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id	title	challengeType	forumTopicId	dashedName
5900f5191000cf542c51002b	Problem 428: Necklace of Circles	5	302098	problem-428-necklace-of-circles

--description--

Let a, b and c be positive numbers.

Let W, X, Y, Z be four collinear points where |WX| = a, |XY| = b, |YZ| = c and |WZ| = a + b + c.

Let C_in be the circle having the diameter XY.

Let C_out be the circle having the diameter WZ.

The triplet (a, b, c) is called a necklace triplet if you can place k ≥ 3 distinct circles C₁, C₂, ..., C_k such that:

C_i has no common interior points with any C_j for 1 ≤ i, j ≤ k and i ≠ j,
C_i is tangent to both C_in and C_out for 1 ≤ i ≤ k,
C_i is tangent to C_i+1 for 1 ≤ i < k, and
C_k is tangent to C₁.

For example, (5, 5, 5) and (4, 3, 21) are necklace triplets, while it can be shown that (2, 2, 5) is not. a visual representation of a necklace triplet

Let T(n) be the number of necklace triplets (a, b, c) such that a, b and c are positive integers, and b ≤ n. For example, T(1) = 9, T(20) = 732 and T(3000) = 438106.

Find T(1 000 000 000).

--hints--

necklace(1000000000) should return 747215561862.

assert.strictEqual(necklace(1000000000), 747215561862);

--seed--

--seed-contents--

function necklace(n) {

  return true;
}

necklace(1000000000)

--solutions--

// solution required

1.8 KiB Raw Blame History