×

# 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
1 year, 9 months ago

Sort by: