MathDB

2

Part of 1978 Romania Team Selection Test

Problems(4)

gcd (number theory)

Source: Romanian TST 1978, Day 1, P2

9/28/2018
Suppose that k,l k,l are natural numbers such that gcd⁡(11m−1,k)=gcd⁡(11m−1,l), \gcd (11m-1,k)=\gcd (11m-1, l) , for any natural number m. m. Prove that there exists an integer n n such that k=11nl. k=11^nl.
number theorygreatest common divisorGCDmodular arithmetic
declaration of f wasn't necessary...

Source: Romanian TST 1978, Day 2, P2

9/30/2018
Prove that there is a function F:N⟶N F:\mathbb{N}\longrightarrow\mathbb{N} satisfying (F∘F)(n)=n2, (F\circ F) (n) =n^2, for all n∈N. n\in\mathbb{N} .
functional equationfunctionalgebra
rotation of a plane

Source: Romanian TST 1978, Day 3, P2

9/30/2018
Points A’,B,C’ A’,B,C’ are arbitrarily taken on edges SA,SB, SA,SB, respectively, SC SC of a tetrahedron SABC. SABC. Plane forrmed by ABC ABC intersects the plane ρ, \rho , formed by A’B’C’, A’B’C’, in a line d. d. Prove that, meanwhile the plane ρ \rho rotates around d, d, the lines AA’,BB’ AA’,BB’ and CC’ CC’ are, and remain concurrent. Find de locus of the respective intersections.
geometry3D geometrytetrahedronrotationLocusPure geometrygeometric transformation
f|_S=F|_S

Source: Romanian TST 1978, Day 4, P2

10/1/2018
Let k k be a natural number. A function f:S:={x1,x2,...,xk}⟶R f:S:=\left\{ x_1,x_2,...,x_k\right\}\longrightarrow\mathbb{R} is said to be additive if, whenever n1x1+n2x2+⋯+nkxk=0, n_1x_1+n_2x_2+\cdots +n_kx_k=0, it holds that n1f(x1)+n2f(x2)+⋯+nkf(xk)=0, n_1f\left( x_1\right)+n_2f\left( x_2\right)+\cdots +n_kf\left( x_k\right)=0, for all natural numbers n1,n2,...,nk. n_1,n_2,...,n_k.
Show that for every additive function and for every finite set of real numbers T, T, there exists a second function, which is a real additive function defined on S∪T S\cup T and which is equal to the former on the restriction S. S.
functionalgebraromania