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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">A. Word Capitalization</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>Capitalization is writing a word with its first letter as a capital letter. Your task is to capitalize the given word.</p><p>Note, that during capitalization all the letters except the first one remains unchanged.</p></div><div class="input-specification"><div class="section-title">Input</div><p>A single line contains a non-empty word. This word consists of lowercase and uppercase English letters. The length of the word will not exceed <span class="tex-span">10<sup class="upper-index">3</sup></span>.</p></div><div class="output-specification"><div class="section-title">Output</div><p>Output the given word after capitalization.</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>ApPLe<br /></pre></div><div class="output"><div class="title">Output</div><pre>ApPLe<br /></pre></div><div class="input"><div class="title">Input</div><pre>konjac<br /></pre></div><div class="output"><div class="title">Output</div><pre>Konjac<br /></pre></div></div></div></div></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">B. Nearest Fraction</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 given three positive integers <span class="tex-span"><i>x</i>, <i>y</i>, <i>n</i></span>. Your task is to find the nearest fraction to fraction <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/d9049b74dfcebaef5774f475b1e3ef4ed39cfc63.png" style="max-width: 100.0%;max-height: 100.0%;" /> whose denominator is no more than <span class="tex-span"><i>n</i></span>. </p><p>Formally, you should find such pair of integers <span class="tex-span"><i>a</i>, <i>b</i></span> <span class="tex-span">(1 ≤ <i>b</i> ≤ <i>n</i>; 0 ≤ <i>a</i>)</span> that the value <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/a4bb27e47b8ebb13e650f4f8a897d41377cf4360.png" style="max-width: 100.0%;max-height: 100.0%;" /> is as minimal as possible.</p><p>If there are multiple "nearest" fractions, choose the one with the minimum denominator. If there are multiple "nearest" fractions with the minimum denominator, choose the one with the minimum numerator.</p></div><div class="input-specification"><div class="section-title">Input</div><p>A single line contains three integers <span class="tex-span"><i>x</i>, <i>y</i>, <i>n</i></span> <span class="tex-span">(1 ≤ <i>x</i>, <i>y</i>, <i>n</i> ≤ 10<sup class="upper-index">5</sup>)</span>.</p></div><div class="output-specification"><div class="section-title">Output</div><p>Print the required fraction in the format "<span class="tex-span"><i>a</i></span>/<span class="tex-span"><i>b</i></span>" (without quotes).</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>3 7 6<br /></pre></div><div class="output"><div class="title">Output</div><pre>2/5<br /></pre></div><div class="input"><div class="title">Input</div><pre>7 2 4<br /></pre></div><div class="output"><div class="title">Output</div><pre>7/2<br /></pre></div></div></div></div></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">C. Rectangle Puzzle</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 given two rectangles on a plane. The centers of both rectangles are located in the origin of coordinates (meaning the center of the rectangle's symmetry). The first rectangle's sides are parallel to the coordinate axes: the length of the side that is parallel to the <span class="tex-span"><i>Ox</i></span> axis, equals <span class="tex-span"><i>w</i></span>, the length of the side that is parallel to the <span class="tex-span"><i>Oy</i></span> axis, equals <span class="tex-span"><i>h</i></span>. The second rectangle can be obtained by rotating the first rectangle relative to the origin of coordinates by angle <span class="tex-span">α</span>.</p><center> <img class="tex-graphics" src="https://espresso.codeforces.com/b997a953b748780107b3f45c318ffd00eca6439a.png" style="max-width: 100.0%;max-height: 100.0%;" /> </center><p>Your task is to find the area of the region which belongs to both given rectangles. This region is shaded in the picture.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains three integers <span class="tex-span"><i>w</i>, <i>h</i>, α</span> <span class="tex-span">(1 ≤ <i>w</i>, <i>h</i> ≤ 10<sup class="upper-index">6</sup>; 0 ≤ α ≤ 180)</span>. Angle <span class="tex-span">α</span> is given in degrees.</p></div><div class="output-specification"><div class="section-title">Output</div><p>In a single line print a real number — the area of the region which belongs to both given rectangles.</p><p>The answer will be considered correct if its relative or absolute error doesn't exceed <span class="tex-span">10<sup class="upper-index"> - 6</sup></span>.</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>1 1 45<br /></pre></div><div class="output"><div class="title">Output</div><pre>0.828427125<br /></pre></div><div class="input"><div class="title">Input</div><pre>6 4 30<br /></pre></div><div class="output"><div class="title">Output</div><pre>19.668384925<br /></pre></div></div></div><div class="note"><div class="section-title">Note</div><p>The second sample has been drawn on the picture above.</p></div></div></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">D. Maximum Xor Secondary</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>Bike loves looking for the second maximum element in the sequence. The second maximum element in the sequence of distinct numbers <span class="tex-span"><i>x</i><sub class="lower-index">1</sub>, <i>x</i><sub class="lower-index">2</sub>, ..., <i>x</i><sub class="lower-index"><i>k</i></sub></span> <span class="tex-span">(<i>k</i> > 1)</span> is such maximum element <span class="tex-span"><i>x</i><sub class="lower-index"><i>j</i></sub></span>, that the following inequality holds: <img align="middle" class="tex-formula" src="https://espresso.codeforces.com/41e9a774451aa769280692592b29103e0a6ecf97.png" style="max-width: 100.0%;max-height: 100.0%;" />.</p><p>The lucky number of the sequence of distinct positive integers <span class="tex-span"><i>x</i><sub class="lower-index">1</sub>, <i>x</i><sub class="lower-index">2</sub>, ..., <i>x</i><sub class="lower-index"><i>k</i></sub></span> <span class="tex-span">(<i>k</i> > 1)</span> is the number that is equal to the bitwise excluding OR of the maximum element of the sequence and the second maximum element of the sequence.</p><p>You've got a sequence of distinct positive integers <span class="tex-span"><i>s</i><sub class="lower-index">1</sub>, <i>s</i><sub class="lower-index">2</sub>, ..., <i>s</i><sub class="lower-index"><i>n</i></sub></span> <span class="tex-span">(<i>n</i> > 1)</span>. Let's denote sequence <span class="tex-span"><i>s</i><sub class="lower-index"><i>l</i></sub>, <i>s</i><sub class="lower-index"><i>l</i> + 1</sub>, ..., <i>s</i><sub class="lower-index"><i>r</i></sub></span> as <span class="tex-span"><i>s</i>[<i>l</i>..<i>r</i>]</span> <span class="tex-span">(1 ≤ <i>l</i> < <i>r</i> ≤ <i>n</i>)</span>. Your task is to find the maximum number among all lucky numbers of sequences <span class="tex-span"><i>s</i>[<i>l</i>..<i>r</i>]</span>.</p><p>Note that as all numbers in sequence <span class="tex-span"><i>s</i></span> are distinct, all the given definitions make sence.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains integer <span class="tex-span"><i>n</i></span> <span class="tex-span">(1 < <i>n</i> ≤ 10<sup class="upper-index">5</sup>)</span>. The second line contains <span class="tex-span"><i>n</i></span> distinct integers <span class="tex-span"><i>s</i><sub class="lower-index">1</sub>, <i>s</i><sub class="lower-index">2</sub>, ..., <i>s</i><sub class="lower-index"><i>n</i></sub></span> <span class="tex-span">(1 ≤ <i>s</i><sub class="lower-index"><i>i</i></sub> ≤ 10<sup class="upper-index">9</sup>)</span>.</p></div><div class="output-specification"><div class="section-title">Output</div><p>Print a single integer — the maximum lucky number among all lucky numbers of sequences <span class="tex-span"><i>s</i>[<i>l</i>..<i>r</i>]</span>.</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>5<br />5 2 1 4 3<br /></pre></div><div class="output"><div class="title">Output</div><pre>7<br /></pre></div><div class="input"><div class="title">Input</div><pre>5<br />9 8 3 5 7<br /></pre></div><div class="output"><div class="title">Output</div><pre>15<br /></pre></div></div></div><div class="note"><div class="section-title">Note</div><p>For the first sample you can choose <span class="tex-span"><i>s</i>[4..5] = {4, 3}</span> and its lucky number is <span class="tex-span">(4 <i>xor</i> 3) = 7</span>. You can also choose <span class="tex-span"><i>s</i>[1..2]</span>.</p><p>For the second sample you must choose <span class="tex-span"><i>s</i>[2..5] = {8, 3, 5, 7}</span>.</p></div></div></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">E. Game on Tree</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>Momiji has got a rooted tree, consisting of <span class="tex-span"><i>n</i></span> nodes. The tree nodes are numbered by integers from <span class="tex-span">1</span> to <span class="tex-span"><i>n</i></span>. The root has number <span class="tex-span">1</span>. Momiji decided to play a game on this tree.</p><p>The game consists of several steps. On each step, Momiji chooses one of the remaining tree nodes (let's denote it by <span class="tex-span"><i>v</i></span>) and removes all the subtree nodes with the root in node <span class="tex-span"><i>v</i></span> from the tree. Node <span class="tex-span"><i>v</i></span> gets deleted as well. The game finishes when the tree has no nodes left. In other words, the game finishes after the step that chooses the node number <span class="tex-span">1</span>.</p><p>Each time Momiji chooses a new node uniformly among all the remaining nodes. Your task is to find the expectation of the number of steps in the described game.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains integer <span class="tex-span"><i>n</i></span> <span class="tex-span">(1 ≤ <i>n</i> ≤ 10<sup class="upper-index">5</sup>)</span> — the number of nodes in the tree. The next <span class="tex-span"><i>n</i> - 1</span> lines contain the tree edges. The <span class="tex-span"><i>i</i></span>-th line contains integers <span class="tex-span"><i>a</i><sub class="lower-index"><i>i</i></sub></span>, <span class="tex-span"><i>b</i><sub class="lower-index"><i>i</i></sub></span> <span class="tex-span">(1 ≤ <i>a</i><sub class="lower-index"><i>i</i></sub>, <i>b</i><sub class="lower-index"><i>i</i></sub> ≤ <i>n</i>; <i>a</i><sub class="lower-index"><i>i</i></sub> ≠ <i>b</i><sub class="lower-index"><i>i</i></sub>)</span> — the numbers of the nodes that are connected by the <span class="tex-span"><i>i</i></span>-th edge.</p><p>It is guaranteed that the given graph is a tree.</p></div><div class="output-specification"><div class="section-title">Output</div><p>Print a single real number — the expectation of the number of steps in the described game.</p><p>The answer will be considered correct if the absolute or relative error doesn't exceed <span class="tex-span">10<sup class="upper-index"> - 6</sup></span>.</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>2<br />1 2<br /></pre></div><div class="output"><div class="title">Output</div><pre>1.50000000000000000000<br /></pre></div><div class="input"><div class="title">Input</div><pre>3<br />1 2<br />1 3<br /></pre></div><div class="output"><div class="title">Output</div><pre>2.00000000000000000000<br /></pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first sample, there are two cases. One is directly remove the root and another is remove the root after one step. Thus the expected steps are: </p><center class="tex-equation"><span class="tex-span">1 × (1 / 2) + 2 × (1 / 2) = 1.5</span></center><p>In the second sample, things get more complex. There are two cases that reduce to the first sample, and one case cleaned at once. Thus the expected steps are: </p><center class="tex-equation"><span class="tex-span">1 × (1 / 3) + (1 + 1.5) × (2 / 3) = (1 / 3) + (5 / 3) = 2</span></center></div></div></div>
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