Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues. Expected questions involve calculating lengths, angles, areas, and volumes of various shapes, understanding geometric theorems, and solving problems related to spatial reasoning.

• Question

  Given two squares, each with a side length of `3` centimeters, and joined together to form a rectangle, what is the perimeter of the resulting rectangle?

  Topics:

  Arithmetic Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Word Problems

• Question

  Samuel wants to tile a room measuring `3` by `4` meters using square tiles with a side length of `25` centimeters. How many tiles does Samuel need?

  Topics:

  Arithmetic Geometry -> Area Calculation Algebra -> Word Problems Units of Measurement

• Question

  A cube with a side length of one meter is cut into cubes with a side length of one centimeter. If all the resulting cubes are placed in a row, what will be the length of the row?

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Arithmetic Algebra -> Word Problems Units of Measurement

• 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

  In a square with side length 1, a finite number of segments parallel to the sides of the square were drawn, with a total length of 18 (they can intersect). Prove that among the parts into which the square is divided by the segments, there is a part with an area of at least 0.01. 
  A. Engenes, A. Brazins 

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Inequalities Proof and Example -> Proof by Contradiction
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 6

• Question

  A regular polygon with 4k sides is divided into parallelograms. Prove that among these parallelograms there are at least k rectangles. Find the sum of the areas of all the rectangles.

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Geometry -> Solid Geometry / Geometry in Space -> Polyhedra -> Regular Polyhedra

• 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 M be a set of points in the plane. O is called a partial center of symmetry if it is possible to remove a point from M such that O is a regular center of symmetry of what remains. How many partial centers of symmetry can a finite set of points in the plane have? 
  V. Prasolov 

  Topics:

  Combinatorics -> Combinatorial Geometry Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Symmetry
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 2 Points 7

• 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