MathDB

3

Part of 2011 All-Russian Olympiad

Problems(6)

2011-gon

Source: All-Russian 2011

5/17/2011
A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?
inductionfloor functioncombinatorics proposedcombinatorics
Prove perimeters the same

Source: All-Russian 2011

5/17/2011
Let ABCABC be an equilateral triangle. A point TT is chosen on ACAC and on arcs ABAB and BCBC of the circumcircle of ABCABC, MM and NN are chosen respectively, so that MTMT is parallel to BCBC and NTNT is parallel to ABAB. Segments ANAN and MTMT intersect at point XX, while CMCM and NTNT intersect in point YY. Prove that the perimeters of the polygons AXYCAXYC and XMBNYXMBNY are the same.
geometryperimetercircumcircleparallelogramgeometric transformationhomothetygeometry proposed
ARO 2011 10-3

Source:

4/26/2011
The graph GG is not 33-coloured. Prove that GG can be divided into two graphs MM and NN such that MM is not 22-coloured and NN is not 11-coloured.
V. Dolnikov
combinatorics proposedcombinatoricsgraph theory
ARO 2011 10-7

Source:

5/6/2011
For positive integers a>b>1a>b>1, define xn=an−1bn−1x_n = \frac {a^n-1}{b^n-1} Find the least dd such that for any a,ba,b, the sequence xnx_n does not contain dd consecutive prime numbers.
V. Senderov
modular arithmeticnumber theoryrelatively primeprime numbersnumber theory proposed
Matching scientists and topics with certain restrictions

Source: ARO 2011 11-3

4/26/2011
There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme.
A. Magazinov
modular arithmeticgraph theoryextremal principlecombinatorics proposedcombinatorics
ARO 2011 11-7

Source:

4/28/2011
Let P(a)P(a) be the largest prime positive divisor of a2+1a^2 + 1. Prove that exist infinitely many positive integers a,b,ca, b, c such that P(a)=P(b)=P(c)P(a)=P(b)=P(c).
A. Golovanov
modular arithmeticalgebrapolynomialcalculusintegrationnumber theory proposednumber theory