Show that the equation \(x^2 +6x-1 = y^2\) has no solutions in integer \(x\) and \(y\).
In a room there are some chairs with 4 legs and some stools with 3 legs. When each chair and stool has one person sitting on it, then in the room there are a total of 39 legs. How many chairs and stools are there in the room?
Prove that, if \(b=a-1\), then \[(a+b)(a^2 +b^2)(a^4 +b^4)\dotsb(a^{32} +b^{32})=a^{64} -b^{64}.\]
Which triangle has the largest area? The dots form a regular grid.
What is the ratio between the red and blue area? All shapes are semicircles.
A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?
Prove that \(\angle ABC > 90^{\circ}\) if and only if the point \(B\) lies inside a circle with diameter \(AC\).
Two circles of radius \(R\) intersect at points \(D\) and \(B\). Let \(F\) and \(G\) be the points of intersection of the middle perpendicular to the segment \(BD\) with these circles lying on one side of the line \(BD\). Prove that \(BD^2 + FG^2 = 4R^2\).
A parallelogram \(ABCD\) and a point \(E\) are given. Through the points \(A, B, C, D\), lines parallel to the straight lines \(EC, ED, EA,EB\), respectively, are drawn. Prove that they intersect at one point.
Prove that a convex \(n\)-gon is regular if and only if it is transformed into itself when it is rotated through an angle of \(360^{\circ}/n\) with respect to some point.