MathDB

2

Part of 2014 Korea National Olympiad

Problems(2)

easy functional equation

Source: Korea National 2014 #2

1/21/2015
Determine all the functions f:R→Rf : \mathbb{R}\rightarrow\mathbb{R} that satisfies the following.
f(xf(x)+f(x)f(y)+y−1)=f(xf(x)+xy)+y−1f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1
functionalgebra proposedalgebrafunctional equationKMOKorea
Find the # of functions

Source: Korean Mathematical Olympiad 2014 #6

1/24/2015
How many one-to-one functions f:{1,2,⋯ ,9}→{1,2,⋯ ,9}f : \{1, 2, \cdots, 9\} \rightarrow \{1, 2, \cdots, 9\} satisfy (i) and (ii)? (i) f(1)>f(2)f(1)>f(2), f(9)<9f(9)<9. (ii) For each i=3,4,⋯ ,8i=3, 4, \cdots, 8, if f(1),⋯ ,f(i−1)f(1), \cdots, f(i-1) are smaller than f(i)f(i), then f(i+1)f(i+1) is also smaller than f(i)f(i).
functioncountingderangementinductioncombinatorics proposedcombinatorics