Let \(O\) be the center of the rectangle \(ABCD\). Find the geometric points of \(M\) for which \(AM \geq OM, BM \geq OM\), \(CM \geq OM\), and \(DM \geq OM\).
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.