Geometry, Plane Geometry, Angle Calculation

This topic focuses on determining the measures of angles within geometric figures (like polygons, triangles) or those formed by intersecting lines, using fundamental geometric properties and theorems (e.g., sum of angles in a triangle, properties of parallel lines).

• 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

  There is a billiard table in the shape of a triangle whose angles are equal to \(90^{\circ}\), \(30^{\circ}\) and \(60^{\circ}\). 

   

  Given a right triangle shaped billiard table, with "pockets" in its corners. One of its acute angles is \(30^{\circ}\). From this corner (the thirty-degree angle) a ball is launched towards the midpoint of the opposite side of the triangle (the median). Prove that if the ball is reflected more than eight times (angle of incidence equals angle of reflection), then eventually the ball will enter the "pocket" located at the 60-degree corner of the triangle.

  Topics:

  Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Reflection

• Question

  Let ABCD be a convex quadrilateral inscribed in a circle such that its diagonals are perpendicular to each other. Let O be the center of the circle. Prove that the broken line AOC divides the quadrilateral into two parts of equal area. 
  V. Varvarkin 

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 3 Points 5

• 5 Degrees on a Clock

  At what time is there a 5-degree angle between the clock hands?

  Topics:

  Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 5

• Question

  Does there exist a convex quadrilateral such that each of its diagonals divides it into two acute triangles?

  Topics:

  Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation

• Perpendicular Hands

  How many times a day do the hour and minute hands lie on the same line, forming an angle of `180^@`? Are any of these lines perpendicular to each other?

  Topics:

  Logic Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation

• Question

  In the plane, a point and `12` lines passing through it are given. Prove that there are two of these lines such that the angle between them is less than `17^@`.

  Topics:

  Combinatorics -> Pigeonhole Principle Geometry -> Plane Geometry -> Angle Calculation

• Question

  Given `12` intersecting lines in the plane. Prove that there exist two of these lines such that the angle between them is less than `17^@`.

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  Topics:

  Combinatorics -> Pigeonhole Principle Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Parallel Translation / Translation

• Pentagon

  In a convex pentagon `ABCDE`, the following holds: `AE=AD`, `AB=AC`, and `angle CAD=angle ABE + angle AEB`.

  Let `AM` be the median to the side `BE` in the triangle `ABE`. Prove that `AM` is half the length of the segment `CD`.

  Topics:

  Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 2 Points 3

• Question

  Given a regular polygon with n vertices. Calculate the number of distinct (non-congruent) triangles whose vertices coincide with the vertices of the polygon.

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  Topics:

  Combinatorics Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4 Question 3