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

  In the plane, a square and a point `P` are given. Prove that it is impossible for the distances from `P` to the vertices of the square to be `1`, `1`, `2`, and `3` centimeters?

  Topics:

  Geometry -> Plane Geometry -> Triangles Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality

• 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

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  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

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  Topics:

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

• Angles

  Calculate the sum of the marked angles:

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 5

• How many triangles?

  How many triangles are there in the picture?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Geometry -> Plane Geometry -> Angle Calculation Number Theory -> Division
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 7

• Length of the Segment

  On the side BC of triangle ABC, a point D is marked. The perimeter of triangle ABC is 15 centimeters, the perimeter of triangle ABD is 12 centimeters, and the perimeter of triangle ACD is 13 centimeters.
  What is the length of segment AD?

  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Equations
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 2