MathDB

3

Part of 2004 All-Russian Olympiad

Problems(6)

on a table there are 2004 boxes

Source: Russian Olympiad 2004, problem 9.3

5/3/2004
On a table there are 2004 boxes, and in each box a ball lies. I know that some the balls are white and that the number of white balls is even. In each case I may point to two arbitrary boxes and ask whether in the box contains at least a white ball lies. After which minimum number of questions I can indicate two boxes for sure, in which white balls lie?
algorithmgroup theoryabstract algebragraph theorycombinatorics unsolvedcombinatorics
natural numbers from 1 to 100 are arranged on a circle

Source: Russian Olympiad 2004, problem 9.7

5/3/2004
The natural numbers from 1 to 100 are arranged on a circle with the characteristic that each number is either larger as their two neighbours or smaller than their two neighbours. A pair of neighbouring numbers is called "good", if you cancel such a pair, the above property remains still valid. What is the smallest possible number of good pairs?
combinatorics unsolvedcombinatorics
quadrilateral which is a cyclic quadrilateral and tangent qu

Source: Russian Olympiad 2004, problem 10.3

5/3/2004
Let ABCD ABCD be a quadrilateral which is a cyclic quadrilateral and a tangent quadrilateral simultaneously. (By a tangent quadrilateral, we mean a quadrilateral that has an incircle.) Let the incircle of the quadrilateral ABCD ABCD touch its sides AB AB, BC BC, CD CD, and DA DA in the points K K, L L, M M, and N N, respectively. The exterior angle bisectors of the angles DAB DAB and ABC ABC intersect each other at a point K′ K^{\prime}. The exterior angle bisectors of the angles ABC ABC and BCD BCD intersect each other at a point L′ L^{\prime}. The exterior angle bisectors of the angles BCD BCD and CDA CDA intersect each other at a point M′ M^{\prime}. The exterior angle bisectors of the angles CDA CDA and DAB DAB intersect each other at a point N′ N^{\prime}. Prove that the straight lines KK′ KK^{\prime}, LL′ LL^{\prime}, MM′ MM^{\prime}, and NN′ NN^{\prime} are concurrent.
geometryincentertrigonometryratiocyclic quadrilateralexterior angleangle bisector
A triangle T is contained inside a point-symmetrical polygon

Source: Russian Olympiad 2004, problem 10.7

5/3/2004
A triangle T T is contained inside a point-symmetrical polygon M. M. The triangle T′ T' is the mirror image of the triangle T T with the reflection at one point P P, which inside the triangle T T lies. Prove that at least one of the vertices of the triangle T′ T' lies in inside or on the boundary of the polygon M. M.
geometrygeometric transformationreflectionsymmetryhomothetyparallelogramgeometry solved
some of these cities are connected by airlines

Source: Russian Olympiad 2004, problem 11.7

5/4/2004
In a country there are several cities; some of these cities are connected by airlines, so that an airline connects exactly two cities in each case and both flight directions are possible. Each airline belongs to one of kk flight companies; two airlines of the same flight company have always a common final point. Show that one can partition all cities in k+2k+2 groups in such a way that two cities from exactly the same group are never connected by an airline with each other.
inductioninequalitiescombinatorics unsolvedcombinatorics
polynomial identity

Source: Russian Olympiad 2004, problem 11.3

5/4/2004
The polynomials P(x) P(x) and Q(x) Q(x) are given. It is known that for a certain polynomial R(x,y) R(x, y) the identity P(x) \minus{} P(y) \equal{} R(x, y) (Q(x) \minus{} Q(y)) applies. Prove that there is a polynomial S(x) S(x) so that P(x) \equal{} S(Q(x))   \forall x.
algebrapolynomialcalculusRussia