MathDB

3

Part of 2000 IMO Shortlist

Problems(3)

Double functional equation

Source: IMO Shortlist 2000, A3

5/9/2007
Find all pairs of functions f:R→R f : \mathbb R \to \mathbb R, g:R→Rg : \mathbb R \to \mathbb R such that f \left( x + g(y) \right) = xf(y) - y f(x) + g(x)  \text{for all } x, y\in\mathbb{R}.
functionalgebrafunctional equationIMO Shortlist
Lines AD, BE, and CF are concurrent

Source: IMO Shortlist 2000, G3

8/10/2008
Let OO be the circumcenter and HH the orthocenter of an acute triangle ABCABC. Show that there exist points DD, EE, and FF on sides BCBC, CACA, and ABAB respectively such that OD+DH=OE+EH=OF+FH OD + DH = OE + EH = OF + FH and the lines ADAD, BEBE, and CFCF are concurrent.
geometrycircumcircleorthocenterTriangleconcurrencyIMO Shortlistgeometry solved
Vertices of a convex polygon if and only if m(S) = f(n)

Source: IMO Shortlist 2000, C3

8/10/2008
Let n≥4 n \geq 4 be a fixed positive integer. Given a set S \equal{} \{P_1, P_2, \ldots, P_n\} of n n points in the plane such that no three are collinear and no four concyclic, let at, a_t, 1≤t≤n, 1 \leq t \leq n, be the number of circles PiPjPk P_iP_jP_k that contain Pt P_t in their interior, and let m(S)=a1+a2+⋯+an.m(S)=a_1+a_2+\cdots + a_n. Prove that there exists a positive integer f(n), f(n), depending only on n, n, such that the points of S S are the vertices of a convex polygon if and only if m(S)=f(n). m(S) = f(n).
geometrycombinatoricscountingcombinatorial geometryIMO Shortlist