MathDB

1

Part of 2010 Saint Petersburg Mathematical Olympiad

Problems(3)

System of equations

Source: St Petersburg Olympiad 2010, Grade 11, P1

9/14/2017
Solve in positives xy=z,yz=x,zx=yx^y=z,y^z=x,z^x=y
algebrasystem of equations
Some polynomials

Source: St Petersburg Olympiad 2010, Grade 10, P1

9/14/2017
f(x)f(x) is square trinomial. Is it always possible to find polynomial g(x)g(x) with fourth degree, such that f(g(x))=0f(g(x))=0 has not roots?
algebrapolynomial
King in chess

Source: St Petersburg Olympiad 2010, Grade 9, P1

9/14/2017
Chess king is standing in some square of chessboard. Every sunday it is moved to one square by diagonal, and every another day it is moved to one square by horisontal or vertical. What maximal numbers of moves can be made ?
combinatorics