MathDB

18

Part of 2021 MOAA

Problems(2)

Team Round P18

Source:

10/16/2021
Let △ABC\triangle ABC be a triangle with side length BC=46BC= 4\sqrt{6}. Denote ω\omega as the circumcircle of △ABC\triangle{ABC}. Point DD lies on ω\omega such that ADAD is the diameter of ω\omega. Let NN be the midpoint of arc BCBC that contains AA. HH is the intersection of the altitudes in △ABC\triangle{ABC} and it is given that HN=HD=6HN = HD= 6. If the area of △ABC\triangle{ABC} can be expressed as abc\frac{a\sqrt{b}}{c}, where a,b,ca,b,c are positive integers with aa and cc relatively prime and bb not divisible by the square of any prime, compute a+b+ca+b+c.
Proposed by Andy Xu
MOAA 2021team
Gunga P18

Source:

10/16/2021
Find the largest positive integer nn such that the number (2n)!(2n)! ends with 1010 more zeroes than the number n!n!.
Proposed by Andy Xu
MOAA 2021Gunga