1.8 KiB
1.8 KiB
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 Cin be the circle having the diameter XY.
Let Cout 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, ..., Ck such that:
- Ci has no common interior points with any Cj for 1 ≤ i, j ≤ k and i ≠ j,
- Ci is tangent to both Cin and Cout for 1 ≤ i ≤ k,
- Ci is tangent to Ci+1 for 1 ≤ i < k, and
- Ck is tangent to C1.
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