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4

Part of 1985 Greece National Olympiad

Problems(2)

classic telescopic, f(x)=4^x/(4^x+2) - 1985 Greece MO X p4

Source:

9/8/2024
Consider function f:R→Rf:\mathbb{R}\to \mathbb{R} with f(x)=4x4x+2,f(x)=\frac{4^x}{4^x+2}, for any x∈Rx\in \mathbb{R} a) Prove that f(x)+f(1−x)=1,f(x)+f(1-x)=1, b) Claculate the sum f(11986)+f(21986)+⋯f(19861986).f\left(\frac{1}{1986} \right)+f\left(\frac{2}{1986} \right)+\cdots f\left(\frac{1986}{1986} \right).
algebraSum
vector spaces problem from 1st Greek MO - 1985

Source: 1985 Greece MO Grade XII p4

9/6/2024
Given the vector spaces V,WV,W with coefficients over a field KK and function ϕ:V→W \phi :V\to W satisfying the relation : φ(λx+y)=λφ(x)+ϕ(y)\varphi(\lambda x+y)= \lambda \varphi(x)+\phi (y) for all x,y∈V,λ∈Kx,y \in V, \lambda \in K. Such a function is called linear.
Let Lφ={x∈V/φ(x)=0}L\varphi=\{x\in V/\varphi(x)=0\} , andM=φ(V)M=\varphi(V) , prove that :
(i) LφL\varphi is subspace of VV and MM is subspace of WW
(ii) Lφ=OL\varphi={O} iff φ\varphi is 1−11-1
(iii) Dimension of VV equals to dimension of LφL\varphi plus dimension of MM
(iv) If θ:R3→R3\theta : \mathbb{R}^3\to\mathbb{R}^3 with θ(x,y,z)=(2x−z,x−y,x−3y+z)\theta(x,y,z)=(2x-z,x-y,x-3y+z), prove that θ\theta is linear function . Find Lθ={x∈R3/θ(x)=0}L\theta=\{x\in {R}^3/\theta(x)=0\} and dimension of M=θ(R3)M=\theta({R}^3).
vectorfunctionlinear algebra