MathDB

2

Part of 2001 Romania Team Selection Test

Problems(3)

Product of injective/surjective functions

Source: Romanian TST 2001

1/16/2011
a) Let f,g:Z→Zf,g:\mathbb{Z}\rightarrow\mathbb{Z} be one to one maps. Show that the function h:Z→Zh:\mathbb{Z}\rightarrow\mathbb{Z} defined by h(x)=f(x)g(x)h(x)=f(x)g(x), for all x∈Zx\in\mathbb{Z}, cannot be a surjective function.
b) Let f:Z→Zf:\mathbb{Z}\rightarrow\mathbb{Z} be a surjective function. Show that there exist surjective functions g,h:Z→Zg,h:\mathbb{Z}\rightarrow\mathbb{Z} such that f(x)=g(x)h(x)f(x)=g(x)h(x), for all x∈Zx\in\mathbb{Z}.
functionpigeonhole principlenumber theoryprime numbersalgebra proposedcombinatorics
Tangents AA',BB',CC',DD' form p with axis of symmetry

Source: Romanian TST 2001

1/16/2011
The vertices A,B,CA,B,C and DD of a square lie outside a circle centred at MM. Let AA′,BB′,CC′,DD′AA',BB',CC',DD' be tangents to the circle. Assume that the segments AA′,BB′,CC′,DD′AA',BB',CC',DD' are the consecutive sides of a quadrilateral pp in which a circle is inscribed. Prove that pp has an axis of symmetry.
symmetrygeometry proposedgeometry
No function satisfies inequality

Source: Romanian TST 2001

1/16/2011
Prove that there is no function f:(0,∞)→(0,∞)f:(0,\infty )\rightarrow (0,\infty) such that f(x+y)≥f(x)+yf(f(x))f(x+y)\ge f(x)+yf(f(x)) for every x,y∈(0,∞)x,y\in (0,\infty ).
functioninequalitiesalgebra unsolvedalgebra