MathDB

15

Part of 2016 ASDAN Math Tournament

Problems(2)

2016 Guts #15

Source:

8/14/2022
Let aa be the least positive integer with 2020 positive divisors and bb be the least positive integer with 1616 positive divisors. What is a+ba+b? (Note that for any integer nn, both 11 and nn are considered divisors of nn.)
2016Guts Round
2016 Team #15

Source:

8/17/2022
Circles ω1\omega_1 and ω2\omega_2 have radii r1<r2r_1<r_2 respectively and intersect at distinct points XX and YY. The common external tangents intersect at point ZZ. The common tangent closer to XX touches ω1\omega_1 and ω2\omega_2 at PP and QQ respectively. Line ZXZX intersects ω1\omega_1 and ω2\omega_2 again at points RR and SS and lines RPRP and SQSQ intersect again at point TT. If XT=8XT=8, XZ=15XZ=15, and XY=12XY=12, then what is r1r2\tfrac{r_1}{r_2}?
2016team test