# International Mathematical Olympiad '61, Second Day

The goal of this set of notes is to improve our problem solving and proof writing skills. You are encouraged to submit a solution to any of these problems, and join in the discussion in #imo-discussion on Saturday 14 at 9:00 pm IST ,8 30 PDT. For more details, see IMO Problems Discussion Group.

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
4 years, 3 months ago

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@Sharky Kesa @Xuming Liang Geometry this time!

- 4 years, 3 months ago