55 lines
2.3 KiB
Markdown
55 lines
2.3 KiB
Markdown
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---
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id: 5900f52e1000cf542c510041
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challengeType: 5
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title: 'Problem 450: Hypocycloid and Lattice points'
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videoUrl: ''
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localeTitle: 问题450:Hypocycloid和Lattice点
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---
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## Description
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<section id="description">内摆线是由在较大圆内滚动的小圆上的点绘制的曲线。以原点为中心,从最右边开始的内摆线的参数方程由下式给出:$ x(t)=(R - r)\ cos(t)+ r \ cos(\ frac {R - r} rt)$ $ y(t)=(R - r)\ sin(t) - r \ sin(\ frac {R - r} rt)$其中R是大圆的半径,r是小圆的半径圈。 <p>设$ C(R,r)$是具有半径为R和r的内摆线上的整数坐标的不同点的集合,并且对应的值为t,使得$ \ sin(t)$和$ \ cos( t)$是有理数。 </p><p>设$ S(R,r)= \ sum _ {(x,y)\ in C(R,r)} | x | + | y | $是$ C(R,r)$中点的x和y坐标的绝对值之和。 </p><p>设$ T(N)= \ sum <em>{R = 3} ^ N \ sum</em> {r = 1} ^ {\ lfloor \ frac {R - 1} 2 \ rfloor} S(R,r)$是$的总和S(R,r)$表示R和r正整数,$ R \ leq N $和$ 2r <R $。 </p><p>给出:C(3,1)= {(3,0),(-1,2),( - 1,0),( - 1,-2)} C(2500,1000)= {(2500 ,0),(772,2376),(772,-2376),(516,1792),(516,-1792),(500,0),(68,504),(68,-504),( -1356,1088),( - 1356,-1088),( - 1500,1000),( - 1500,-1000)} </p><p>注意:( - 625,0)不是C(2500,1000)的元素,因为$ \ sin(t)$不是t的相应值的有理数。 </p><p> S(3,1)=(| 3 | + | 0 |)+(| -1 | + | 2 |)+(| -1 | + | 0 |)+(| -1 | + | -2 |) = 10 </p><p> T(3)= 10; T(10)= 524; T(100)= 580442; T(103)= 583108600。 </p><p>求T(106)。 </p></section>
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## Instructions
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undefined
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>euler450()</code>应该返回583333163984220900。
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testString: 'assert.strictEqual(euler450(), 583333163984220900, "<code>euler450()</code> should return 583333163984220900.");'
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function euler450() {
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// Good luck!
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return true;
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}
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euler450();
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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// solution required
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```
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</section>
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