We solve problems

Problem #PRU-30432

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Skeletons of multifaceted figures

11–12 3.0

a) A piece of wire that is 120 cm long is given. Is it possible, without breaking the wire, to make a cube frame with sides of 10 cm?

b) What is the smallest number of times it will be necessary to break the wire in order to still produce the required frame?

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 #PRU-31097

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Skeletons of multifaceted figures

11–13 3.0

a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?

b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).

Problem #PRU-32032

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality (other)

13–14 3.0

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

Problem #PRU-32038

Problems Methods Examples and counterexamples. Constructive proofs Geometry Plane geometry Lines, segments, rays, and angles Measurement of segments and angles. Adjacent angles.

11–13 2.0

Is it possible to draw five lines from one point on a plane so that there are exactly four acute angles among the angles formed by them? Angles between not only neighboring rays, but between any two rays, can be considered.

Problem #PRU-32095

Problems Geometry Plane geometry Visual geometry

10–13 2.0

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

Problem #PRU-32124

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Solid geometry Tetrahedron and pyramid Particular cases of tetrahedra Regular tetrahedron

13–15 3.0

A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.

Problem #PRU-32900

Problems Geometry Solid geometry Visual geometry in space

12–12 2.0

Does the number of 1999 occur in the Pascal triangle?

Problem #PRU-32902

Problems Geometry Solid geometry Visual geometry in space

12–12 2.0

How many times greater is the sum of the numbers in the hundred and first line of the Pascal triangle than the sum of the numbers in the hundredth line?

Problem #PRU-32903

Problems Geometry Solid geometry Visual geometry in space

12–12 2.0

Let’s put plus and minus signs in the 99th line of Pascal’s triangle. Between the first and second number there is a minus sign, between the second and the third there is a plus sign, between the third and the fourth there is a minus sign, then again a plus sign, and so on. Find the value of the resulting expression.