MathDB

1

Part of 2008 Romania Team Selection Test

Problems(5)

No subsets divisible with n+1

Source: Romanian TST 1 2008, Problem 1

5/1/2008
Let n n be an integer, n≥2 n\geq 2. Find all sets A A with n n integer elements such that the sum of any nonempty subset of A A is not divisible by n\plus{}1.
modular arithmeticnumber theoryrelatively primecombinatorics proposedcombinatorics
Tangent circles and concyclic points

Source: Romanian TST 3 2008, Problem 1

6/7/2008
Let ABC ABC be a triangle with ∡BAC<∡ACB \measuredangle{BAC} < \measuredangle{ACB}. Let D D, E E be points on the sides AC AC and AB AB, such that the angles ACB ACB and BED BED are congruent. If F F lies in the interior of the quadrilateral BCDE BCDE such that the circumcircle of triangle BCF BCF is tangent to the circumcircle of DEF DEF and the circumcircle of BEF BEF is tangent to the circumcircle of CDF CDF, prove that the points A A, C C, E E, F F are concyclic. Author: Cosmin Pohoata
geometrycircumcirclegeometry proposed
Maximum value of a sum of square roots

Source: Romanian TST 2 2008, Problem 1

6/7/2008
Let n≥3 n \geq 3 be an odd integer. Determine the maximum value of \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|}, where xi x_{i} are positive real numbers from the interval [0,1] [0,1].
functioninequalities proposedinequalities
&quot;Updating&quot; China TST 2002

Source: Romanian TST 4 2008, Problem 1

6/13/2008
Let ABCD ABCD be a convex quadrilateral and let O∈AC∩BD O \in AC \cap BD, P∈AB∩CD P \in AB \cap CD, Q∈BC∩DA Q \in BC \cap DA. If R R is the orthogonal projection of O O on the line PQ PQ prove that the orthogonal projections of R R on the sidelines of ABCD ABCD are concyclic.
Gaussgeometry proposedgeometry
A permutation with distinct differences

Source: Romanian TST 5 2008, Problem 1

6/13/2008
Let n n be a nonzero positive integer. Find n n such that there exists a permutation σ∈Sn \sigma \in S_{n} such that \left| \{ |\sigma(k) \minus{} k| \ : \ k \in \overline{1, n} \}\right | = n.
inductiongraph theoryalgebra proposedalgebra