55 lines
1.1 KiB
Markdown
55 lines
1.1 KiB
Markdown
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---
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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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f1,n(x,y,z) = xn+1 + yn+1 − zn+1f2,n(x,y,z) = (xy + yz + zx)\*(xn-1 + yn-1 − zn-1)f3,n(x,y,z) = xyz\*(xn-2 + yn-2 − zn-2)
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and their combination
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fn(x,y,z) = f1,n(x,y,z) + f2,n(x,y,z) − f3,n(x,y,z)
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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 a / b with
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0 < a < b ≤ k and there is (at least) one integer n, so that fn(x,y,z) = 0.
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Let s(x,y,z) = x + y + z.
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Let t = 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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`euler180()` should return 285196020571078980.
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```js
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assert.strictEqual(euler180(), 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 euler180() {
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return true;
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}
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euler180();
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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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