MathDB

4

Part of 2011 Akdeniz University MO

Problems(2)

Geometry (Classic)

Source:

1/31/2016
Let an acute-angled triangle ABCABC's circumcircle is SS. SS's tangent from BB and CC intersects at point MM. A line, lies MM and parallel to [AB][AB] intersects with SS at points DD and EE, intersect with [AC][AC] at point FF. Prove that [DF]=[FE][DF]=[FE]
geometrycircumcircle
Arithmetic sequence

Source:

1/29/2016
ana_n sequence is a arithmetic sequence with all terms be positive integers. (for ana_n non-constant sequence) Let pnp_n is greatest prime divisor of ana_n. Prove that (anpn)(\frac{a_n}{p_n}) sequence is infinity.
Note: If we find a M>0M>0 constant such that xn≤Mx_n \leq M for all n∈Nn \in {\mathbb N}'s, (xn)(x_n) sequence is non-infinite, but we can't find MM, (xn)(x_n) sequence is infinity
arithmetic sequencenumber theory