MathDB

2001 Tuymaada Olympiad

Part of Tuymaada Olympiad

Subcontests

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The unit square is divided into 10^12 squares..

Unit square ABCDABCD is divided into 101210^{12} smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal ACAC does not exceed 1500.
Proposed by A. Kanel-Belov

Coloring of the positive reals : possible or not?

Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.)
Proposed by A. Golovanov

1-100 contained in union of N geometric progressions, prove N >=31

Natural numbers 1,2,3,..,1001, 2, 3,.., 100 are contained in the union of NN geometric progressions (not necessarily with integer denominations). Prove that N≥31N \ge 31
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A convex quadrilateral, bisectors, circumcircles..

ABCDABCD is a convex quadrilateral; half-lines DADA and CBCB meet at point QQ; half-lines BABA and CDCD meet at point PP. It is known that ∠AQB=∠APD\angle AQB=\angle APD. The bisector of angle ∠AQB\angle AQB meets the sides ABAB and CDCD of the quadrilateral at points XX and YY, respectively; the bisector of angle ∠APD\angle APD meets the sides ADAD and BCBC at points ZZ and TT, respectively. The circumcircles of triangles ZQTZQT and XPYXPY meet at point KK inside the quadrilateral. Prove that KK lies on the diagonal ACAC.
Proposed by S. Berlov

Wanted : 3 quadratic polynomials such that P(x)+Q(y)=R(z).

Do there exist quadratic trinomials P,  Q,  RP, \ \ Q, \ \ R such that for every integers xx and yy an integer zz exists satisfying P(x)+Q(y)=R(z)?P(x)+Q(y)=R(z)?
Proposed by A. Golovanov

concyclic points, Tuymaada juniors 2001

Let ABC be an acute isosceles triangle (AB=BCAB=BC) inscribed in a circle with center OO . The line through the midpoint of the chord ABAB and point OO intersects the line ACAC at LL and the circle at the point PP. Let the bisector of angle BACBAC intersects the circle at point KK. Lines ABAB and PKPK intersect at point DD. Prove that the points L,B,DL,B,D and PP lie on the same circle.
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Diophantine equation : (a^2,b^2)+(a,bc)+(b,ac)+(c,ab)=199.

Solve the equation (a2,b2)+(a,bc)+(b,ac)+(c,ab)=199.(a^{2},b^{2})+(a,bc)+(b,ac)+(c,ab)=199. in positive integers. (Here (x,y)(x,y) denotes the greatest common divisor of xx and yy.)
Proposed by S. Berlov

An arrangement of non-zero numbers in a square.

Non-zero numbers are arranged in n×nn \times n square (n>2n>2). Every number is exactly kk times less than the sum of all the other numbers in the same cross (i.e., 2n−22n-2 numbers written in the same row or column with this number). Find all possible kk.
Proposed by D. Rostovsky, A. Khrabrov, S. Berlov

integers in an inifite chessboard so that 101 divides any sum of 10 numbers

Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any 1010 numbers in a row vertically or horizontal, would be divisible by 101101?
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Let us play volleyball, shall we?

Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored nn-th has xnx_{n} points (n=1,…,10n=1, \dots, 10), prove that x1+2x2+⋯+10x10≥165x_{1}+2x_{2}+\dots+10x_{10}\geq 165.
Proposed by D. Teryoshin

16 players in a chess tournament, 15 get first place, points of 16th ?

1616 chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given 11 point, for a draw 0.50.5 points, for defeat 00 points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

A special partition of the positive integers.

All positive integers are distributed among two disjoint sets N1N_{1} and N2N_{2} such that no difference of two numbers belonging to the same set is a prime greater than 100.
Find all such distributions.
Proposed by N. Sedrakyan