Enclosing the Center
If we randomly select 3 vertices of a regular hexagon to form a triangle, what are the respective probabilities that the center of the hexagon is i) inside, ii) outside, iii) on the edge of the triangle?
Lucas Sequences
A Lucas Sequence is made from two starting terms and a recursive formula for the next term using the two previous ones.
Solid Angles
Let’s consider the 3D version of regular planar angles, called solid angles. If an angle can be thought of as a portion of a full circle, then a solid angle is a portion of a full sphere.
Integer Coefficients
Consider expanding this expression: f(x) = (ax+b)(cx+d).
Divisible Sums
Problem of the Week #77: Monday January 20th, 2025As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability…
Line Stitching
Consider all 1 unit long line segments that have an end point on the line y=0 and the other on x=0.
Inscribed Perimeters
If a quadrilateral has 1 vertex on each side of a square of side 1, its smallest possible perimeter is 2sqrt2.
Symmetrical Sets
A set of 3 real numbers satisfies the symmetry that any number added to the product of the other 2 will result in 2. There are 2 such sets: all 1s and all -2s.
Distances to Vertices
In the square below, points that are closer to the center than one of the vertices is coloured yellow. The area of the yellow region is 1/2 of the area of the square.
Number Line Mirrors
Starting with some points on a number line, if we put a mirror somewhere on the number line, it will reflect all existing points in itself so that the resulting set of points are symmetrical around it.