chore(i18n,curriculum): update translations (#43661)
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@ -1,6 +1,6 @@
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---
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id: 5900f3f51000cf542c50ff08
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title: 'Problem 137: Fibonacci golden nuggets'
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title: '问题 137：斐波那契金块'
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challengeType: 5
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forumTopicId: 301765
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dashedName: problem-137-fibonacci-golden-nuggets
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@ -8,30 +8,34 @@ dashedName: problem-137-fibonacci-golden-nuggets
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# --description--
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Consider the infinite polynomial series AF(x) = xF1 + x2F2 + x3F3 + ..., where Fk is the kth term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... ; that is, Fk = Fk−1 + Fk−2, F1 = 1 and F2 = 1.
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考虑无穷级数 $A_{F}(x) = xF_1 + x^2F_2 + x^3F_3 + \ldots$，其中 $F_k$ 是斐波那契数列 $1, 1, 2, 3, 5, 8, \ldots$ 的第 $k$ 项；即 $F_k = F_{k − 1} + F_{k − 2}, F_1 = 1$，$F_2 = 1$。
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For this problem we shall be interested in values of x for which AF(x) is a positive integer.
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在这个问题中，我们关注的是那些使得 $A_{F}(x)$ 为正整数的 $x$ 的值。
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Surprisingly AF(1/2)
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令人惊讶的是：
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=
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$$\begin{align} A_F(\frac{1}{2}) & = (\frac{1}{2}) × 1 + {(\frac{1}{2})}^2 × 1 + {(\frac{1}{2})}^3 × 2 + {(\frac{1}{2})}^4 × 3 + {(\frac{1}{2})}^5 × 5 + \cdots \\\\ & = \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \cdots \\\\ & = 2 \end{align}$$
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(1/2).1 + (1/2)2.1 + (1/2)3.2 + (1/2)4.3 + (1/2)5.5 + ...
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前五个对应的自然数 $x$ 如下。
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= 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ...
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| $x$ | $A_F(x)$ |
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| --------------------------- | -------- |
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| $\sqrt{2} − 1$ | $1$ |
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| $\frac{1}{2}$ | $2$ |
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| $\frac{\sqrt{13} − 2}{3}$ | $3$ |
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| $\frac{\sqrt{89} − 5}{8}$ | $4$ |
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| $\frac{\sqrt{34} − 3}{5}$ | $5$ |
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= 2 The corresponding values of x for the first five natural numbers are shown below.
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当 $x$ 是有理数时，我们称 $A_F(x)$ 是一个金砖，因为这样的数字逐渐变得稀少；例如，第 10 个金砖是 74049690。
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xAF(x) √2−11 1/22 (√13−2)/33 (√89−5)/84 (√34−3)/55
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We shall call AF(x) a golden nugget if x is rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690. Find the 15th golden nugget.
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请求出第 15 个金砖。
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# --hints--
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`euler137()` should return 1120149658760.
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`goldenNugget()` 应该返回 `1120149658760`。
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```js
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assert.strictEqual(euler137(), 1120149658760);
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assert.strictEqual(goldenNugget(), 1120149658760);
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```
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# --seed--
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@ -39,12 +43,12 @@ assert.strictEqual(euler137(), 1120149658760);
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## --seed-contents--
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```js
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function euler137() {
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function goldenNugget() {
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return true;
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}
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euler137();
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goldenNugget();
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```
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# --solutions--
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---
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id: 5900f37a1000cf542c50fe8d
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title: 'Problem 14: Longest Collatz sequence'
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title: '问题 14：最长考拉兹序列'
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challengeType: 5
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forumTopicId: 301768
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dashedName: problem-14-longest-collatz-sequence
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@ -8,61 +8,61 @@ dashedName: problem-14-longest-collatz-sequence
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# --description--
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The following iterative sequence is defined for the set of positive integers:
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对正整数集定义如下迭代序列：
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<div style='padding-left: 4em;'><var>n</var> → <var>n</var>/2 (<var>n</var> is even)</div>
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<div style='padding-left: 4em;'><var>n</var> → <var>n</var> / 2（<var>n</var> 为偶数）</div>
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<div style='padding-left: 4em;'><var>n</var> → 3<var>n</var> + 1 (<var>n</var> is odd)</div>
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<div style='padding-left: 4em;'><var>n</var> → 3<var>n</var> + 1（<var>n</var> 为奇数）</div>
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Using the rule above and starting with 13, we generate the following sequence:
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从 13 开始使用上述规则，我们可以得到如下序列：
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<div style='text-align: center;'>13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1</div>
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It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.
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可以看出，该序列（从 13 开始到 1 结束）共包含 10 项。 虽然考拉兹猜想尚未得到证明，但是该猜想认为以任意数字开始，序列均到 1 结束。
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Which starting number, under the given `limit`, produces the longest chain?
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求出在小于 `limit` 的数中，哪个可以产生最长的考拉兹序列？
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**Note:** Once the chain starts the terms are allowed to go above one million.
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**注意：** 序列中数字的个数允许超过一百万。
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# --hints--
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`longestCollatzSequence(14)` should return a number.
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`longestCollatzSequence(14)` 应该返回一个数字。
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```js
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assert(typeof longestCollatzSequence(14) === 'number');
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```
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`longestCollatzSequence(14)` should return 9.
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`longestCollatzSequence(14)` 应该返回 9。
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```js
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assert.strictEqual(longestCollatzSequence(14), 9);
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```
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`longestCollatzSequence(5847)` should return 3711.
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`longestCollatzSequence(5847)` 应该返回 3711。
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```js
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assert.strictEqual(longestCollatzSequence(5847), 3711);
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```
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`longestCollatzSequence(46500)` should return 35655.
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`longestCollatzSequence(46500)` 应该返回 35655。
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```js
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assert.strictEqual(longestCollatzSequence(46500), 35655);
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```
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`longestCollatzSequence(54512)` should return 52527.
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`longestCollatzSequence(54512)` 应该返回 52527。
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```js
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assert.strictEqual(longestCollatzSequence(54512), 52527);
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```
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`longestCollatzSequence(100000)` should return 77031.
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`longestCollatzSequence(100000)` 应该返回 77031。
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```js
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assert.strictEqual(longestCollatzSequence(100000), 77031);
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```
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`longestCollatzSequence(1000000)` should return 837799.
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`longestCollatzSequence(1000000)` 应该返回 837799。
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```js
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assert.strictEqual(longestCollatzSequence(1000000), 837799);
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---
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id: 5900f3f91000cf542c50ff0b
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title: 'Problem 141: Investigating progressive numbers, n, which are also square'
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title: '问题 141：累进平方数 n'
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challengeType: 5
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forumTopicId: 301770
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dashedName: problem-141-investigating-progressive-numbers-n-which-are-also-square
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@ -8,22 +8,22 @@ dashedName: problem-141-investigating-progressive-numbers-n-which-are-also-squar
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# --description--
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A positive integer, n, is divided by d and the quotient and remainder are q and r respectively. In addition d, q, and r are consecutive positive integer terms in a geometric sequence, but not necessarily in that order.
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一个正整数 $n$ 除以 $d$ 后得到商 $q$ 和余数 $r$。 同时 $d$，$q$ 和 $r$ 是一个等比数列中三个连续的正整数项，但顺序不要求一致。
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For example, 58 divided by 6 has quotient 9 and remainder 4. It can also be seen that 4, 6, 9 are consecutive terms in a geometric sequence (common ratio 3/2).
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例如，58 除以 6 后得到商 9 和余数 4。 可以发现，4、6、9 构成一个等比数列的连续三项（公比为 $\frac{3}{2}$）。
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We will call such numbers, n, progressive.
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我们称这样的数字 $n$ 为累进数。
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Some progressive numbers, such as 9 and 10404 = 1022, happen to also be perfect squares. The sum of all progressive perfect squares below one hundred thousand is 124657.
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一些累进数，如 9 和 10404 = ${102}^2$，同时也是完全平方数。 所有小于十万的累进平方数之和为 124657。
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Find the sum of all progressive perfect squares below one trillion (1012).
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请求出所有小于一万亿（${10}^{12}$）累进平方数之和。
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# --hints--
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`euler141()` should return 878454337159.
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`progressivePerfectSquares()` 应该返回 `878454337159`。
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```js
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assert.strictEqual(euler141(), 878454337159);
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assert.strictEqual(progressivePerfectSquares(), 878454337159);
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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 euler141() {
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function progressivePerfectSquares() {
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return true;
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}
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euler141();
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progressivePerfectSquares();
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```
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# --solutions--
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---
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id: 5900f3fa1000cf542c50ff0d
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title: 'Problem 142: Perfect Square Collection'
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title: '问题 142：完全平方数合集'
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challengeType: 5
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forumTopicId: 301771
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dashedName: problem-142-perfect-square-collection
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# --description--
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Find the smallest x + y + z with integers x > y > z > 0 such that x + y, x − y, x + z, x − z, y + z, y − z are all perfect squares.
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请找出最小的 $x + y + z$，其中整数 $x > y > z > 0$ 需要满足 $x + y$、$x − y$、$x + z$、$x − z$、$y + z$、$y − z$ 均为完全平方数。
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# --hints--
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`euler142()` should return 1006193.
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`perfectSquareCollection()` 应该返回 `1006193`。
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```js
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assert.strictEqual(euler142(), 1006193);
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assert.strictEqual(perfectSquareCollection(), 1006193);
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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 euler142() {
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function perfectSquareCollection() {
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return true;
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}
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euler142();
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perfectSquareCollection();
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```
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# --solutions--
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