MathDB

Problem 3

Part of 2005 Federal Math Competition of S&M

Problems(3)

sum √x>=sum xy if x+y+z=3 (Serbia MO 2005 1st Grade P3)

Source:

4/11/2021
If x,y,zx,y,z are nonnegative numbers with x+y+z=3x+y+z=3, prove that x+y+z≥xy+yz+xz.\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.
inequalitiesInequality
configuration in triangle with radii

Source: Serbia MO 2005 2nd Grade P3

4/11/2021
In a triangle ABCABC, DD is the orthogonal projection of the incenter II onto BCBC. Line DIDI meets the incircle again at EE. Line AEAE intersects side BCBC at point FF. Suppose that the segment IO is parallel to BCBC, where OO is the circumcenter of △ABC\triangle ABC. If RR is the circumradius and rr the inradius of the triangle, prove that EF=2(R−2r)EF=2(R-2r).
geometryTriangle
FE, floor of polynomial

Source: Serbia MO 2005 3&4th Grades P3

4/11/2021
Determine all polynomials pp with real coefficients for which p(0)=0p(0)=0 and f(f(n))+n=4f(n)for all n∈N,f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,where f(n)=⌊p(n)⌋f(n)=\lfloor p(n)\rfloor.
functional equationfealgebrapolynomial