\nopagenumbers
\input amssym.def
\input amssym.tex
\centerline{\bf ``TUYMAADA--2006''}
\medskip
\centerline{\it Senior league}
\medskip
\centerline{\tt First day}
\medskip
1. Seven different odd primes are given. Is it possible that the
difference of 8th powers of every two of them is divisible by each of the
remained numbers?
\rightline{({\it F. Petrov, K. Sukhov}\/)}
2. A sequence which is infinite in both directions is called
{\it Fibonacci-type sequence} if each of its terms is the sum of the two
preceding terms. How many Fibonacci-type sequences contain two successive
positive integral terms not exceeding $N$? (We do not distinguish between
sequences differing only by a shift of indices.)
\rightline{({\it I.Pevzner}\/)}
3. Points $A$ and $B$ are given in the plane. Line $l$ goes through $B$.
Consider an arbitrary circle $\omega$ touching $\ell$ at $B$ so that $A$
lies outside $\omega$. Tangents to $\omega$ from $A$ touch $\omega$ at
$X$ and $Y$.
Prove that line $XY$ passes through a fixed point independent on the
choice of $\omega$.
\rightline{({\it F.Bakharev}\/)}
4. Find all the
functions $f\colon (0, \infty)\to (0, \infty)$ such that $f(x+1)=f(x)+1$ and
$f\left({1\over f(x)}\right)={1\over x}$ for all positive $x$.
\rightline{({\it P. Volkmann}\/)}
\medskip
\centerline{\tt Second day}
\medskip
5. 100 boxers of different strength participate in the Boxing Championship
of Dirtytrickland. Each of them fights each other once. Several boxers
formed a plot: each of them put a leaden horseshoe in his boxing-glove
during one of his fights. When just one of two boxers has a horseshoe,
he wins; otherwise, the stronger boxer wins. It turned out after
the championship that three boxers won more fights than any of the three
strongest participants.
What is the minimum possible number of plotters?
\rightline{({\it N.Kalinin}\/)}
6. $H$ is the orthocentre of an acute triangle $ABC$ and $M$ is the point
of intersection of its medians. $B_1$ is the midpoint of arc $AC$ of
the circumcircle of $ABC$. It is known that $B_1M$ is equal to the radius
of the circumcircle. Prove that $BM\geq BH$.
\rightline{({\it F.Bakharev}\/)}
7. The {\it corner} consists of all the squares of the first row and the
first column of $n\times (n-1)$ cardboard rectangle (that is, the corner contains
$2n-2$ squares). All the squares of an infinite squared plane are coloured
by $k$ colours so that all the squares covered by the corner in any
position are of different colour (the corner can be rotated and turned
upside down). For what minimum $k$ is it possible?
\rightline{({\it S. Berlov}\/)}
8. {\it Set of exponents} of a positive integer is the unordered list of
exponents of primes appearing in its factorization. For example,
the numbers $180=2^2\cdot 3^2\cdot 5^1$ and $882=3^2\cdot 2^1\cdot 7^2$
have the same set of exponents 1, 2, 2. Two increasing arithmetical
progressions $(a_n)$ and $(b_n)$ are such that the numbers $(a_n)$ and
$(b_n)$ have the same set of exponents for each $n$. Prove that the
progressions are proportional.
\rightline{({\it A. Golovanov}\/)}
\end