MathDB

1

Part of 2018 Korea Winter Program Practice Test

Problems(2)

Functional Equation

Source: 2018 Korea Winter Program Practice Test 1 #1

1/7/2018
Find all functions f:R→Rf : \mathbb{R} \rightarrow \mathbb{R} satisfying the following conditions : 1) f(x+y)−f(x)−f(y)∈{0,1}f(x+y)-f(x)-f(y) \in \{0,1\} for all x,y∈Rx,y \in \mathbb{R} 2) ⌊f(x)⌋=⌊x⌋\lfloor f(x) \rfloor = \lfloor x \rfloor for all real xx.
functional equationfunctionfloor functionalgebra
Mixtilinear circle property

Source: 2018 Korea Winter Program Practice Test #5

1/7/2018
Let ΔABC\Delta ABC be a triangle with circumcenter OO and circumcircle ww. Let SS be the center of the circle which is tangent with ABAB, ACAC, and ww (in the inside), and let the circle meet ww at point KK. Let the circle with diameter ASAS meet ww at TT. If MM is the midpoint of BCBC, show that K,T,M,OK,T,M,O are concyclic.
mixtilinear incirclegeometrycircumcircle