Geometry, Plane Geometry

Plane Geometry concerns figures and shapes on a flat, two-dimensional surface. It covers properties of points, lines, angles, polygons (like triangles and quadrilaterals), and circles. Questions typically involve proofs, constructions, and calculations related to these elements.

Area Calculation Triangles Circles Symmetry Angle Calculation Pythagorean Theorem Triangle Inequality

• Question

  Is there a quadrilateral that can be cut into `6` parts by two straight cuts? Justify your answer or provide an example.

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  Topics:

  Geometry -> Plane Geometry Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems

• Area of the Shape

  Given a grid paper where the area of each square is one unit area. Find the area of the shape (in unit areas)

   

  

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  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 7

• Question

  The plane is divided by n lines and circles.

  Prove that the resulting map can be colored with two colors such that any two adjacent regions (separated by a segment or an arc) are colored with different colors.

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  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Induction (Mathematical Induction) Geometry -> Plane Geometry Proof and Example Combinatorics -> Colorings
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4

• Question

  Inside a square with side length 1, `n>=101` points are marked, such that no three are collinear. A triangle is called marked if its vertices are marked points. Prove that the area of one of the marked triangles is less than `1/100`

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  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4

• Question

  Given an infinite grid whose vertices are colored with two colors, blue and red. Prove that there exist two horizontal lines and two vertical lines such that their four intersection points are colored with the same color.

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  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry Combinatorics -> Colorings
  Sources:
  • Zebra Exercises, 2018-2019, Exercise 1 Question 1

• Hexagon

  In the diagram, there is a regular hexagon. By what factor is the area of the white region larger than the area of the shaded region?

  (A regular hexagon is a hexagon where all sides are equal and all angles are equal.)

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 6

• Products of Areas

  In the figure, there is a rectangle and a point inside it. Two segments are drawn through the point, parallel to the sides of the rectangle, dividing the rectangle into 4 smaller rectangles.
  Prove that the product of the areas of the shaded rectangles inside the rectangle is equal to the product of the areas of the unshaded rectangles inside the rectangle.

  

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  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Equations
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 4

• Puzzle

  You have a puzzle where each piece looks like this:

  

  
  The following shape is made up of 12 pieces. What is its perimeter?

  

  Topics:

  Arithmetic Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Word Problems
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 Question 6

• Another Gray Area

  In the figure, a rectangle of size `3 times 5` and a line that crosses it diagonally. Find the shaded area.

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage A, Grades 5-6 Question 3

• The Parallelogram

  In the drawing, there is a parallelogram with its diagonals drawn and the midpoints of two of its sides connected to opposite vertices.
  Which area is larger: the shaded or the striped?

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 7