MathDB

3

Part of 2015 Poland - Second Round

Problems(2)

Number theory

Source: Polish National Olympiad 2015 2nd round, 3rd problem

3/3/2015
Let an=∣n(n+1)−19∣a_{n}=|n(n+1)-19| for n=0,1,2,...n=0, 1, 2, ... and n≠4n \neq 4. Prove that if for every k<nk<n we have gcd⁡(an,ak)=1\gcd(a_{n}, a_{k})=1, then ana_{n} is a prime number.
number theory
Geometry problem

Source: Polish National Olympiad 2015 2nd round, 6th problem

3/3/2015
Let ABCABC be a triangle. Let KK be a midpoint of BCBC and MM be a point on the segment ABAB. L=KM∩ACL=KM \cap AC and CC lies on the segment ACAC between AA and LL. Let NN be a midpoint of MLML. ANAN cuts circumcircle of ΔABC\Delta ABC in SS and S≠NS \neq N. Prove that circumcircle of ΔKSN\Delta KSN is tangent to BCBC.
geometrycircumcircle