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 \(\Delta P_1P_2P_3\) a point \(P\) is given. Let \(Q_1, Q_2,\) and \(Q_3\) respectively be the intersections of \(PP_1, PP_2,\) and \(PP_3\) with the opposing edges of \(\Delta P_1P_2P_3\). Prove that among the ratios \(PP_1 / PQ_1, PP_2/PQ_2,\) and \(PP_3/PQ_3\) there exists at least one not larger than 2 and at least one not smaller than 2.

5. (CZS) Construct a triangle \(ABC\) if the following elements are given: \(AC = b, AB = c,\) and \(AMB = \omega (\omega < 90^{\circ})\), where \(M\) is the midpoint of \(BC\). Prove that the construction has a solution if and only if \[b \tan{\dfrac{\omega}{2}} \leq c < b\]. In what case does equality hold?

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

This is part of the set International Mathematical Olympiads

Note by Sualeh Asif
3 years, 2 months ago

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@Sharky Kesa @Xuming Liang Geometry this time!

Sualeh Asif - 3 years, 2 months ago

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