Problems

The edges of a cube are assigned with integer values. For each vertex we look at the numbers corresponding to the three edges coming from this vertex and add them up. In case we get 8 equal results we call such cube “cute”. Are there any “cute” cubes with the following numbers corresponding to the edges:

(a) \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\);

(b) \(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6\)?

The area of a rectangle is 1 cm\(^2\). Can its perimeter be greater than 1 km?

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.

Let \(ABC\) and \(A_1B_1C_1\) be two triangles with the following properties: \(AB = A_1B_1\), \(AC = A_1C_1\), and angles \(\angle BAC = \angle B_1A_1C_1\). Then the triangles \(ABC\) and \(A_1B_1C_1\) are congruent. This is usually known as the “side-angle-side" criterion for congruence.

In the triangle \(\triangle ABC\) the sides \(AC\) and \(BC\) are equal. Prove that the angles \(\angle CAB\) and \(\angle CBA\) are equal.

Point \(A\) is the centre of a circle and points \(B,C,D\) lie on that circle. The segment \(BD\) is a diameter of the circle. Show that \(\angle CAD = 2 \angle CBD\). This is a special case of the inscribed angle theorem.