4

Part of 2020 Final Mathematical Cup

Problems(2)

Tangent Circles

Source: 2nd Final Mathematical Cup Junior Division P4 and Senior Division P2 (2020)

ABC

be a triangle such that

\measuredangle BAC = 60^{\circ}

D

E

be the feet of the perpendicular from

A

to the bisectors of the external angles of

B

C

ABC

, respectively. Let

O

be the circumcenter of the triangle

ABC

. Prove that circumcircle of the triangle

BOC

has exactly one point in common with the circumcircle of

ADE

geometrytangent circlesperpendicular

Relatively Prime Implies Prime

Source: 2nd Final Mathematical Cup Senior Division P4 (2020)

Find all positive integers

n

such that for all positive integers

m

1<m<n

, relatively prime to

n

m

must be a prime number.

number theoryrelatively prime