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.

Solid Geometry / Geometry in Space Trigonometry Spherical Geometry Plane Geometry Vectors

• 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

• STEALING THE BELL-ROPES

  Two men broke into a church tower one night to steal the bell-ropes. The two ropes passed through holes in the wooden ceiling high above them, and they lost no time in climbing to the top. Then one man drew his knife and cut the rope above his head, in consequence of which he fell to the floor and was badly injured. His fellow-thief called out that it served him right for being such a fool. He said that he should have done as he was doing, upon which he cut the rope below the place at which he held on. Then, to his dismay, he found that he was in no better plight, for, after hanging on as long as his strength lasted, he was compelled to let go and fall beside his comrade. Here they were both found the next morning with their limbs broken. How far did they fall? One of the ropes when they found it was just touching the floor, and when you pulled the end to the wall, keeping the rope taut, it touched a point just three inches above the floor, and the wall was four feet from the rope when it hung at rest. How long was the rope from floor to ceiling?
  Topics:

  Algebra Geometry -> Plane Geometry -> Pythagorean Theorem
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 179

• 5 Degrees on a Clock

  At what time is there a 5-degree angle between the clock hands?

  Topics:

  Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 5

• Question

  Does there exist a convex quadrilateral such that each of its diagonals divides it into two acute triangles?

  Topics:

  Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation

• Perpendicular Hands

  How many times a day do the hour and minute hands lie on the same line, forming an angle of `180^@`? Are any of these lines perpendicular to each other?

  Topics:

  Logic Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation

• Question

  In the plane, a point and `12` lines passing through it are given. Prove that there are two of these lines such that the angle between them is less than `17^@`.

  Topics:

  Combinatorics -> Pigeonhole Principle Geometry -> Plane Geometry -> Angle Calculation

• Question

  Given `12` intersecting lines in the plane. Prove that there exist two of these lines such that the angle between them is less than `17^@`.

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  Topics:

  Combinatorics -> Pigeonhole Principle Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Parallel Translation / Translation

• Pentagon

  In a convex pentagon `ABCDE`, the following holds: `AE=AD`, `AB=AC`, and `angle CAD=angle ABE + angle AEB`.

  Let `AM` be the median to the side `BE` in the triangle `ABE`. Prove that `AM` is half the length of the segment `CD`.

  Topics:

  Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation
  Sources:
  • Tournament of Towns, 1983-1984, Fall, Practice Version, Grades 9-10 Question 2 Points 3

• Question

  Given a regular polygon with n vertices. Calculate the number of distinct (non-congruent) triangles whose vertices coincide with the vertices of the polygon.

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  Topics:

  Combinatorics Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4 Question 3

• Clock Angle

  How many minutes after 7:00 will the angle between the hour and minute hands be exactly one degree for the first time?
  Note: The clock hands move continuously at a constant speed.

  Topics:

  Arithmetic Algebra -> Word Problems -> Motion Problems Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 3
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 3