Problems

Age
Difficulty
Found: 10

The point \(A\) is fixed on a circle. Find the locus of the point \(X\) which divides the chords that end at point \(A\) in a \(1:2\) ratio, starting from the point \(A\).

Two circles of radius \(R\) touch at point \(E\). On one of them, point \(B\) is chosen and on the other point \(D\) is chosen. These points have a property of \(\angle BED = 90^{\circ}\). Prove that \(BD = 2R\).

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 \(BD^2 + FG^2 = 4R^2\).

Inside the rectangle \(ABCD\), the point \(E\) is taken. Prove that there exists a convex quadrilateral with perpendicular diagonals of lengths \(AB\) and \(BC\) whose sides are equal to \(AE\), \(BE\), \(CE\), \(DE\).

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.

Prove that the triangle \(ABC\) is regular if and only if, by turning it by \(60^{\circ}\) (either clockwise or anticlockwise) with respect to point A, its vertex B moves to \(C\).

Two perpendicular straight lines are drawn through the centre of the square. Prove that their intersection points with the sides of a square form a square.