2

Part of 2015 Czech-Polish-Slovak Junior Match

Problems(2)

sum of pairs of 1, 2,...., 30 a perfect square?

Source: Czech-Polish-Slovak Junior Match 2015, Individual p2 CPSJ

Decide if the vertices of a regular

30

-gon can be numbered by numbers

1, 2,.., 30

in such a way that the sum of the numbers of every two neighboring to be a square of a certain natural number.

number theorySumPerfect Square

checkers on a 8x8 board minus the middle 2x2 square

Source: Czech-Polish-Slovak Junior Match 2015, Team p2 CPSJ

We removed the middle square of

2 \times 2

8 \times 8

board. a) How many checkers can be placed on the remaining

60

boxes so that there are no two not jeopardize? b) How many at least checkers can be placed on the board so that they are at risk all

60

squares? (A lady is threatening the box she stands on, as well as any box she can get to in one move without going over any of the four removed boxes.)