2015 Final tour - Azerbaijan in lower age category

1) a, b, and c are positive real numbers that abc=1/8. Prove a^2+b^2+c^2+a^2*b^2+a^2*c^2+b^2*c^2>=15/16 (2 points)

2) a, b and c are sides of triangle. Prove that area of triangle isn't more than (a^2+b^2+c^2)/6. (4 points)

3) Find all P(x) polynomials that has real coefficient, which for all real numbers of x this equation must be true: P(P(x))=(x^2+x+1)*P(x) (6 points)

4) Natural number M has 6 natural divisors. If sum of this divisors is 3500, find all numbers M. (8 points)

5) ABCD is convex quadrilateral. Angle BAD=90 degree, measure of angle BAC=2*(measure of angle BDC) and (measure of angle DBA)+(measure of angle DCB)=180. Find the measure of angle DBA. (10 points)

[[Category:National Olympiads]