MathDB

3

Part of 2001 District Olympiad

Problems(6)

Romania District Olympiad 2001 - VII Grade

Source:

3/12/2011
Consider a triangle ΔABC\Delta ABC and three points D,E,FD,E,F such that: BB and EE are on different side of the line ACAC, CC and DD are on different sides of ABAB, AA and FF are on the same side of the line BCBC. Also ΔADB∼ΔCEA∼ΔCFB\Delta ADB \sim \Delta CEA \sim \Delta CFB. Let MM be the middle point of AFAF. Prove that:
a)ΔBDF∼ΔFEC\Delta BDF \sim \Delta FEC. b) MM is the middle point of DEDE.
Dan Branzei
symmetrygeometryparallelogramgeometry proposed
Romania District Olympiad 2001 - VIII Grade

Source:

3/12/2011
Consider four points A,B,C,DA,B,C,D not in the same plane such that
AB=BD=CD=AC=2AD=22BC=aAB=BD=CD=AC=\sqrt{2} AD=\frac{\sqrt{2}}{2}BC=a
Prove that:
a)There is a point M∈[BC]M\in [BC] such that MA=MB=MC=MDMA=MB=MC=MD. b)2m(∢(AD,BC))=3m(∢((ABC),(BCD)))2m(\sphericalangle(AD,BC))=3m(\sphericalangle((ABC),(BCD))) c)6(d(A,CD))2=7(d(A,(BCD)))26(d(A,CD))^2=7(d(A,(BCD)))^2
Ion Trandafir
geometry proposedgeometry
Romania District Olympiad 2001 - Grade IX

Source:

3/12/2011
Conside a positive odd integer kk and let n1<n2<…<nkn_1<n_2<\ldots<n_k be kk positive odd integers. Prove that:
n12−n22+n32−n42+…+nk2≥2k2−1n_1^2-n_2^2+n_3^2-n_4^2+\ldots+n_k^2\ge 2k^2-1
Titu Andreescu
number theory proposednumber theory
Romania District Olympiad 2001 - Grade X

Source:

3/16/2011
Consider an inscriptible polygon ABCDEABCDE. Let H1,H2,H3,H4,H5H_1,H_2,H_3,H_4,H_5 be the orthocenters of the triangles ABC,BCD,CDE,DEA,EABABC,BCD,CDE,DEA,EAB and let M1,M2,M3,M4,M5M_1,M_2,M_3,M_4,M_5 be the midpoints of DE,EA,AB,BCDE,EA,AB,BC and CDCD, respectively. Prove that the lines H1M1,H2M2,H3M3,H4M4,H5M5H_1M_1,H_2M_2,H_3M_3,H_4M_4,H_5M_5 have a common point.
Dinu Serbanescu
geometrycircumcirclegeometry proposed
Romania District Olympiad 2001 - Grade XII

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3/16/2011
Consider a continuous function f:[0,1]→Rf:[0,1]\rightarrow \mathbb{R} such that for any third degree polynomial function P:[0,1]→[0,1]P:[0,1]\to [0,1], we have
∫01f(P(x))dx=0\int_0^1f(P(x))dx=0
Prove that f(x)=0, (∀)x∈[0,1]f(x)=0,\ (\forall)x\in [0,1].
Mihai Piticari
functionalgebrapolynomialintegrationcalculusreal analysisreal analysis unsolved
Romania District Olympiad 2001 - Grade XI

Source:

3/16/2011
Let f:R→Rf:\mathbb{R}\to \mathbb{R} a function which transforms any closed bounded interval in a closed bounded interval and any open bounded interval in an open bounded interval.
Prove that ff is continuous.
Mihai Piticari
functionreal analysisreal analysis unsolved