MathDB

3

Part of 2015 Costa Rica - Final Round

Problems(2)

functional with fractional part , {f(x)} sin^2 x+{x} cos(f(x))cosx=f(x)=f(f(x))

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 3.3

9/23/2021
Indicate (justifying your answer) if there exists a function f:R→Rf: R \to R such that for all x∈Rx \in R fulfills that
i) {f(x))}sin⁡2x+{x}cos(f(x))cosx=f(x)\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x) ii) f(f(x))=f(x)f (f(x)) = f(x)
where {m}\{m\} denotes the fractional part of mm. That is, {2.657}=0.657\{2.657\} = 0.657, and {−1.75}=0.25\{-1.75\} = 0.25.
algebrafractional partfunctional equationfunctional
partition a set X of n people into two sets Y and Z, A knows B ...

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 C2 p3

9/30/2021
In a set XX of n people, some know each other and others do not, where the relationship to know is symmetric; that is, if A A knows B B. then B B knows A A. On the other hand, given any4 4 people: A,B,CA, B, C and DD: if AA knows BB, BB knows CC and CC knows DD, then it happens at least one of the following three: AA knows C,BC, B knows DD or AA knows DD. Prove that XX can be partition into two sets YY and ZZ so that all elements of YY know all those of ZZ or no element in YY knows any in ZZ.
combinatorics