MathDB

2

Part of 2018 Saint Petersburg Mathematical Olympiad

Problems(3)

Square from cards

Source: St Petersburg Olympiad 2018, Grade 11, P2

6/21/2018
Vasya has 100100 cards of 33 colors, and there are not more than 5050 cards of same color. Prove that he can create 10×1010\times 10 square, such that every cards of same color have not common side.
combinatorics
Coloring vertices

Source: St Petersburg Olympiad 2018, Grade 10, P2

6/22/2018
Color every vertex of 20082008-gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call 20082008-gon as interesting. Convex 20092009-gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting 20082008-gon. Prove, that if we remove marked vertex, then we get interesting 20082008-gon too.
combinatoricsgeometry
Sum divided by member

Source: St Petersburg Olympiad 2018, Grade 9, P2

7/13/2018
n>1n>1 is odd number. There are numbers n,n+1,n+2,...,2n−1n,n+1,n+2,...,2n-1 on the blackboard. Prove that we can erase one number, such that the sum of all numbers will be not divided any number on the blackboard.
number theory