MathDB

6

Part of 1990 IMO Longlists

Problems(2)

Show that the fraction isn't an integer - ILL 1990 COL2

Source:

9/18/2010
Let function f:Z≥00→Nf : \mathbb Z_{\geq 0}^0 \to \mathbb N satisfy the following conditions:
(i) f(0,0,0)=1; f(0, 0, 0) = 1;
(ii) f(x,y,z)=f(x−1,y,z)+f(x,y−1,z)+f(x,y,z−1);f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);
(iii) when applying above relation iteratively, if any of x′,y′,zx', y', z' is negative, then f(x′,y′,z′)=0.f(x', y', z') = 0.
Prove that if x,y,zx, y, z are the side lengths of a triangle, then (f(x,y,z))kf(mx,my,mz)\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)} is not an integer for any integers k,m>1.k, m > 1.
functionalgebrafunctional equationtriangle inequalityIMO ShortlistIMO Longlist
Geometric Inequality on the lengths - ILL 1990 YUG1

Source:

9/19/2010
Let S,TS, T be the circumcenter and centroid of triangle ABCABC, respectively. MM is a point in the plane of triangle ABCABC such that 90∘≤∠SMT<180∘90^\circ \leq \angle SMT < 180^\circ. A1,B1,C1A_1, B_1, C_1 are the intersections of AM,BM,CMAM, BM, CM with the circumcircle of triangle ABCABC respectively. Prove that MA1+MB1+MC1≥MA+MB+MC.MA_1 + MB_1 + MC_1 \geq MA + MB + MC.
inequalitiesgeometrycircumcirclegeometry proposed