MathDB

3

Part of 2021 Romania Team Selection Test

Problems(3)

Romania TST 2021 Day 1 P3

Source:

5/15/2021
The external bisectors of the angles of the convex quadrilateral ABCDABCD intersect each other in E,F,GE,F,G and HH such that A∈EH, B∈EF, C∈FG, D∈GHA\in EH, \ B\in EF, \ C\in FG, \ D\in GH. We know that the perpendiculars from EE to ABAB, from FF to BCBC and from GG to CDCD are concurrent. Prove that ABCDABCD is cyclic.
geometryRomanian TSTTSTcyclic quadrilateral
Romania TST 2021 Day 2 P3

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6/13/2021
Let P\mathcal{P} be a convex quadrilateral. Consider a point XX inside P.\mathcal{P}. Let M,N,P,QM,N,P,Q be the projections of XX on the sides of P.\mathcal{P}. We know that M,N,P,QM,N,P,Q all sit on a circle of center L.L. Let JJ and KK be the midpoints of the diagonals of P.\mathcal{P}. Prove that J,KJ,K and LL lie on a line.
geometryquadrilateralromaniaRomanian TSTTST
Romania TST 2021 Day 3 P3

Source:

6/14/2021
Let α\alpha be a real number in the interval (0,1).(0,1). Prove that there exists a sequence (εn)n≥1(\varepsilon_n)_{n\geq 1} where each term is either 00 or 11 such that the sequence (sn)n≥1(s_n)_{n\geq 1} sn=ε1n(n+1)+ε2(n+1)(n+2)+...+εn(2n−1)2ns_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}verifies the inequality 0≤α−2nsn≤2n+10\leq \alpha-2ns_n\leq\frac{2}{n+1} for any n≥2.n\geq 2.
algebraSequenceromaniaRomanian TSTTST