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.
Solve the equations in integers:
a) \(3x^2 + 5y^2 = 345\);
b) \(1 + x + x^2 + x^3 = 2^y\).
In honor of the March 8 holiday, a competition of performances was organized. Two performances reached the final. \(N\) students of the 5th grade played in the first one and \(n\) students of the 4th grade played in the second one. The performance was attended by \(2n\) mothers of all \(2n\) students. The best performance is chosen by a vote of the mothers. It is known that half of the mothers vote honestly, i.e. for the performance that was truly better and the mothers of the other half in any case vote for the performance in which their child participates.
a) Find the probability of the best performance winning by a majority of votes.
b) The same question but this time more than two performances made it to the final.
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.
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?