MathDB

1

Part of 2009 Saint Petersburg Mathematical Olympiad

Problems(3)

Sets of values.

Source: St Petersburg Olympiad 2009, Grade 11, P1

8/30/2017
f(x)=ax2+bx+c;a,b,cf(x)=ax^2+bx+c;a,b,c are reals. M={f(2n)∣n is integer},N={f(2n+1)∣n is integer}M=\{f(2n)|n \text{ is integer}\},N=\{f(2n+1)|n \text{ is integer}\} Prove that M=NM=N or M \cap N = \O
algebra
GCD of degrees

Source: St Petersburg Olympiad 2009, Grade 10, P1

8/30/2017
x,yx,y are naturals. GCM(x7,y4)∗GCM(x8,y5)=xyGCM(x^7,y^4)*GCM(x^8,y^5)=xy Prove that xyxy is cube
number theorygreatest common divisor
Division

Source: St Petersburg Olympiad 2009, Grade 9, P1

8/30/2017
b,cb,c are naturals. b∣c+1b|c+1 Prove that exists such natural x,y,zx,y,z that x+y=bz,xy=czx+y=bz,xy=cz
number theory