MathDB

5

Part of 2011 Saint Petersburg Mathematical Olympiad

Problems(2)

Working with divisors

Source: St Petersburg Olympiad 2011, Grade 11, P5

9/15/2017
Let M(n)M(n) and m(n)m(n) are maximal and minimal proper divisors of nn Natural number n>1000n>1000 is on the board. Every minute we replace our number with n+M(n)−m(n)n+M(n)-m(n). If we get prime, then process is stopped. Prove that after some moves we will get number, that is not divisible by 1717
number theory
Area of convex quadrilatera

Source: St Petersburg Olympiad 2011, Grade 9, P5

9/15/2017
ABCDABCD - convex quadrilateral. ∠A+∠D=150,∠B<150,∠C<150\angle A+ \angle D=150, \angle B<150, \angle C<150 Prove, that area ABCDABCD is greater than 14(AB∗CD+AB∗BC+BC∗CD)\frac{1}{4}(AB*CD+AB*BC+BC*CD)
geometry