chore(i18n,learn): processed translations (#44805)
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---
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id: 5900f4b71000cf542c50ffc9
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title: 'Problem 330: Euler''s Number'
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title: '問題 330:歐拉數'
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challengeType: 5
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forumTopicId: 301988
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dashedName: problem-330-eulers-number
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@ -8,30 +8,28 @@ dashedName: problem-330-eulers-number
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# --description--
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An infinite sequence of real numbers a(n) is defined for all integers n as follows:
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對於所有的整數 $n$,一個無限實數序列 $a(n)$ 定義如下:
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<!-- TODO Use MathJax and re-write from projecteuler.net -->
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$$ a(n) = \begin{cases} 1 & n < 0 \\\\ \displaystyle \sum_{i = 1}^{\infty} \frac{a(n - 1)}{i!} & n \ge 0 \end{cases} $$
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For example,a(0) = 11! + 12! + 13! + ... = e − 1 a(1) = e − 11! + 12! + 13! + ... = 2e − 3 a(2) = 2e − 31! + e − 12! + 13! + ... = 72 e − 6
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例如,
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with e = 2.7182818... being Euler's constant.
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$$\begin{align} & a(0) = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = e − 1 \\\\ & a(1) = \frac{e − 1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = 2e − 3 \\\\ & a(2) = \frac{2e − 3}{1!} + \frac{e − 1}{2!} + \frac{1}{3!} + \ldots = \frac{7}{2} e − 6 \end{align}$$
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It can be shown that a(n) is of the form
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其中,$e = 2.7182818\ldots$ 是歐拉常數。
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A(n) e + B(n)n! for integers A(n) and B(n).
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可以看出,$a(n)$ 可以寫成 $\displaystyle\frac{A(n)e + B(n)}{n!}$ 這樣的形式,其中 $A(n)$ 和 $B(n)$ 均是整數。
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For example a(10) =
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例如,$\displaystyle a(10) = \frac{328161643e − 652694486}{10!}$。
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328161643 e − 65269448610!.
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Find A(109) + B(109) and give your answer mod 77 777 777.
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求解 $A({10}^9)$ + $B({10}^9)$ 並給出答案 $\bmod 77\\,777\\,777$。
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# --hints--
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`euler330()` should return 15955822.
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`eulersNumber()` 應該返回 `15955822`。
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```js
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assert.strictEqual(euler330(), 15955822);
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assert.strictEqual(eulersNumber(), 15955822);
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```
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# --seed--
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@ -39,12 +37,12 @@ assert.strictEqual(euler330(), 15955822);
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## --seed-contents--
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```js
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function euler330() {
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function eulersNumber() {
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return true;
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}
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euler330();
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eulersNumber();
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```
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# --solutions--
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@ -1,6 +1,6 @@
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---
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id: 5900f4b71000cf542c50ffc9
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title: 'Problem 330: Euler''s Number'
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title: '问题 330:欧拉数'
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challengeType: 5
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forumTopicId: 301988
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dashedName: problem-330-eulers-number
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@ -8,30 +8,28 @@ dashedName: problem-330-eulers-number
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# --description--
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An infinite sequence of real numbers a(n) is defined for all integers n as follows:
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对于所有的整数 $n$,一个无限实数序列 $a(n)$ 定义如下:
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<!-- TODO Use MathJax and re-write from projecteuler.net -->
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$$ a(n) = \begin{cases} 1 & n < 0 \\\\ \displaystyle \sum_{i = 1}^{\infty} \frac{a(n - 1)}{i!} & n \ge 0 \end{cases} $$
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For example,a(0) = 11! + 12! + 13! + ... = e − 1 a(1) = e − 11! + 12! + 13! + ... = 2e − 3 a(2) = 2e − 31! + e − 12! + 13! + ... = 72 e − 6
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例如,
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with e = 2.7182818... being Euler's constant.
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$$\begin{align} & a(0) = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = e − 1 \\\\ & a(1) = \frac{e − 1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = 2e − 3 \\\\ & a(2) = \frac{2e − 3}{1!} + \frac{e − 1}{2!} + \frac{1}{3!} + \ldots = \frac{7}{2} e − 6 \end{align}$$
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It can be shown that a(n) is of the form
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其中,$e = 2.7182818\ldots$ 是欧拉常数。
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A(n) e + B(n)n! for integers A(n) and B(n).
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可以看出,$a(n)$ 可以写成 $\displaystyle\frac{A(n)e + B(n)}{n!}$ 这样的形式,其中 $A(n)$ 和 $B(n)$ 均是整数。
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For example a(10) =
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例如,$\displaystyle a(10) = \frac{328161643e − 652694486}{10!}$。
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328161643 e − 65269448610!.
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Find A(109) + B(109) and give your answer mod 77 777 777.
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求解 $A({10}^9)$ + $B({10}^9)$ 并给出答案 $\bmod 77\\,777\\,777$。
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# --hints--
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`euler330()` should return 15955822.
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`eulersNumber()` 应该返回 `15955822`。
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```js
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assert.strictEqual(euler330(), 15955822);
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assert.strictEqual(eulersNumber(), 15955822);
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```
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# --seed--
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@ -39,12 +37,12 @@ assert.strictEqual(euler330(), 15955822);
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## --seed-contents--
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```js
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function euler330() {
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function eulersNumber() {
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return true;
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}
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euler330();
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eulersNumber();
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```
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# --solutions--
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@ -27,6 +27,7 @@ Reescreva o código de forma que o array global `bookList` não seja alterado em
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`bookList` não deve ser alterado e precisa permanecer igual a `["The Hound of the Baskervilles", "On The Electrodynamics of Moving Bodies", "Philosophiæ Naturalis Principia Mathematica", "Disquisitiones Arithmeticae"]`.
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```js
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add(bookList, "Test");
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assert(
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JSON.stringify(bookList) ===
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JSON.stringify([
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);
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```
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`newBookList` deve ser igual a `["The Hound of the Baskervilles", "On The Electrodynamics of Moving Bodies", "Philosophiæ Naturalis Principia Mathematica", "Disquisitiones Arithmeticae", "A Brief History of Time"]`.
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`add(bookList, "A Brief History of Time")` deve retornar `["The Hound of the Baskervilles", "On The Electrodynamics of Moving Bodies", "Philosophiæ Naturalis Principia Mathematica", "Disquisitiones Arithmeticae", "A Brief History of Time"]`.
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```js
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assert(
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JSON.stringify(newBookList) ===
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JSON.stringify(add(bookList, "A Brief History of Time")) ===
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JSON.stringify([
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'The Hound of the Baskervilles',
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'On The Electrodynamics of Moving Bodies',
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);
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```
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`newerBookList` deve ser igual a `["The Hound of the Baskervilles", "Philosophiæ Naturalis Principia Mathematica", "Disquisitiones Arithmeticae"]`.
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`remove(bookList, "On The Electrodynamics of Moving Bodies")` deve retornar `["The Hound of the Baskervilles", "Philosophiæ Naturalis Principia Mathematica", "Disquisitiones Arithmeticae"]`.
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```js
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assert(
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JSON.stringify(newerBookList) ===
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JSON.stringify(remove(bookList, 'On The Electrodynamics of Moving Bodies')) ===
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JSON.stringify([
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'The Hound of the Baskervilles',
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'Philosophiæ Naturalis Principia Mathematica',
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);
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```
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`newestBookList` deve ser igual a `["The Hound of the Baskervilles", "Philosophiæ Naturalis Principia Mathematica", "Disquisitiones Arithmeticae", "A Brief History of Time"]`.
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`remove(add(bookList, "A Brief History of Time"), "On The Electrodynamics of Moving Bodies");` deve ser igual a `["The Hound of the Baskervilles", "Philosophiæ Naturalis Principia Mathematica", "Disquisitiones Arithmeticae", "A Brief History of Time"]`.
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```js
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assert(
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JSON.stringify(newestBookList) ===
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JSON.stringify(remove(add(bookList, 'A Brief History of Time'), 'On The Electrodynamics of Moving Bodies')) ===
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JSON.stringify([
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'The Hound of the Baskervilles',
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'Philosophiæ Naturalis Principia Mathematica',
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// Change code above this line
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}
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}
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const newBookList = add(bookList, 'A Brief History of Time');
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const newerBookList = remove(bookList, 'On The Electrodynamics of Moving Bodies');
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const newestBookList = remove(add(bookList, 'A Brief History of Time'), 'On The Electrodynamics of Moving Bodies');
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console.log(bookList);
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```
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# --solutions--
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}
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return bookListCopy;
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}
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const newBookList = add(bookList, 'A Brief History of Time');
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const newerBookList = remove(bookList, 'On The Electrodynamics of Moving Bodies');
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const newestBookList = remove(add(bookList, 'A Brief History of Time'), 'On The Electrodynamics of Moving Bodies');
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```
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