5

Part of 2014 Czech-Polish-Slovak Junior Match

Problems(2)

largest possible sum of internal angles of n polygons a square is divdided

Source: Czech-Polish-Slovak Junior Match 2014, Individual p5 CPSJ

A square is given. Lines divide it into

n

polygons. What is he the largest possible sum of the internal angles of all polygons?

SumanglesPolygonsSqaurecombinatorial geometrycombinatorics

contructing positive integers when (a + b + 1) divides (a^2 + b^2 + 1) ?

Source: Czech-Polish-Slovak Junior Match 2014, Team p5 CPSJ

There is the number

1

on the board at the beginning. If the number

a

is written on the board, then we can also write a natural number

b

a + b + 1

is a divisor of

a^2 + b^2 + 1

. Can any positive integer appear on the board after a certain time? Justify your answer.

gamenumber theorydividesdivisor