a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?
b) The same problem, but for a regular polygon with 12 sides.
The area of a rectangle is 1 cm\(^2\). Can its perimeter be greater than 1 km?
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
a) An axisymmetric convex 101-gon is given. Prove that its axis of symmetry passes through one of its vertices.
b) What can be said about the case of a decagon?
Prove that the bisectors of a triangle intersect at one point.
Let \(ABCD\) be a square and let \(X\) be any point on side \(BC\) between \(B\) and \(C\). Let \(Y\) be the point on line \(CD\) such that \(BX=YD\) and \(D\) is between \(C\) and \(Y\). Prove that the midpoint of \(XY\) lies on diagonal \(BD\).
Let \(ABCD\) be a trapezium such that \(AB\) is parallel to \(CD\). Let \(E\) be the intersection of diagonals \(AC\) and \(BD\). Suppose that \(AB=BE\) and \(AC=DE\). Prove that the internal angle bisector of \(\angle BAC\) is perpendicular to \(AD\).
Let \(ABC\) be an isosceles triangle with \(AB=AC\). Point \(D\) lies on side \(AC\) such that \(BD\) is the angle bisector of \(\angle ABC\). Point \(E\) lies on side \(BC\) between \(B\) and \(C\) such that \(BE=CD\). Prove that \(DE\) is parallel to \(AB\).
\(ABCD\) is a rectangle with side lengths \(AB=CD=1\) and \(BC=DA=2\). Let \(M\) be the midpoint of \(AD\). Point \(P\) lies on the opposite side of line \(MB\) to \(A\), such that triangle \(MBP\) is equilateral. Find the value of \(\angle PCB\).
A rectangular sheet of paper is folded so that one corner lies on top of the corner diagonally opposite. The resulting shape is a pentagon whose area is \(20\%\) one-sheet-thick, and \(80\%\) two-sheets-thick. Determine the ratio of the two sides of the original sheet of paper.