Problems

Construct the triangle ABC by the medians \(m_a, m_b\) and \(m_c\).

Solve the equations \(x^2 = 14 + y^2\) in integers.

Many maths problems begin with the question “Is it possible…?”. In these kinds of problems, what you need to do depends on what you think is true.

• If you believe it is possible, then you must give an example that really satisfies the conditions in the problem.

• If you believe it is not possible, then you must explain clearly why it cannot be done.

When trying to build an example, it often helps to ask yourself extra questions to narrow things down: “How could it be possible?”, or “What properties must a correct example have?”.

On the other hand, if you have been trying to build an example for a while and nothing works, perhaps the answer is that it is impossible. In that case, look for a property that any example would need to have — and then show why that property cannot actually happen. Let’s see some examples!

Cut a square into two equal:
1. Triangles.
2. Pentagons
3. Hexagons.

Cut the following figure into two equal parts.

Suppose that a rectangle can be divided into \(13\) equal smaller squares. What could be the side lengths of this rectangle?

Daniel has drawn on a sheet of paper a circle and a dot inside it. Show that he can cut a circle into two parts which can be used to make a circle in which the marked point would be the center.

A square \(4 \times 4\) is called magic if all the numbers from 1 to 16 can be written into its cells in such a way that the sums of numbers in columns, rows and two diagonals are equal to each other. Sixth-grader Edwin began to make a magic square and written the number 1 in certain cell. His younger brother Theo decided to help him and put the numbers \(2\) and \(3\) in the cells adjacent to the number \(1\). Is it possible for Edwin to finish the magic square after such help?

Is it possible to cut such a hole in \(10\times 10 \,\,cm^2\) piece of paper, though which you can step?

Cut a square into \(3\) parts which you can use to construct a triangle with angles less than \(90^{\circ}\) and three different sides.