Inscribed squares: Symmetric case

I recently heard about the “inscribed square” problem which states that on any closed, continuous curve without self intersections there should exist four points which are the vertices of a (non-degenerate) square. This problem, first raised by Otto Toeplitz in 1911, is still unsolved today. Many particular cases are solved (convex curves, smooth curves, etc.). Nevertheless, things quickly get complicated as one gets close to the original problems where only continuity is assumed.

I will show below a quick argument for a very particular case. Suppose the curve is symmetric with respect to the origin. Since the curve is simple, the origin does not lie on the curve. Consider the curve , rotated with degrees around the origin. Then curves must intersect. Indeed, by continuity, there exists a point which is closest to the origin on the curve and a point furthest away from the origin on the same curve. The rotated curve has the same minimal and maximal distances to the origin. Therefore, point lies in the interior of (or on the boundary) and point lies outside of (or on the boundary). In either case, going from to on we must intersect the curve .

If is a point of intersection, it lies both on and on its rotation with degrees. Moreover, the symmetric points also lie on by symmetry. Therefore, we have four points at equal distance from the origin, for which the rays from the origin form equal angles. These points are the vertices of a square!

IMC 2013 Problem 9

Problem 9. Does there exist an infinite set consisting of positive integers such that for any , with the sum is square-free?

Circles in a square SEEMOUS 2010

Given a square, we consider some circles inside the square such that the sum of lengths of all circles is equal to twice the perimeter of the square.

i) What is the minimum number of circles which satisfy the given hypothesis?

ii) Prove that there exist infinitely many lines which intersect at least three of the given circles.

SEEMOUS 2010, Problem 2

Matrices

1. Show that for any integer there exists a square matrix with elements such that it’s determinant is .
2. Show that there exists a matrix having real entries such that the determinant of any of its submatrices is not a rational number. (a submatrix is obtained from the initial matrix by removing the same number of lines and columns )

Circumscribed around the same circle

A triangle and a square are circumscribed around the unit circle. Prove that the common area of these two figures is at least 3.4.

Build a house in the forest

There is a forest having a square shape with edge of 1000 meters. This forest contains 4500 oak trees having 0.5 meters in diameter. The owner of the forest wants to know if he can find a place having a rectangular place of dimensions 20 meters by 10 meters which does not intersect any of the oak trees, because he wants to build a house there. He meets a mathematician, and asks him if he can deduce the existence of such a place without going out in the field to measure things on the spot. After a few minutes and calculations, the mathematician says that he can find such a place with certainty. How did he do that?

Freddy Flinstone’s car

Freddy Flinstone has a strange car, with square wheels. He wants you to design a road for him, such that he could ride his car perfectly smooth (i.e. the center of his wheels travel in a horizontal line). Can you design such a road?

Square inscribed in a triangle

1. Prove that given any triangle, we can inscribe a square in it.

2.0 Prove that there exists a square with maximum area inscribed in a given triangle.
2.1 Prove that if a square which lies inside a triangle has maximum area then two of its vertices lie on the same edge of the triangle.
2.2 Prove that if a square lies inside a triangle , then .

3. Now, we have a square inscribed in a triangle . Prove that the incenter of lies inside .