Cut $2003$ disjoint rectangles from an acute-angled triangle $ABC$, such that any of them has a parallel side to $AB$ and the sum of their areas is maximal.

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$

Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.

Is it true that for any permulation $a_1,a_2.....,a_{2002}$ of $1,2....,2002$ there are positive integers $m,n$ of the same parity such that $0<m<n<2003$ and $a_m+a_n=2a_{\frac {m+n}{2}}$

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$