Area-Bisecting Lines
Every line going through the center of a square bisects its area. Does every line going through the center of an equilateral triangle bisect its area too?
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.
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.
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.
Inscribed Perimeters
If a quadrilateral has 1 vertex on each side of a square of side 1, its smallest possible perimeter is 2sqrt2.
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.
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…
Integer Coefficients
Consider expanding this expression: f(x) = (ax+b)(cx+d).
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.
Lucas Sequences
A Lucas Sequence is made from two starting terms and a recursive formula for the next term using the two previous ones.