Reply to Re: I gave up to the Computers National Olympiad
If you don't have an account, just leave the password field blank.
Ok. Hope you'll understand my "English math".
1. The three points X(5,4,0), Y(3,0,2), Z(1,8,4), are the vertex of a cube. Find it's center.
2. m is a natural number, and the function F:M2(C)->M2(C), F(X)=X^m.
Prove that F is surjective if and only if m=1.
3. f:R->R is a function, and there exists L>0, so |f(x)-f(y)|>=L|x-y|, for every x,y, that belongs to R.
Prove that f is surjective if and only if f is continous.
4. Find all the continous functions g:R->R, with the following property: for every x that belongs to R, there exists a,b that belongs to (0,1), a+b=1, so g(x)=a*g(ax)+b*g(bx).(a and b aren't necessary the same for every x)
The first one was way too easy and the secod one was quite easy too.(I took 7/7 on both of them). The third one wasn't difficult, but it seems that I wasn't 100% rigurous(only 6.5/7). The last one was pretty difficult, but I managed to obtain 5 points(and I think the special prize I obtained was for this). I won't tell you yet how I solved them, I'll let you think a while on them.
1. The three points X(5,4,0), Y(3,0,2), Z(1,8,4), are the vertex of a cube. Find it's center.
2. m is a natural number, and the function F:M2(C)->M2(C), F(X)=X^m.
Prove that F is surjective if and only if m=1.
3. f:R->R is a function, and there exists L>0, so |f(x)-f(y)|>=L|x-y|, for every x,y, that belongs to R.
Prove that f is surjective if and only if f is continous.
4. Find all the continous functions g:R->R, with the following property: for every x that belongs to R, there exists a,b that belongs to (0,1), a+b=1, so g(x)=a*g(ax)+b*g(bx).(a and b aren't necessary the same for every x)
The first one was way too easy and the secod one was quite easy too.(I took 7/7 on both of them). The third one wasn't difficult, but it seems that I wasn't 100% rigurous(only 6.5/7). The last one was pretty difficult, but I managed to obtain 5 points(and I think the special prize I obtained was for this). I won't tell you yet how I solved them, I'll let you think a while on them.
