MathDB

2006 Tuymaada Olympiad

Part of Tuymaada Olympiad

Subcontests

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hard function

Find all functions f:(0,∞)→(0,∞)f: (0,\infty)\rightarrow(0,\infty) with the following properties: f(x+1)=f(x)+1f(x+1)=f(x)+1 and f(1f(x))=1xf\left(\frac{1}{f(x)}\right)=\frac{1}{x}.
Proposed by P. Volkmann

set of exponents and arithmetical progressions

For a positive integer, we define it's set of exponents the unordered list of all the exponents of the primes, in it`s decomposition. For example, 18=2⋅3218=2\cdot 3^{2} has it`s set of exponents 1,21,2 and 300=22⋅3⋅52300=2^{2}\cdot 3\cdot 5^{2} has it`s set of exponents 1,2,21,2,2. There are given two arithmetical progressions (an)n\big(a_{n}\big)_{n} and (bn)n\big(b_{n}\big)_{n}, such that for any positive integer nn, ana_{n} and bnb_{n} have the same set of exponents. Prove that the progressions are proportional (that is, there is kk such that an=kbna_{n}=kb_{n} for any nn).
Proposed by A. Golovanov
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a line and a fixed point

A line dd is given in the plane. Let B∈dB\in d and AA another point, not on dd, and such that ABAB is not perpendicular on dd. Let ω\omega be a variable circle touching dd at BB and letting AA outside, and XX and YY the points on ω\omega such that AXAX and AYAY are tangent to the circle. Prove that the line XYXY passes through a fixed point.
Proposed by F. Bakharev

colouring squares

From a n×(n−1)n\times (n-1) rectangle divided into unit squares, we cut the corner, which consists of the first row and the first column. (that is, the corner has 2n−22n-2 unit squares). For the following, when we say corner we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with kk colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of kk.
Proposed by S. Berlov
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fibonacci sequences

We call a sequence of integers a Fibonacci-type sequence if it is infinite in both ways and an=an−1+an−2a_{n}=a_{n-1}+a_{n-2} for any n∈Zn\in\mathbb{Z}. How many Fibonacci-type sequences can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than NN ? (two sequences are considered to be the same if they differ only by shifting of indices)
Proposed by I. Pevzner

an inequality in triangle

Let ABCABC be a triangle, GG it`s centroid, HH it`s orthocenter, and MM the midpoint of the arc AC^\widehat{AC} (not containing BB). It is known that MG=RMG=R, where RR is the radius of the circumcircle. Prove that BG≥BHBG\geq BH.
Proposed by F. Bakharev
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