MathDB

2

Part of 2015 Middle European Mathematical Olympiad

Problems(2)

f(x^2yf(x)) functional equation on nonzero reals

Source: MEMO 2015, problem T-2

8/28/2015
Determine all functions f:R∖{0}→R∖{0}f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\setminus\{0\} such that f(x2yf(x))+f(1)=x2f(x)+f(y)f(x^2yf(x))+f(1)=x^2f(x)+f(y) holds for all nonzero real numbers xx and yy.
algebrafunctional equation
Minimal number of inner diagonals iff none of them intersect

Source: MEMO 2015, problem I-2.

8/27/2015
Let n≥3n\ge 3 be an integer. An inner diagonal of a simple nn-gon is a diagonal that is contained in the nn-gon. Denote by D(P)D(P) the number of all inner diagonals of a simple nn-gon PP and by D(n)D(n) the least possible value of D(Q)D(Q), where QQ is a simple nn-gon. Prove that no two inner diagonals of PP intersect (except possibly at a common endpoint) if and only if D(P)=D(n)D(P)=D(n).
Remark: A simple nn-gon is a non-self-intersecting polygon with nn vertices. A polygon is not necessarily convex.
combinatoricspolygondiagonalscombinatorial geometry