MathDB

4

Part of 1978 Romania Team Selection Test

Problems(4)

rhombus.

Source: Romanian TST 1978, Day 1, P4

9/28/2018
Diagonals AC AC and BD BD of a convex quadrilateral ABCD ABCD intersect a point O. O. Prove that if triangles OAB,OBC,OCD OAB,OBC,OCD and ODA ODA have the same perimeter, then ABCD ABCD is a rhombus. What happens if O O is some other point inside the quadrilateral?
geometryperimeterrhombusPure geometry
nice geometry involving distances

Source: Romanian TST 1978, Day 2, P4

9/30/2018
Let M \mathcal{M} a set of 3n≥3 3n\ge 3 planar points such that the maximum distance between two of these points is 1 1 . Prove that:
a) among any four points,there are two aparted by a distance at most 12. \frac{1}{\sqrt{2}} . b) for n=2 n=2 and any ϵ>0, \epsilon >0, it is possible that 12 12 or 15 15 of the distances between points from M \mathcal{M} lie in the interval (1−ϵ,1]; (1-\epsilon , 1]; but any 13 13 of the distances can´t be found all in the interval (12,1]. \left(\frac{1}{\sqrt 2} ,1\right]. c) there exists a circle of diameter 6 \sqrt{6} that contains M. \mathcal{M} . d) some two points of M \mathcal{M} are on a distance not exceeding 43n−3. \frac{4}{3\sqrt n-\sqrt 3} .
geometryalgebra
trigonometric equation in two variables

Source: Romanian TST 1978, Day 3, P4

9/30/2018
Solve the equation sin⁡xsin⁡2x⋯sin⁡nx+cos⁡xcos⁡2x⋯cos⁡nx=1, \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, for n∈N n\in\mathbb{N} and x∈R. x\in\mathbb{R} .
trigonometrytrigonometric equationequationsalgebra
Another discrete geometric problem

Source: Romanian TST 1978, Day 4, P4

10/1/2018
Let be some points on a plane, no three collinear. We associate a positive or a negative value to every segment formed by these. Prove that the number of points, the number of segments with negative associated value, and the number of triangles that has a negative product of the values of its sides, share the same parity.
geometry