2018-09-30 22:01:58 +00:00
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---
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id: 5900f4311000cf542c50ff43
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title: 'Problem 195: Inscribed circles of triangles with one angle of 60 degrees'
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2020-11-27 18:02:05 +00:00
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challengeType: 5
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2019-08-05 16:17:33 +00:00
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forumTopicId: 301833
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2021-01-13 02:31:00 +00:00
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dashedName: problem-195-inscribed-circles-of-triangles-with-one-angle-of-60-degrees
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2018-09-30 22:01:58 +00:00
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---
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2020-11-27 18:02:05 +00:00
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# --description--
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2021-07-15 13:52:14 +00:00
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Let's call an integer sided triangle with exactly one angle of 60° a 60° triangle.
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2020-11-27 18:02:05 +00:00
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2021-07-15 13:52:14 +00:00
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Let $r$ be the radius of the inscribed circle of such a 60° triangle.
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2020-11-27 18:02:05 +00:00
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2021-07-15 13:52:14 +00:00
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There are 1234 60° triangles for which $r ≤ 100$.
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2018-09-30 22:01:58 +00:00
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2021-07-15 13:52:14 +00:00
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Let $T(n)$ be the number of 60° triangles for which $r ≤ n$, so $T(100) = 1234$, $T(1000) = 22767$, and $T(10000) = 359912$.
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2018-09-30 22:01:58 +00:00
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2021-07-15 13:52:14 +00:00
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Find $T(1053779)$.
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2018-09-30 22:01:58 +00:00
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2020-11-27 18:02:05 +00:00
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# --hints--
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2018-09-30 22:01:58 +00:00
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2021-07-15 13:52:14 +00:00
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`inscribedCirclesOfTriangles()` should return `75085391`.
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2018-09-30 22:01:58 +00:00
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2020-11-27 18:02:05 +00:00
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```js
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2021-07-15 13:52:14 +00:00
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assert.strictEqual(inscribedCirclesOfTriangles(), 75085391);
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2018-09-30 22:01:58 +00:00
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```
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2020-11-27 18:02:05 +00:00
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# --seed--
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2018-09-30 22:01:58 +00:00
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2020-11-27 18:02:05 +00:00
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## --seed-contents--
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2018-09-30 22:01:58 +00:00
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```js
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2021-07-15 13:52:14 +00:00
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function inscribedCirclesOfTriangles() {
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2020-09-15 16:57:40 +00:00
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2018-09-30 22:01:58 +00:00
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return true;
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}
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2021-07-15 13:52:14 +00:00
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inscribedCirclesOfTriangles();
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2018-09-30 22:01:58 +00:00
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```
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2020-11-27 18:02:05 +00:00
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# --solutions--
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2018-09-30 22:01:58 +00:00
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```js
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// solution required
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```
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