We solve problems

Problems Algebra Rational functions Rational equations

Solve this equation: \[(x+2010)(x+2011)(x+2012)=(x+2011)(x+2012)(x+2013).\]

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Relations between sides and angles of a triangle (other)

13–15 2.0

Two intersecting circles of radius \(R\) are given, and the distance between their centers is greater than \(R\). Prove that \(\angle ECD = 3\angle CAD\).

Problem #PRU-57808

Problems Geometry Plane geometry Plane transformations Parallel translation

13–15 4.0

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

Problem #PRU-57809

Problems Geometry Plane geometry Plane transformations Parallel translation

13–15 3.0

Two circles of radius \(R\) intersect at points \(B\) and \(D\). Consider the perpendicular bisector of the segment \(BD\). This line meets the two circles again at points \(F\) and \(G\), both chosen on the same side of \(BD\). Prove that \[BD^2 + FG^2 = 4R^2.\]

Problem #PRU-57836

Problems Geometry Solid geometry Space transformations Isometry Point symmetry

13–15 3.0

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.

Problem #PRU-57866

Problems Geometry Solid geometry Space transformations Isometry Point symmetry Reflectional symmetry

13–15 2.0

Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.

Problem #PRU-57915

Problems Geometry Plane geometry Plane transformations Isometry of the plane Rotations

13–15 3.0

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.

Problem #PRU-57917

Problems Geometry Plane geometry Polygons Regular polygons

13–15 3.0

Prove that the midpoints of the sides of a regular polygon form a regular polygon.

Problem #PRU-57918

Problems Geometry Plane geometry Plane transformations Isometry of the plane Rotations

13–16 2.0

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.

Problem #PRU-57976

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations

14–16 3.0

Two circles \(c\) and \(d\) are tangent at point \(B\). Two straight lines intersecting the first circle at points \(D\) and \(E\) and the second circle at points \(G\) and \(F\) are drawn through the point \(B\). Prove that \(ED\) is parallel to \(FG\).