Geometry, Vectors

Vectors are quantities that have both magnitude (length) and direction. They are often used in geometry to represent displacements, forces, or velocities. Questions may involve vector addition, subtraction, scalar multiplication, and their applications in geometric problems and proofs.

• Question

  In space, there are 30 non-degenerate vectors. Prove that there are at least 2 such that the angle between them is no greater than 45 degrees. 
  A. Tulpigo

  Topics:

  Geometry -> Trigonometry Geometry -> Spherical Geometry Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry -> Angle Calculation Geometry -> Vectors
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 4

• Question

  Given a convex quadrilateral ABCD. Each of its sides is divided into K equal parts. Points on side AB are connected to corresponding points on CD, and points on BC are connected to points on DA, creating K² smaller quadrilaterals. From these, K quadrilaterals are chosen such that any two quadrilaterals are separated by at least one line connecting AB and CD, and one line connecting BC and DA. Prove that the sum of the areas of these quadrilaterals is S_ABCD/K. 
  By A. Angans. 

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Geometry -> Vectors
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 5

• Question

  Consider an arbitrary hexagon and denote the midpoints of its sides by `M_1,M_2,M_3,M_4,M_5,M_6`. Prove that the segments `M_1M_2`, `M_3M_4` and `M_5M_6` can form a triangle, even without rotating these segments.

  Topics:

  Geometry -> Vectors

• Question

  Given two parallelograms: `ABCD` and `A^'B^'C^'D^'`. Let `A^″ ` be the midpoint of the segment `A\A^'`, `B^″ ` be the midpoint of the segment `B\B^'`, and so on. Prove that `A^″B^″ C^″ D^″ ` is a parallelogram.

  Topics:

  Geometry -> Plane Geometry Geometry -> Vectors

• Where is the point?

  In a convex hexagon ABCDEF, triangles ACE and BDF are congruent and regular. Show that the three segments connecting the midpoints of opposite sides of the hexagon intersect at one point.

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  Topics:

  Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Vectors
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 3