MathDB

4

Part of 2007 All-Russian Olympiad

Problems(4)

$2007$ points on a circle

Source: All-Russian 2007

5/4/2007
A. Akopyan, A. Akopyan, A. Akopyan, I. Bogdanov A conjurer Arutyun and his assistant Amayak are going to show following super-trick. A circle is drawn on the board in the room. Spectators mark 20072007 points on this circle, after that Amayak removes one of them. Then Arutyun comes to the room and shows a semicircle, to which the removed point belonged. Explain, how Arutyun and Amayak may show this super-trick.
algorithmcombinatoricscircleRussiagametrick
a bisector of an acute triangle

Source: All-Russian 2007

5/4/2007
BB1BB_{1} is a bisector of an acute triangle ABCABC. A perpendicular from B1B_{1} to BCBC meets a smaller arc BCBC of a circumcircle of ABCABC in a point KK. A perpendicular from BB to AKAK meets ACAC in a point LL. BB1BB_{1} meets arc ACAC in TT. Prove that KK, LL, TT are collinear. V. Astakhov
geometrycircumcircleAsymptotecyclic quadrilateralangle bisectorgeometry proposed
sequence of $N$ (decimal) digits on a board

Source: All-Russian 2007

5/4/2007
Arutyun and Amayak show another effective trick. A spectator writes down on a board a sequence of NN (decimal) digits. Amayak closes two adjacent digits by a black disc. Then Arutyun comes and says both closed digits (and their order). For which minimal NN they may show such a trick? K. Knop, O. Leontieva
algorithmcombinatorics unsolvedcombinatorics
prove sequence contains an integer

Source: All-Russian 2007

5/4/2007
An infinite sequence (xn)(x_{n}) is defined by its first term x1>1x_{1}>1, which is a rational number, and the relation xn+1=xn+1⌊xn⌋x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor} for all positive integers nn. Prove that this sequence contains an integer. A. Golovanov
floor functionalgebra proposedalgebra