Problems

Construct the triangle $ABC$ along the side $a$, the height $h_a$ and the angle $A$.

Construct a right-angled triangle along the leg and the hypotenuse.

Construct a circle with a given centre, tangent to a given circle.

Construct a straight line passing through a given point and tangent to a given circle.

Three segments whose lengths are equal to $a,b$ and $c$ are given. Construct a segment of length: a) $ab/c$; b) $\sqrt{ab}$.

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.

A wide variety of questions in mathematics starts with the question ’Is it possible...?’. In such problems you would either present an example, in case the described situation is possible, or rigorously prove that the situation is impossible, with the help of counterexample or by any other means. Sometimes the border between what seems should be possible and impossible is not immediately obvious, therefore you have to be cautious and verify that your example (or counterexample) satisfies the conditions stated in the problem. When you are asked the question whether something is possible or not and you suspect it is actually possible, it is always useful to ask more questions to gather additional information to narrow the possible answers. You can ask for example "How is it possible"? Or "$\mathbf{W}\mathbf{h}\mathbf{i}\mathbf{c}\mathbf{h}$ properties should the correct construction satisfy"?