-even test-example-line-2">4 6 3 4 5</div><div class="test-example-line test-example-line-even test-example-line-2">4 6 5 3 4</div><div class="test-example-line test-example-line-even test-example-line-2">0 </div><div class="test-example-line test-example-line-even test-example-line-2">1 5</div><div class="test-example-line test-example-line-even test-example-line-2">1 6</div><div class="test-example-line test-example-line-even test-example-line-2">0 </div><div class="test-example-line test-example-line-odd test-example-line-3">2 1</div><div class="test-example-line test-example-line-odd test-example-line-3">4 3</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">0 </div><div class="test-example-line test-example-line-odd test-example-line-3">1 1</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
0 0 0 0 0 2
|
||
0 0 0 0 0 0
|
||
0 0
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first test case of the first sample, it is optimal:</p><ul> <li> Get a potion of the first type by buying and mixing $$$2$$$, $$$3$$$ and $$$4$$$; </li><li> a potion of the second type can only be obtained by purchasing it; </li><li> Nastya already has an unlimited number of potions of the third type; </li><li> a potion of the fourth type is more profitable to buy than to buy and mix other potions; </li><li> a potion of the fifth type can only be obtained by purchasing it. </li></ul></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">F. Lisa and the Martians</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>512 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>Lisa was kidnapped by martians! It okay, because she has watched a lot of TV shows about aliens, so she knows what awaits her. Let's call integer <span class="tex-font-style-it">martian</span> if it is <span class="tex-font-style-bf">a non-negative integer</span> and <span class="tex-font-style-bf">strictly less than</span> $$$2^k$$$, for example, when $$$k = 12$$$, the numbers $$$51$$$, $$$1960$$$, $$$0$$$ are <span class="tex-font-style-it">martian</span>, and the numbers $$$\pi$$$, $$$-1$$$, $$$\frac{21}{8}$$$, $$$4096$$$ are not.</p><p>The aliens will give Lisa $$$n$$$ <span class="tex-font-style-it">martian</span> numbers $$$a_1, a_2, \ldots, a_n$$$. Then they will ask her to name any <span class="tex-font-style-it">martian</span> number $$$x$$$. After that, Lisa will select a pair of numbers $$$a_i, a_j$$$ ($$$i \neq j$$$) in the given sequence and count $$$(a_i \oplus x) \& (a_j \oplus x)$$$. The operation $$$\oplus$$$ means <a href="http://tiny.cc/xor_wiki">Bitwise exclusive OR</a>, the operation $$$\&$$$ means <a href="http://tiny.cc/and_wiki ">Bitwise And</a>. For example, $$$(5 \oplus 17) \& (23 \oplus 17) = (00101_2 \oplus 10001_2) \& (10111_2 \oplus 10001_2) = 10100_2 \& 00110_2 = 00100_2 = 4$$$.</p><p>Lisa is sure that the higher the calculated value, the higher her chances of returning home. Help the girl choose such $$$i, j, x$$$ that maximize the calculated value.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — number of testcases.</p><p>Each testcase is described by two lines.</p><p>The first line contains integers $$$n, k$$$ ($$$2 \le n \le 2 \cdot 10^5$$$, $$$1 \le k \le 30$$$) — the length of the sequence of <span class="tex-font-style-it">martian</span> numbers and the value of $$$k$$$.</p><p>The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i < 2^k$$$) — a sequence of <span class="tex-font-style-it">martian</span> numbers.</p><p>It is guaranteed that the sum of $$$n$$$ over all testcases <span class="tex-font-style-bf">does not exceed</span> $$$2 \cdot 10^5$$$.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each testcase, print three integers $$$i, j, x$$$ ($$$1 \le i, j \le n$$$, $$$i \neq j$$$, $$$0 \le x < 2^k$$$). The value of $$$(a_i \oplus x) \& (a_j \oplus x)$$$ should be the maximum possible.</p><p>If there are several solutions, you can print any one.</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">10</div><div class="test-example-line test-example-line-odd test-example-line-1">5 4</div><div class="test-example-line test-example-line-odd test-example-line-1">3 9 1 4 13</div><div class="test-example-line test-example-line-even test-example-line-2">3 1</div><div class="test-example-line test-example-line-even test-example-line-2">1 0 1</div><div class="test-example-line test-example-line-odd test-example-line-3">6 12</div><div class="test-example-line test-example-line-odd test-example-line-3">144 1580 1024 100 9 13</div><div class="test-example-line test-example-line-even test-example-line-4">4 3</div><div class="test-example-line test-example-line-even test-example-line-4">7 3 0 4</div><div class="test-example-line test-example-line-odd test-example-line-5">3 2</div><div class="test-example-line test-example-line-odd test-example-line-5">0 0 1</div><div class="test-example-line test-example-line-even test-example-line-6">2 4</div><div class="test-example-line test-example-line-even test-example-line-6">12 2</div><div class="test-example-line test-example-line-odd test-example-line-7">9 4</div><div class="test-example-line test-example-line-odd test-example-line-7">6 14 9 4 4 4 5 10 2</div><div class="test-example-line test-example-line-even test-example-line-8">2 1</div><div class="test-example-line test-example-line-even test-example-line-8">1 0</div><div class="test-example-line test-example-line-odd test-example-line-9">2 4</div><div class="test-example-line test-example-line-odd test-example-line-9">11 4</div><div class="test-example-line test-example-line-even test-example-line-10">9 4</div><div class="test-example-line test-example-line-even test-example-line-10">2 11 10 1 6 9 11 0 5</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
1 3 14
|
||
1 3 0
|
||
5 6 4082
|
||
2 3 7
|
||
1 2 3
|
||
1 2 15
|
||
4 5 11
|
||
1 2 0
|
||
1 2 0
|
||
2 7 4
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>First testcase: $$$(3 \oplus 14) \& (1 \oplus 14) = (0011_2 \oplus 1110_2) \& (0001_2 \oplus 1110_2) = 1101_2 = 1101_2 \& 1111_2 = 1101_2 = 13$$$.</p><p>Second testcase: $$$(1 \oplus 0) \& (1 \oplus 0) = 1$$$.</p><p>Third testcase: $$$(9 \oplus 4082) \& (13 \oplus 4082) = 4091$$$.</p><p>Fourth testcase: $$$(3 \oplus 7) \& (0 \oplus 7) = 4$$$.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">G. Vlad and the Mountains</div><div class="time-limit"><div class="property-title">time limit per test</div>5 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>Vlad decided to go on a trip to the mountains. He plans to move between $$$n$$$ mountains, some of which are connected by roads. The $$$i$$$-th mountain has a height of $$$h_i$$$.</p><p>If there is a road between mountains $$$i$$$ and $$$j$$$, Vlad can move from mountain $$$i$$$ to mountain $$$j$$$ by spending $$$h_j - h_i$$$ units of energy. If his energy drops below zero during the transition, he will not be able to move from mountain $$$i$$$ to mountain $$$j$$$. Note that $$$h_j - h_i$$$ can be negative and then the energy will be restored.</p><p>Vlad wants to consider different route options, so he asks you to answer the following queries: is it possible to construct some route starting at mountain $$$a$$$ and ending at mountain $$$b$$$, given that he initially has $$$e$$$ units of energy?</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line of the input contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.</p><p>The descriptions of the test cases follow.</p><p>The first line of each test case contains two numbers $$$n$$$ and $$$m$$$ ($$$2 \le n \le 2 \cdot 10^5$$$, $$$1 \le m \le \min(\frac{n\cdot(n - 1)}{2}, 2 \cdot 10^5)$$$) — the number of mountains and the number of roads between them, respectively.</p><p>The second line contains $$$n$$$ integers $$$h_1, h_2, h_3, \dots, h_n$$$ ($$$1 \le h_i \le 10^9$$$) — the heights of the mountains.</p><p>The next $$$m$$$ lines contain two integers $$$u$$$ and $$$v$$$ ($$$1 \le u, v \le n$$$, $$$u \ne v$$$) — the numbers of the mountains connected by a road. It is guaranteed that no road appears twice.</p><p>The next line contains an integer $$$q$$$ ($$$1 \le q \le 2 \cdot 10^5$$$) — the number of queries.</p><p>The following $$$q$$$ lines contain three numbers $$$a$$$, $$$b$$$, and $$$e$$$ ($$$1 \le a, b \le n$$$, $$$0 \le e \le 10^9$$$) — the initial and final mountains of the route, and the amount of energy, respectively.</p><p>It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$. The same guarantee applies to $$$m$$$ and $$$q$$$.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each query, output "<span class="tex-font-style-tt">YES</span>" if Vlad can construct a route from mountain $$$a$$$ to mountain $$$b$$$, 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><p>In the examples below, the answers for different test cases are separated by an empty line, which you do not need to output.</p></div><div class="sample-tests"><div class="section-title">Examples</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">2</div><div class="test-example-line test-example-line-odd test-example-line-1">7 7</div><div class="test-example-line test-example-line-odd test-example-line-1">1 5 3 4 2 4 1</div><div class="test-example-line test-example-line-odd test-example-line-1">1 4</div><div class="test-example-line test-example-line-odd test-example-line-1">4 3</div><div class="test-example-line test-example-line-odd test-example-line-1">3 6</div><div class="test-example-line test-example-line-odd test-example-line-1">3 2</div><div class="test-example-line test-example-line-odd test-example-line-1">2 5</div><div class="test-example-line test-example-line-odd test-example-line-1">5 6</div><div class="test-example-line test-example-line-odd test-example-line-1">5 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 1 3</div><div class="test-example-line test-example-line-odd test-example-line-1">6 2 0</div><div class="test-example-line test-example-line-odd test-example-line-1">4 7 0</div><div class="test-example-line test-example-line-odd test-example-line-1">1 7 4</div><div class="test-example-line test-example-line-odd test-example-line-1">1 7 2</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">4 7 6 2 5 1</div><div class="test-example-line test-example-line-even test-example-line-2">1 3</div><div class="test-example-line test-example-line-even test-example-line-2">5 3</div><div class="test-example-line test-example-line-even test-example-line-2">1 5</div><div class="test-example-line test-example-line-even test-example-line-2">2 4</div><div class="test-example-line test-example-line-even test-example-line-2">6 2</div><div class="test-example-line test-example-line-even test-example-line-2">5</div><div class="test-example-line test-example-line-even test-example-line-2">1 5 1</div><div class="test-example-line test-example-line-even test-example-line-2">1 3 1</div><div class="test-example-line test-example-line-even test-example-line-2">1 2 1000</div><div class="test-example-line test-example-line-even test-example-line-2">6 2 6</div><div class="test-example-line test-example-line-even test-example-line-2">6 2 5</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
YES
|
||
NO
|
||
YES
|
||
YES
|
||
NO
|
||
|
||
YES
|
||
NO
|
||
NO
|
||
YES
|
||
NO
|
||
</pre></div><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">2</div><div class="test-example-line test-example-line-odd test-example-line-1">3 2</div><div class="test-example-line test-example-line-odd test-example-line-1">1 3 9</div><div class="test-example-line test-example-line-odd test-example-line-1">1 2</div><div class="test-example-line test-example-line-odd test-example-line-1">2 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">1 1 1</div><div class="test-example-line test-example-line-odd test-example-line-1">3 2 2</div><div class="test-example-line test-example-line-odd test-example-line-1">1 1 2</div><div class="test-example-line test-example-line-odd test-example-line-1">3 3 0</div><div class="test-example-line test-example-line-odd test-example-line-1">1 2 1</div><div class="test-example-line test-example-line-even test-example-line-2">3 3</div><div class="test-example-line test-example-line-even test-example-line-2">1 4 1</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">2 3</div><div class="test-example-line test-example-line-even test-example-line-2">1 3</div><div class="test-example-line test-example-line-even test-example-line-2">5</div><div class="test-example-line test-example-line-even test-example-line-2">3 3 9</div><div class="test-example-line test-example-line-even test-example-line-2">1 3 6</div><div class="test-example-line test-example-line-even test-example-line-2">1 1 2</div><div class="test-example-line test-example-line-even test-example-line-2">3 3 6</div><div class="test-example-line test-example-line-even test-example-line-2">3 3 4</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
YES
|
||
YES
|
||
YES
|
||
YES
|
||
NO
|
||
|
||
YES
|
||
YES
|
||
YES
|
||
YES
|
||
YES
|
||
</pre></div><div class="input"><div class="title">Input</div><pre><div class="test-example-line test-example-line-even test-example-line-0">1</div><div class="test-example-line test-example-line-odd test-example-line-1">6 10</div><div class="test-example-line test-example-line-odd test-example-line-1">7 9 2 10 8 6</div><div class="test-example-line test-example-line-odd test-example-line-1">4 2</div><div class="test-example-line test-example-line-odd test-example-line-1">6 1</div><div class="test-example-line test-example-line-odd test-example-line-1">4 5</div><div class="test-example-line test-example-line-odd test-example-line-1">3 5</div><div class="test-example-line test-example-line-odd test-example-line-1">6 4</div><div class="test-example-line test-example-line-odd test-example-line-1">1 3</div><div class="test-example-line test-example-line-odd test-example-line-1">2 6</div><div class="test-example-line test-example-line-odd test-example-line-1">6 5</div><div class="test-example-line test-example-line-odd test-example-line-1">1 2</div><div class="test-example-line test-example-line-odd test-example-line-1">3 6</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">4 4 8</div><div class="test-example-line test-example-line-odd test-example-line-1">3 3 1</div><div class="test-example-line test-example-line-odd test-example-line-1">5 5 9</div><div class="test-example-line test-example-line-odd test-example-line-1">2 1 7</div><div class="test-example-line test-example-line-odd test-example-line-1">6 6 10</div></pre></div><div class="output"><div class="title">Output</div><pre>
|
||
YES
|
||
YES
|
||
YES
|
||
YES
|
||
YES
|
||
</pre></div></div></div></div><p> </p></div>
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