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

• Sets in the Plane

  A. Does there exist a set A in the plane such that its intersection with every circle contains exactly two points?

  B. Does there exist a set B in the plane such that its intersection with every circle of radius 1 contains exactly two points?

  Topics:

  Geometry -> Plane Geometry -> Circles Proof and Example -> Constructing an Example / Counterexample Set Theory Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems
  Sources:
  • Grossman Math Olympiad, 2006 Question 3

• Hexagonal Tiling

  Given two types of tiles. The shape of each tile of the first type is a regular hexagon with a side of length 1. The shape of each tile of the second type is a regular hexagon with a side of length 2. An unlimited supply of tiles of each type is given. Is it possible to tile the entire plane using these tiles, using both types of tiles?

  Topics:

  Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Grossman Math Olympiad, 2006 Question 4

• Triangle Side Lengths

  Let `n > 2` be an integer, and let ` t_1,t_2,...,t_n` be positive real numbers such that

  `(t_1+t_2+...+t_n)(1/t_1 + 1/t_2 + ... + 1/t_n) < n^2+1`

  Prove that for all i,j,k such that `1<=i<j<k<=n`, the triple of numbers `t_i,t_j,t_k` are the side lengths of a triangle.

  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Grossman Math Olympiad, 2006 Question 5

• A Walk in the Plane

  Given a Cartesian coordinate system x-y in the plane. You need to get from the point (1,0) to the point (2006,2005), where in each step you are allowed to move one unit up (in the positive direction of y) or one unit to the right (in the positive direction of the x-axis).

  a. In how many different paths can the task be performed?

  b. In how many different paths can the task be performed if it is forbidden at any stage to pass through a point on the line x=y?

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Binomial Coefficients and Pascal's Triangle Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Reflection
  Sources:
  • Grossman Math Olympiad, 2006 Question 7

• How a Wheel Turns

  All 6 wheels in the diagram rotate as they touch each other without slipping. The diameter of the leftmost wheel is 15.7 cm and it makes 12 revolutions per minute.

  It is known that the smallest wheel makes one revolution per second.

  What is the diameter of the smallest wheel?

  

  Topics:

  Arithmetic Geometry -> Plane Geometry -> Circles Algebra -> Word Problems
  Sources:
  • Bar-Ilan's weekly mathriddle competition, 2024 Question 7

• Paths in a Triangular Park

  In a park, there are 3 straight paths that form a triangle (there are no additional paths). The entrances to the park are at the midpoints of the paths, and a lamp hangs at each vertex of the triangle. From each entrance, the shortest walking distance along the park's paths to the lamp at the opposite vertex was measured. It turned out that 2 out of the 3 distances are equal to each other. Is the triangle necessarily isosceles?

  

  Topics:

  Geometry -> Plane Geometry -> Triangles Proof and Example -> Constructing an Example / Counterexample Geometry -> Plane Geometry -> Triangle Inequality
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 3

• Question

  Inside a square ABCD with side length 1, a point E is marked, and outside the square, a point F is marked, such that triangles ABE and DAF are equilateral. Calculate the area of the pentagon CBEFD.
  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 6

• DEFECTIVE OBSERVATION

  Our observation of little things is frequently defective, and our memories very liable to lapse. A certain judge recently remarked in a case that he had no recollection whatever of putting the wedding-ring on his wife's finger. Can you correctly answer these questions without having the coins in sight? On which side of a penny is the date given? Some people are so unobservant that, although they are handling the coin nearly every day of their lives, they are at a loss to answer this simple question. If I lay a penny flat on the table, how many other pennies can I place around it, every one also lying flat on the table, so that they all touch the first one? The geometrician will, of course, give the answer at once, and not need to make any experiment. He will also know that, since all circles are similar, the same answer will necessarily apply to any coin. The next question is a most interesting one to ask a company, each person writing down his answer on a slip of paper, so that no one shall be helped by the answers of others. What is the greatest number of three-penny-pieces that may be laid flat on the surface of a half-crown, so that no piece lies on another or overlaps the surface of the half-crown? It is amazing what a variety of different answers one gets to this question. Very few people will be found to give the correct number. Of course the answer must be given without looking at the coins.
  Topics:

  Geometry -> Plane Geometry -> Circles Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 28

• THE RAILWAY STATION CLOCK

  A clock hangs on the wall of a railway station, `71` ft. `9` in. long and `10` ft. `4` in. high. Those are the dimensions of the wall, not of the clock! While waiting for a train we noticed that the hands of the clock were pointing in opposite directions, and were parallel to one of the diagonals of the wall. What was the exact time?
  Topics:

  Algebra -> Word Problems Arithmetic -> Fractions Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 65

• THE THREE VILLAGES

  I set out the other day to ride in a motor-car from Acrefield to Butterford, but by mistake I took the road going via Cheesebury, which is nearer Acrefield than Butterford, and is twelve miles to the left of the direct road I should have travelled. After arriving at Butterford I found that I had gone thirty-five miles. What are the three distances between these villages, each being a whole number of miles? I may mention that the three roads are quite straight.
  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Word Problems Geometry -> Plane Geometry -> Pythagorean Theorem
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 69