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-
The Orange Path
In the illustration, there is an orange path surrounding a blue square. The area of the path is 44% of the area of the square.
What is the width of the orange path as a percentage relative to the side length of the blue square?
Sources:Topics:Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Word Problems Arithmetic -> Percentages -
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.
Sources:
What is the length of segment AD? -
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.Sources:Topics:Arithmetic Algebra -> Word Problems -> Motion Problems Geometry -> Plane Geometry -> Angle Calculation -
Star of David in a Circle
Eight points are given on a circle. In how many different ways can a Star of David be drawn with vertices at these points?
Note: A Star of David is a figure obtained when two triangles intersect and their sides create exactly 6 intersection points.
Sources: -
Area of the Shape
In the image, the area of the semicircle is equal to 1. Find the area of the larger shape, given that all the curved lines in the image are quarter circles.
Sources:
-
Where is the point?
In a convex hexagon ABCDEF, triangles ACE and BDF are congruent and regular. Show that the three segments connecting the midpoints of opposite sides of the hexagon intersect at one point.
Sources:Topics:Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Vectors- Gillis Mathematical Olympiad, 2019-2020 Question 3
-
Cyclic Quadrilaterals
Given two triangles ACE, BDF
intersecting at 6 points: G,H,I,J,K,L
as shown in the figure. It is given that in each of the quadrilaterals
EFGI, DELH, CDKG, BCJL, ABIK a circle can be inscribed.
Is it possible that a circle can also be inscribed in quadrilateral FAHJ?
Sources:Topics:Geometry -> Solid Geometry / Geometry in Space Geometry -> Plane Geometry -> Circles Algebra -> Equations Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation- Gillis Mathematical Olympiad, 2019-2020 Question 5
-
Inscribed Circle in a Triangle
Inside a triangle there is a point P, whose distances from the lines containing the sides of the triangle are `d_a,d_b,d_c`. Let R denote the radius of the circumscribed circle of the triangle and r the radius of the inscribed circle in the triangle. Show that `sqrt(d_a)+sqrt(d_b)+sqrt(d_3)<= sqrt (2R+5r) `.
Sources:Topics:Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Circles Algebra -> Inequalities- Gillis Mathematical Olympiad, 2019-2020 Question 7
-
Triangles on Lines
Six congruent isosceles triangles are arranged as shown in the figure.
Show that points C, F, and M are collinear.
Sources:Topics:Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Triangles -> Triangle Congruence Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Parallel Translation / Translation- Gillis Mathematical Olympiad, 2018-2019 Question 3
-
Enlarging a Rectangle
A rectangle is given in the plane. Is it possible that after each side of the rectangle is increased by 1 cm, the area increases by 1 square meter? Provide an example or prove that it is impossible.
(If the rectangle is 1x5, it becomes 2x6 and no side can be 0)
Sources:- Grossman Math Olympiad, 2017, Juniors Question 1