MathDB

3

Part of 2010 Czech-Polish-Slovak Match

Problems(2)

Choosing squares in an array that do not form a rectangle

Source: Czech-Polish-Slovak Match, 2010

8/8/2011
Let pp be a prime number. Prove that from a p2×p2p^2\times p^2 array of squares, we can select p3p^3 of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
geometryrectanglecalculuslinear algebramatrixanalytic geometrynumber theory
Conditions on sides & diagonals => ABCD is a parallelogram

Source: Czech-Polish-Slovak Match, 2010

8/8/2011
Let ABCDABCD be a convex quadrilateral for which AB+CD=2⋅ACandBC+DA=2⋅BD. AB+CD=\sqrt{2}\cdot AC\qquad\text{and}\qquad BC+DA=\sqrt{2}\cdot BD. Prove that ABCDABCD is a parallelogram.
geometryparallelogramgeometric transformationreflectioninequalitiestriangle inequalitygeometry unsolved