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

  `120` identical spheres are arranged in the shape of a triangular pyramid. How many layers are there in the pyramid?

  Note: This is a pyramid, which is a three-dimensional shape, and not a triangle in a plane.

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  Topics:

  Geometry -> Solid Geometry / Geometry in Space Arithmetic Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence Number Theory -> Triangular Numbers

• Question

  A cube with a side of one meter is cut into cubes with a side of one centimeter. If we put all the resulting cubes in a row, what will be the length of the row (in kilometers)?

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Arithmetic Units of Measurement

• Question

  Can you fit a rectangle of size `5xx5` into a rectangle of size `4xx6`?

  Topics:

  Geometry -> Area Calculation

• Question

  Shlomi has a flat box with a size of `5xx5` centimeters. Shlomi claims that any rectangle that can be stored in this box must have all its sides smaller than 5 centimeters. Is he right?

  Topics:

  Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Pythagorean Theorem

• Question

  In a quadrilateral, the lengths of all diagonals and all sides are less than 1. Prove that the quadrilateral can be covered by a circle with a radius of 0.9.

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  Topics:

  Geometry -> Plane Geometry -> Circles Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Tournament of Towns, 1981-1982, Spring

• 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}`

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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.

  

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  Topics:

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

• 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