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-58085

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Geometry Plane geometry Polygons Inscribed and circumscribed polygons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 4.0

A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).

Problem #PRU-65304

Problems Probability and statistics Probability theory Continuous distribution Algebra Mean values Geometry Plane geometry Polygons Polygons (other)

14–16 4.0

James furiously cuts a rectangular sheet of paper with scissors. Every second he cuts a random piece by an unsystematic rectilinear cut into two parts.

a) Find the mathematical expectation of the number of sides of a polygon (made from a piece of paper) that James randomly picks up after an hour of such work.

b) Solve the same problem if at first the piece of paper had the form of an arbitrary polygon.

Problem #PRU-73727

Problems Number Theory Divisibility Divisibility of a number. General properties Geometry Solid geometry Lines and planes in space Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Out of the given numbers 1, 2, 3, ..., 1000, find the largest number \(m\) that has this property: no matter which \(m\) of these numbers you delete, among the remaining \(1000 - m\) numbers there are two, of which one is divisible by the other.

Problem #PRU-7653

Problems Geometry Plane geometry Area Area ratio Finding the area by rearrangment

12–14 4.0

In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Polygons (other)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.

Problem #PRU-116087

Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let \(C_{1}\) be the point of intersection, further from the vertex \(C\), of the circles constructed from the medians \(AM_{1}\) and \(BM_{2}\). Points \(A_{1}\) and \(B_{1}\) are defined similarly. Prove that the lines \(AA_{1}\), \(BB_{1}\) and \(CC_{1}\) intersect at the same point.

Problem #PRU-57006

Problems Geometry Plane geometry Quadrilateral Circumscribed quadrilaterals

13–16 4.0

Prove that a convex quadrilateral \(ICEF\) can have a circle inscribed into it if and only if \(IC+EH = CE+IF\).

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 \(BD\).