47 lines
1.0 KiB
Markdown
47 lines
1.0 KiB
Markdown
---
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id: 5900f5071000cf542c510018
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title: 'Problem 410: Circle and tangent line'
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challengeType: 5
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forumTopicId: 302079
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dashedName: problem-410-circle-and-tangent-line
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---
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# --description--
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Let $C$ be the circle with radius $r$, $x^2 + y^2 = r^2$. We choose two points $P(a, b)$ and $Q(-a, c)$ so that the line passing through $P$ and $Q$ is tangent to $C$.
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For example, the quadruplet $(r, a, b, c) = (2, 6, 2, -7)$ satisfies this property.
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Let $F(R, X)$ be the number of the integer quadruplets $(r, a, b, c)$ with this property, and with $0 < r ≤ R$ and $0 < a ≤ X$.
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We can verify that $F(1, 5) = 10$, $F(2, 10) = 52$ and $F(10, 100) = 3384$.
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Find $F({10}^8, {10}^9) + F({10}^9, {10}^8)$.
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# --hints--
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`circleAndTangentLine()` should return `799999783589946600`.
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```js
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assert.strictEqual(circleAndTangentLine(), 799999783589946600);
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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 circleAndTangentLine() {
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return true;
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}
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circleAndTangentLine();
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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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