MathDB

1

Part of 2009 Germany Team Selection Test

Problems(5)

Germany: Three lines AA', BB' and CC' have a common point.

Source: German TST 4, P1, 2009, Exam set by Christian Reiher

7/18/2009
Let I I be the incircle centre of triangle ABC ABC and ω \omega be a circle within the same triangle with centre I. I. The perpendicular rays from I I on the sides BC‾,CA‾ \overline{BC}, \overline{CA} and AB‾ \overline{AB} meets ω \omega in A′,B′ A', B' and C′. C'. Show that the three lines AA′,BB′ AA', BB' and CC′ CC' have a common point.
geometrygeometry unsolved
German chordal quadrilateral

Source: German TST 3, P1, 2009, Exam set by Gunther Vogel

7/18/2009
Let ABCD ABCD be a chordal/cyclic quadrilateral. Consider points P,Q P,Q on AB AB and R,S R,S on CD CD with \overline{AP}: \overline{PB} \equal{} \overline{CS}: \overline{SD},   \overline{AQ}: \overline{QB} \equal{} \overline{CR}: \overline{RD}. How to choose P,Q,R,S P,Q,R,S such that \overline{PR} \cdot \overline{AB} \plus{} \overline{QS} \cdot \overline{CD} is minimal?
geometry unsolvedgeometry
Consider cubes of edge length 5 composed of 125 cubes

Source: German TST 5, P1, 2009

7/18/2009
Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer n n is called "representable" if there is a big cube with exactly n n small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances 1,2,3,4 1,2,3,4 (zero counts as even number). (a) What is the smallest and biggest representable number? (b) Construct 45 representable numbers.
geometry3D geometryalgebra unsolvedalgebra
Product Prod^n_{i=1} A_i+k is a power for each n in N

Source: German TST 7, P1, 2009, Exam set by Christian Reiher

7/18/2009
For which n≥2,n∈N n \geq 2, n \in \mathbb{N} are there positive integers A1,A2,…,An A_1, A_2, \ldots, A_n which are not the same pairwise and have the property that the product \prod^n_{i \equal{} 1} (A_i \plus{} k) is a power for each natural number k. k.
number theoryprime numbersnumber theory unsolved
2^{m-1} can be divided by 127m without residue

Source: VAIMO 1, German Pre-TST 2009

7/16/2011
Let p>7p > 7 be a prime which leaves residue 1 when divided by 6. Let m=2p−1,m=2^p-1, then prove 2m−1−12^{m-1}-1 can be divided by 127m127m without residue.
number theory unsolvednumber theory