We solve problems

Problem #PRU-100430

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Discrete Mathematics Set theory and logic Logic and set theory

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

Problem #PRU-100454

Problems Discrete Mathematics Set theory and logic Mathematical logic Paradoxes Geometry Plane geometry Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

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

Problem #PRU-7646

Problems Geometry Plane geometry Area Triangle area

10–12 3.0

The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is \(18\), find the area of the small blue triangle.

Problem #PRU-7650

Problems Geometry Plane geometry Area Area of a circle, sector, segment

11–13 3.0

On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?

Problem #PRU-88235

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Various dissection problems

10–13 3.0

A square piece of paper is cut into 6 pieces, each of which is a convex polygon. 5 of the pieces are lost, leaving only one piece in the form of a regular octagon (see the drawing). Is it possible to reconstruct the original square using just this information?

Problem #PRU-30290

Problems Number Theory Divisibility Odd and even numbers Geometry Plane geometry Polygons

10–12 2.0

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?

Problem #PRU-30911

Problems Fun Problems Word Problems Solving problems by inequalities. Going through the cases. Geometry Solid geometry Volume Volume of a parallelepiped

11–12 2.0

Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?

Problem #NZO-000003

Problems Geometry Plane geometry

11–14

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 \(BD\).

Problem #NZO-000004

Problems Geometry Plane geometry

11–14

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 \(AD\).

Problem #NZO-000005

Problems Geometry Plane geometry

11–14

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 \(AB\).