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

  Semicircles are constructed on the legs and hypotenuse of a right-angled triangle, as shown in the figure. Which area is larger—the hatched area or the shaded area?

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Circles Geometry -> Plane Geometry -> Pythagorean Theorem

• Question

  A square is divided into several convex polygons (more than `1`), each of which has a different number of sides. Prove that among these polygons there is a triangle.

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Combinatorics -> Graph Theory Geometry -> Plane Geometry -> Triangles Proof and Example -> Proof by Contradiction Geometry -> Solid Geometry / Geometry in Space -> Polyhedra Minimum and Maximum Problems / Optimization Problems

• Question

  Is it possible to cut a triangle into four convex shapes: a triangle, a quadrilateral, a pentagon, and a hexagon?

  Topics:

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

• Question

  How many triangles are there in the image?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  A plane is colored with two colors (that is, every point on the plane is colored with one of these two colors). Prove that there exist two points on the plane at a distance of `1` such that they are both the same color.

  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Combinatorial Geometry Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Proof and Example -> Proof by Contradiction

• Question

  Let the sides of a triangle be a, b, c and the lengths of the corresponding medians be `m_a , m_b, m_c`. Show that

  `sum_{cyc} m_a / a >= {3( m_a + m_b + m_c)} /{a + b + c}`

  Show hint

  Log in here to receive the hint
  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 1

• Triangles in a Hexagon

  How many triangles are in the picture? Count all the triangles formed by the lines.

  

  Show hint

  Log in here to receive the hint
  Topics:

  Combinatorics Geometry -> Plane Geometry -> Triangles
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 3

• Product of Areas

  In the diagram, there is a quadrilateral with perpendicular diagonals. Prove that the product of the areas of the shaded regions within the quadrilateral is equal to the product of the areas of the unshaded regions within the quadrilateral.

  

  Show hint

  Log in here to receive the hint
  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 3

• Right Triangles and a Square

  Given a large number of congruent right triangles.

  The side lengths of each triangle are 3, 4, and 5.

  What is the maximum number of such triangles that can be placed inside a 20×20 square, such that they do not overlap?

  Show hint

  Log in here to receive the hint
  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Pythagorean Theorem
  Sources:
  • Gillis Mathematical Olympiad, 2016-2017 Question 3

• Hexagon and Triangle

  A regular hexagon and an equilateral triangle have the same perimeter. The area of the triangle is known to be 60. Find the area of the hexagon.

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 3-4 Question 7
  • Young Mathematician Olympiad, 2018-2019, Stage B, Grades 5-6 Question 5