International Mathematical Olympiad '61, Second Day

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Here is the next note in the collection, A whole lot of interesting Olympiad Geometry! Post any progress you make and Enjoy!

4. (GDR) In the interior of ΔP1P2P3\Delta P_1P_2P_3 a point PP is given. Let Q1,Q2,Q_1, Q_2, and Q3Q_3 respectively be the intersections of PP1,PP2,PP_1, PP_2, and PP3PP_3 with the opposing edges of ΔP1P2P3\Delta P_1P_2P_3. Prove that among the ratios PP1/PQ1,PP2/PQ2,PP_1 / PQ_1, PP_2/PQ_2, and PP3/PQ3PP_3/PQ_3 there exists at least one not larger than 2 and at least one not smaller than 2.

5. (CZS) Construct a triangle ABCABC if the following elements are given: AC=b,AB=c,AC = b, AB = c, and AMB=ω(ω<90∘)AMB = \omega (\omega < 90^{\circ}), where MM is the midpoint of BCBC. Prove that the construction has a solution if and only if btan⁡ω2≤c<bb \tan{\dfrac{\omega}{2}} \leq c < b. In what case does equality hold?

6. (ROM) A plane ϵ\epsilon is given and on one side of the plane three noncollinear points A,B,A, B, and CC such that the plane determined by them is not parallel to ϵ\epsilon. Three arbitrary points A', B', and C′C' in ϵ\epsilon are selected. Let L,M,L, M, and NN be the midpoints of AA', BB', and CC′CC', and GG the centroid of ΔLMN\Delta LMN. Find the locus of all points obtained for G as A', B', and C′C' are varied (independently of each other) across ϵ\epsilon

This is part of the set International Mathematical Olympiads
#Geometry #Trigonometry #TriangleProperties #InternationalMathOlympiad(IMO) #MadMath

Note by Sualeh Asif
5 years, 7 months ago

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Comments

@Sharky Kesa @Xuming Liang Geometry this time!

Sualeh Asif - 5 years, 7 months ago
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