32fac23a2d
* fix: clean-up Project Euler 161-180 * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
59 lines
1.3 KiB
Markdown
59 lines
1.3 KiB
Markdown
---
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id: 5900f4201000cf542c50ff33
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title: 'Problem 180: Rational zeros of a function of three variables'
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challengeType: 5
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forumTopicId: 301816
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dashedName: problem-180-rational-zeros-of-a-function-of-three-variables
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---
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# --description--
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For any integer $n$, consider the three functions
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$$\begin{align}
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& f_{1,n}(x,y,z) = x^{n + 1} + y^{n + 1} − z^{n + 1}\\\\
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& f_{2,n}(x,y,z) = (xy + yz + zx) \times (x^{n - 1} + y^{n - 1} − z^{n - 1})\\\\
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& f_{3,n}(x,y,z) = xyz \times (x^{n - 2} + y^{n - 2} − z^{n - 2})
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\end{align}$$
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and their combination
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$$\begin{align}
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& f_n(x,y,z) = f_{1,n}(x,y,z) + f_{2,n}(x,y,z) − f_{3,n}(x,y,z)
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\end{align}$$
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We call $(x,y,z)$ a golden triple of order $k$ if $x$, $y$, and $z$ are all rational numbers of the form $\frac{a}{b}$ with $0 < a < b ≤ k$ and there is (at least) one integer $n$, so that $f_n(x,y,z) = 0$.
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Let $s(x,y,z) = x + y + z$.
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Let $t = \frac{u}{v}$ be the sum of all distinct $s(x,y,z)$ for all golden triples $(x,y,z)$ of order 35. All the $s(x,y,z)$ and $t$ must be in reduced form.
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Find $u + v$.
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# --hints--
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`rationalZeros()` should return `285196020571078980`.
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```js
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assert.strictEqual(rationalZeros(), 285196020571078980);
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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 rationalZeros() {
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return true;
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}
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rationalZeros();
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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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