Problems

There is a triangle with side lengths \(a\), \(b\) and \(c\). Can you form a triangle with side lengths \(\frac{a}{b}\), \(\frac{b}{c}\) and \(\frac{c}{a}\)? Does it depend on what \(a\), \(b\) and \(c\) are? Give a proof if it is always possible or never possible. Otherwise, construct examples to show the dependence on \(a\), \(b\) and \(c\).
Recall that a triangle can be drawn with side lengths \(x\), \(y\) and \(z\) if and only if \(x+y>z\), \(y+z>x\) and \(z+x>y\).

There is a triangle with side lengths \(a\), \(b\) and \(c\). Does there exist a triangle with side lengths \(|a-b|\), \(|b-c|\) and \(|c-a|\)? Does it depend on what \(a\), \(b\) and \(c\) are?
Recall that a triangle can be formed with side lengths \(x\), \(y\) and \(z\) if and only if all the inequalities \(x+y>z\), \(y+z>x\) and \(z+x>y\) hold.

There is a triangle with side lenghts \(a\), \(b\) and \(c\). Does there exist a triangle with sides of lengths \(a^2+bc\), \(b^2+ca\) and \(c^2+ab\)? Does it depend on the values of \(a\), \(b\) and \(c\)?

Is it possible to draw \(K_5\) without intersecting edges on a Möbius band? Recall that \(K_5\) is the complete graph on \(5\) vertices. That is, \(5\) points with an edge between every pair of different points.

A circle is inscribed in a triangle (that is, the circle touches the sides of the triangle on the inside). Let the radius of the circle be \(r\) and the perimeter of the triangle be \(p\). Prove that the area of the triangle is \(\frac{pr}{2}\).

Let \(A\), \(B\), \(C\) and \(D\) be four points labelled clockwise on the circumference of a circle. The diagonals \(AC\) and \(BD\) intersect at the centre \(O\) of the circle. What can be deduced about the quadrilateral \(ABCD\)?

Consider the 7 different tetrominoes. Is it possible to cover a \(4\times7\) rectangle with exactly one copy of each of the tetrominoes? If it is possible, provide an example layout. If it is not possible, prove that it’s impossible.

We allow rotation of the tetrominoes, but not reflection. This means that we consider \(S\) and \(Z\) as different, as well as \(L\) and \(J\).

Let \(ABCDE\) be a regular pentagon. The point \(G\) is the midpoint of \(CD\), the point \(F\) is the midpoint of \(AE\). The lines \(EG\) and \(BF\) intersect at the point \(H\). Find the angle \(EHF\).

A paper band of constant width is tied into a simple knot and tightened. Prove that the knot has the shape of a regular polygon.

Prove the triangle inequality: in any triangle \(ABC\) the side \(AB < AC+ BC\).