MathDB

3

Part of 2006 Germany Team Selection Test

Problems(4)

And if you interchange the digits...

Source: 1st German pre-TST 2006, problem 3

12/5/2005
Is the following statement true? For each positive integer nn, we can find eight nonnegative integers aa, bb, cc, dd, ee, ff, gg, hh such that n=2a−2b2c−2d⋅2e−2f2g−2hn=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}.
modular arithmeticnumber theory proposednumber theory
Set meets any plane at a finite but nonzero number of points

Source: 4th German TST 2006, written on 1 April 2006, problem 3; Engel, PSS, 12.3.2, problem 29

4/3/2006
Does there exist a set M M of points in space such that every plane intersects M M at a finite but nonzero number of points?
geometry proposedgeometry
Old inequality in new clothing

Source: 6th German TST 2006, 21 may 2006, problem 3

12/29/2006
Let nn be a positive integer, and let b1b_{1}, b2b_{2}, ..., bnb_{n} be nn positive reals. Set a1=b1b1+b2+...+bna_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}} and ak=b1+b2+...+bkb1+b2+...+bk−1a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}} for every k>1k>1. Prove the inequality a1+a2+...+an≤1a1+1a2+...+1ana_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}.
inequalitiesinductioninequalities proposed
cyclic quadrilateral and diagonals: OY perp. XY

Source: china

10/28/2004
The diagonals ACAC and BDBD of a cyclic quadrilateral ABCDABCD meet at a point XX. The circumcircles of triangles ABXABX and CDXCDX meet at a point YY (apart from XX). Let OO be the center of the circumcircle of the quadrilateral ABCDABCD. Assume that the points OO, XX, YY are all distinct. Show that OYOY is perpendicular to XYXY.
geometrycircumcirclecyclic quadrilateralgeometry solved