MYTS 2018 - Grade 9 - Round 2

Solve √(8 x+3)=9 x²+10 x+9/4.

Prove that 2017.49 $<$ √(1+2/2²)+√(1+2/3²)+⋯+√(1+2/2018²) $<$2018.

ABCD is a parallelogram and O is the intersection of its diagonals. On the ray opposite to CD take M, and on the ray opposite to DC take N, so CM+DN=CD. Let I=AM∩BN and P=OI∩AB. Prove AP=CM.

Find all pairs of primes (p,q) for which 3 p²+25 pq+3 q² is a power of 31.

Four pebble piles initially contain 10,20,30,40. Each move takes one pebble from each of three piles and puts all three into the fourth pile. Can finitely many such moves make all four piles equal? Explain.

Ngoan wants to assign a positive integer to every vertex of a convex 2018-gon so that at least one assigned number is even and the sum at every three consecutive vertices is odd. Is this possible? Explain.