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---
id: 5900f4f21000cf542c510005
title: 'Problem 390: Triangles with non rational sides and integral area'
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challengeType: 5
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forumTopicId: 302055
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dashedName: problem-390-triangles-with-non-rational-sides-and-integral-area
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---
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# --description--
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Consider the triangle with sides $\sqrt{5}$, $\sqrt{65}$ and $\sqrt{68}$. It can be shown that this triangle has area 9.
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$S(n)$ is the sum of the areas of all triangles with sides $\sqrt{1 + b^2}$, $\sqrt{1 + c^2}$ and $\sqrt{b^2 + c^2}$ (for positive integers $b$ and $c$) that have an integral area not exceeding $n$.
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The example triangle has $b = 2$ and $c = 8$.
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$S({10}^6) = 18\\,018\\,206$.
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Find $S({10}^{10})$.
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# --hints--
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`nonRationalSidesAndIntegralArea()` should return `2919133642971` .
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```js
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assert.strictEqual(nonRationalSidesAndIntegralArea(), 2919133642971);
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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 nonRationalSidesAndIntegralArea() {
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return true;
}
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nonRationalSidesAndIntegralArea();
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```
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# --solutions--
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```js
// solution required
```