1.3 KiB
1.3 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f4201000cf542c50ff33 | Problem 180: Rational zeros of a function of three variables | 5 | 301816 | problem-180-rational-zeros-of-a-function-of-three-variables |
--description--
For any integer n, consider the three functions
\begin{align}
& f_{1,n}(x,y,z) = x^{n + 1} + y^{n + 1} − z^{n + 1}\\\\
& f_{2,n}(x,y,z) = (xy + yz + zx) \times (x^{n - 1} + y^{n - 1} − z^{n - 1})\\\\
& f_{3,n}(x,y,z) = xyz \times (x^{n - 2} + y^{n - 2} − z^{n - 2})
\end{align}$$
and their combination
$$\begin{align}
& f_n(x,y,z) = f_{1,n}(x,y,z) + f_{2,n}(x,y,z) − f_{3,n}(x,y,z)
\end{align}$$
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 $f_n(x,y,z) = 0$.
Let $s(x,y,z) = x + y + z$.
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.
Find $u + v$.
# --hints--
`rationalZeros()` should return `285196020571078980`.
```js
assert.strictEqual(rationalZeros(), 285196020571078980);
```
# --seed--
## --seed-contents--
```js
function rationalZeros() {
return true;
}
rationalZeros();
```
# --solutions--
```js
// solution required
```