Problems

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.

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

Cut the following figure into two equal parts.

Find all rectangles that can be cut into \(13\) equal squares.

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.