47 lines
1.2 KiB
Markdown
47 lines
1.2 KiB
Markdown
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---
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id: 5900f48a1000cf542c50ff9c
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title: 'Problem 285: Pythagorean odds'
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challengeType: 5
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forumTopicId: 301936
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dashedName: problem-285-pythagorean-odds
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---
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# --description--
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Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval \[0,1] with uniform distribution.
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The square root of the sum (k·a+1)2 + (k·b+1)2 is then computed and rounded to the nearest integer. If the result is equal to k, he scores k points; otherwise he scores nothing.
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For example, if k = 6, a = 0.2 and b = 0.85, then (k·a+1)2 + (k·b+1)2 = 42.05. The square root of 42.05 is 6.484... and when rounded to the nearest integer, it becomes 6. This is equal to k, so he scores 6 points.
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It can be shown that if he plays 10 turns with k = 1, k = 2, ..., k = 10, the expected value of his total score, rounded to five decimal places, is 10.20914.
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If he plays 105 turns with k = 1, k = 2, k = 3, ..., k = 105, what is the expected value of his total score, rounded to five decimal places?
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# --hints--
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`euler285()` should return 157055.80999.
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```js
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assert.strictEqual(euler285(), 157055.80999);
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```
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# --seed--
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## --seed-contents--
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```js
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function euler285() {
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return true;
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}
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euler285();
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```
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# --solutions--
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```js
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// solution required
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```
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